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- $( <MM> <PROOF_ASST> THEOREM=areaquad LOC_AFTER=?
- * The area of a quadrilateral with two sides which are parallel to the y-axis in RR X. RR
- is its width multiplied by the average height of it's top minus the average height of it's bottom.
- (Contributed by Jon Pennant, 31-Mar-2019.)
- h1::areaquad.1 |- A e. RR
- h2::areaquad.2 |- B e. RR
- h3::areaquad.3 |- C e. RR
- h4::areaquad.4 |- D e. RR
- h5::areaquad.5 |- E e. RR
- h6::areaquad.6 |- F e. RR
- h7::areaquad.7 |- A < B
- h8::areaquad.8 |- C <_ E
- h9::areaquad.9 |- D <_ F
- h10::areaquad.10 |- U = ( C + ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) )
- h11::areaquad.11 |- V = ( E + ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) )
- h12::areaquad.12 |- S = { <. x , y >. | ( x e. ( A [,] B ) /\ y e. ( U [,] V ) ) }
- *236 is broken
- *10a:1:idi |- A e. RR
- 10ac:1:recni |- A e. CC
- 10ad:1:a1i |- ( x e. RR -> A e. RR )
- 10ae:1:rexri |- A e. RR*
- *10b:2:idi |- B e. RR
- 10bc:2:recni |- B e. CC
- 10bd:2:a1i |- ( x e. RR -> B e. RR )
- 10be:2:rexri |- B e. RR*
- *10c:3:idi |- C e. RR
- 10cc:3:recni |- C e. CC
- 10ce:10cc:a1i |- ( x e. RR -> C e. CC )
- *10d:4:idi |- D e. RR
- 10dc:4:recni |- D e. CC
- 10de:10dc:a1i |- ( x e. RR -> D e. CC )
- *10e:5:idi |- E e. RR
- 10ec:5:recni |- E e. CC
- 10ee:10ec:a1i |- ( x e. RR -> E e. CC )
- *10f:6:idi |- F e. RR
- 10fc:6:recni |- F e. CC
- 10fe:10fc:a1i |- ( x e. RR -> F e. CC )
- *11a:7:idi |- A < B
- 11aa:1,2:leloei |- ( A <_ B <-> ( A < B \/ A = B ) )
- 11ab::orc |- ( A < B -> ( A < B \/ A = B ) )
- 11ac:7,11ab:ax-mp |- ( A < B \/ A = B )
- 11ad:11ac,11aa:mpbir |- A <_ B
- 11b:8:idi |- C <_ E
- 11c:9:idi |- D <_ F
- 206::0re |- 0 e. RR
- 206ccc:206:recni |- 0 e. CC
- 28a::1red |- ( x e. RR -> 1 e. RR )
- 28ab::1cnd |- ( x e. RR -> 1 e. CC )
- 11e::tru |- T.
- 11f::2cn |- 2 e. CC
- 11g::2ne0 |- 2 =/= 0
- 11h:10ac,11f:mulcli |- ( A x. 2 ) e. CC
- 11hc:10cc,11f:mulcli |- ( C x. 2 ) e. CC
- 11he:10ec,11f:mulcli |- ( E x. 2 ) e. CC
- 11i::2cnne0 |- ( 2 e. CC /\ 2 =/= 0 )
- 11j:10fc,10ec:subcli |- ( F - E ) e. CC
- 11k:11j:a1i |- ( x e. ( A [,] B ) -> ( F - E ) e. CC )
- 11fe:10fc,10ec:addcli |- ( F + E ) e. CC
- 11dc:10dc,10cc:addcli |- ( D + C ) e. CC
- *12a:10:idi |- U = ( C + ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) )
- *12b:11:idi |- V = ( E + ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) )
- *12c:: |- U = ( ( C x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) + ( D x. ( ( x - A ) / ( B - A ) ) ) )
- *12d:: |- V = ( ( E x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) + ( F x. ( ( x - A ) / ( B - A ) ) ) )
- *13:12:idi |- S = { <. x , y >. | ( x e. ( A [,] B ) /\ y e. ( U [,] V ) ) }
- *things to prove
- dmarea::dmarea |- ( S e. dom area <-> ( S C_ ( RR X. RR ) /\ A. x e. RR ( S " { x } ) e. ( `' vol " RR )
- /\ ( x e. RR |-> ( vol ` ( S " { x } ) ) ) e. L^1 ) )
- areaval::areaval |- ( S e. dom area -> ( area ` S ) = S. RR ( vol ` ( S " { x } ) ) _d x )
- *get A,B in RR
- 14::iccssre |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
- 15:1,2,14:mp2an |- ( A [,] B ) C_ RR
- 15a::axresscn |- RR C_ CC
- 15b:15,15a:sstri |- ( A [,] B ) C_ CC
- 15c:15b:a1i |- ( x e. RR -> ( A [,] B ) C_ CC )
- 15d::dfss3 |- ( ( A [,] B ) C_ RR <-> A. x e. ( A [,] B ) x e. RR )
- 15e:15,15d:mpbi |- A. x e. ( A [,] B ) x e. RR
- 15f:15e:rspec |- ( x e. ( A [,] B ) -> x e. RR )
- *get U,V in RR
- 16:2,1:resubcli |- ( B - A ) e. RR
- 16a:16:recni |- ( B - A ) e. CC
- 16ab:10ac,16a:mulcli |- ( A x. ( B - A ) ) e. CC
- 16ac:10bc:sqcli |- ( B ^ 2 ) e. CC
- 16ad:10ac:sqcli |- ( A ^ 2 ) e. CC
- 16ae:16ac,16ad:subcli |- ( ( B ^ 2 ) - ( A ^ 2 ) ) e. CC
- 16af:16ae,11f,11g:divcli |- ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) e. CC
- 16c::resubcl |- ( ( x e. RR /\ A e. RR ) -> ( x - A ) e. RR )
- 16ca:1,16c:mpan2 |- ( x e. RR -> ( x - A ) e. RR )
- 16cb:15f,16ca:syl |- ( x e. ( A [,] B ) -> ( x - A ) e. RR )
- 17::ltne |- ( ( A e. RR /\ A < B ) -> B =/= A )
- 18:1,7,17:mp2an |- B =/= A
- d2:10bc:a1i |- ( A e. RR -> B e. CC )
- d3:10ac:a1i |- ( A e. RR -> A e. CC )
- d4:18:a1i |- ( A e. RR -> B =/= A )
- 23:d2,d3,d4:subne0d |- ( A e. RR -> ( B - A ) =/= 0 )
- 24:1,23:ax-mp |- ( B - A ) =/= 0
- 25::redivcl |- ( ( ( x - A ) e. RR /\ ( B - A ) e. RR /\ ( B - A ) =/= 0 ) -> ( ( x - A ) / ( B - A ) ) e. RR )
- 26a:16:a1i |- ( x e. RR -> ( B - A ) e. RR )
- 26aa:15f,26a:syl |- ( x e. ( A [,] B ) -> ( B - A ) e. RR )
- 26b:24:a1i |- ( x e. RR -> ( B - A ) =/= 0 )
- 26c:3:a1i |- ( x e. RR -> C e. RR )
- 26e:5:a1i |- ( x e. RR -> E e. RR )
- 26d:4:a1i |- ( x e. RR -> D e. RR )
- 26f:6:a1i |- ( x e. RR -> F e. RR )
- 27:16ca,26a,26b,25:syl3anc |- ( x e. RR -> ( ( x - A ) / ( B - A ) ) e. RR )
- 27a:15f,27:syl |- ( x e. ( A [,] B ) -> ( ( x - A ) / ( B - A ) ) e. RR )
- 27ab:27:recnd |- ( x e. RR -> ( ( x - A ) / ( B - A ) ) e. CC )
- 27b:27a:recnd |- ( x e. ( A [,] B ) -> ( ( x - A ) / ( B - A ) ) e. CC )
- 28::resubcl |- ( ( 1 e. RR /\ ( ( x - A ) / ( B - A ) ) e. RR ) -> ( 1 - ( ( x - A ) / ( B - A ) ) ) e. RR )
- 29:28a,27,28:syl2anc |- ( x e. RR -> ( 1 - ( ( x - A ) / ( B - A ) ) ) e. RR )
- 29c:29:recnd |- ( x e. RR -> ( 1 - ( ( x - A ) / ( B - A ) ) ) e. CC )
- 30c:26c,29:remulcld |- ( x e. RR -> ( C x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) e. RR )
- 30ca:30c:recnd |- ( x e. RR -> ( C x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) e. CC )
- 30e:26e,29:remulcld |- ( x e. RR -> ( E x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) e. RR )
- 30d:26d,27:remulcld |- ( x e. RR -> ( D x. ( ( x - A ) / ( B - A ) ) ) e. RR )
- 30f:26f,27:remulcld |- ( x e. RR -> ( F x. ( ( x - A ) / ( B - A ) ) ) e. RR )
- *convert U and V to a form suited to inequalities
- 12aa::subdi |- ( ( ( ( x - A ) / ( B - A ) ) e. CC /\ D e. CC /\ C e. CC )
- -> ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) )
- = ( ( ( ( x - A ) / ( B - A ) ) x. D ) - ( ( ( x - A ) / ( B - A ) ) x. C ) ) )
- 12ab:27ab,10de,10ce,12aa:syl3anc |- ( x e. RR
- -> ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) )
- = ( ( ( ( x - A ) / ( B - A ) ) x. D ) - ( ( ( x - A ) / ( B - A ) ) x. C ) ) )
- 12ac:12ab:oveq2d |- ( x e. RR
- -> ( C + ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) )
- = ( C + ( ( ( ( x - A ) / ( B - A ) ) x. D ) - ( ( ( x - A ) / ( B - A ) ) x. C ) ) ) )
- 12ad:10,12ac:syl5eq |- ( x e. RR -> U
- = ( C + ( ( ( ( x - A ) / ( B - A ) ) x. D ) - ( ( ( x - A ) / ( B - A ) ) x. C ) ) ) )
- 12ae::addsub12 |- ( ( C e. CC /\ ( ( ( x - A ) / ( B - A ) ) x. D ) e. CC /\ ( ( ( x - A ) / ( B - A ) ) x. C ) e. CC )
- -> ( C + ( ( ( ( x - A ) / ( B - A ) ) x. D ) - ( ( ( x - A ) / ( B - A ) ) x. C ) ) )
- = ( ( ( ( x - A ) / ( B - A ) ) x. D ) + ( C - ( ( ( x - A ) / ( B - A ) ) x. C ) ) ) )
- 12af:27ab,10de:mulcld |- ( x e. RR -> ( ( ( x - A ) / ( B - A ) ) x. D ) e. CC )
- 12ag:27ab,10ce:mulcld |- ( x e. RR -> ( ( ( x - A ) / ( B - A ) ) x. C ) e. CC )
- 12ah:10ce,12af,12ag,12ae:syl3anc |- ( x e. RR
- -> ( C + ( ( ( ( x - A ) / ( B - A ) ) x. D ) - ( ( ( x - A ) / ( B - A ) ) x. C ) ) )
- = ( ( ( ( x - A ) / ( B - A ) ) x. D ) + ( C - ( ( ( x - A ) / ( B - A ) ) x. C ) ) ) )
- 12ai:12ad,12ah:eqtrd |- ( x e. RR -> U = ( ( ( ( x - A ) / ( B - A ) ) x. D ) + ( C - ( ( ( x - A ) / ( B - A ) ) x. C ) ) ) )
- 12ak:10cc:mulid2i |- ( 1 x. C ) = C
- 12al::oveq1 |- ( ( 1 x. C ) = C -> ( ( 1 x. C ) - ( ( ( x - A ) / ( B - A ) ) x. C ) ) = ( C - ( ( ( x - A ) / ( B - A ) ) x. C ) ) )
- 12am:12ak,12al:ax-mp |- ( ( 1 x. C ) - ( ( ( x - A ) / ( B - A ) ) x. C ) ) = ( C - ( ( ( x - A ) / ( B - A ) ) x. C ) )
- 12an::subdir |- ( ( 1 e. CC /\ ( ( x - A ) / ( B - A ) ) e. CC /\ C e. CC )
- -> ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. C ) = ( ( 1 x. C ) - ( ( ( x - A ) / ( B - A ) ) x. C ) ) )
- 12ao:28ab,27ab,10ce,12an:syl3anc |- ( x e. RR -> ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. C ) = ( ( 1 x. C ) - ( ( ( x - A ) / ( B - A ) ) x. C ) ) )
- 12ap:12ao,12am:syl6eq |- ( x e. RR -> ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. C ) = ( C - ( ( ( x - A ) / ( B - A ) ) x. C ) ) )
- 12aq:12ap:oveq2d |- ( x e. RR -> ( ( ( ( x - A ) / ( B - A ) ) x. D ) + ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. C ) )
- = ( ( ( ( x - A ) / ( B - A ) ) x. D ) + ( C - ( ( ( x - A ) / ( B - A ) ) x. C ) ) ) )
- 12ar:12ai,12aq:eqtr4d |- ( x e. RR -> U = ( ( ( ( x - A ) / ( B - A ) ) x. D ) + ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. C ) ) )
- 12as::addcomgi |- ( ( ( ( x - A ) / ( B - A ) ) x. D ) + ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. C ) )
- = ( ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. C ) + ( ( ( x - A ) / ( B - A ) ) x. D ) )
- 12au:12ar,12as:syl6eq |- ( x e. RR -> U =
- ( ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. C ) + ( ( ( x - A ) / ( B - A ) ) x. D ) ) )
- 12av:29c,10ce:mulcomd |- ( x e. RR
- -> ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. C ) = ( C x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) )
- 12aw:27ab,10de:mulcomd |- ( x e. RR -> ( ( ( x - A ) / ( B - A ) ) x. D ) = ( D x. ( ( x - A ) / ( B - A ) ) ) )
- 12ax:12av,12aw:oveq12d |- ( x e. RR -> ( ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. C ) + ( ( ( x - A ) / ( B - A ) ) x. D ) )
- = ( ( C x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) + ( D x. ( ( x - A ) / ( B - A ) ) ) ) )
- 12ay:12au,12ax:eqtrd |- ( x e. RR -> U = ( ( C x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) + ( D x. ( ( x - A ) / ( B - A ) ) ) ) )
- 12az:15f,12ay:syl |- ( x e. ( A [,] B ) -> U = ( ( C x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) + ( D x. ( ( x - A ) / ( B - A ) ) ) ) )
- 12ba::subdi |- ( ( ( ( x - A ) / ( B - A ) ) e. CC /\ F e. CC /\ E e. CC )
- -> ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) )
- = ( ( ( ( x - A ) / ( B - A ) ) x. F ) - ( ( ( x - A ) / ( B - A ) ) x. E ) ) )
- 12bb:27ab,10fe,10ee,12ba:syl3anc |- ( x e. RR
- -> ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) )
- = ( ( ( ( x - A ) / ( B - A ) ) x. F ) - ( ( ( x - A ) / ( B - A ) ) x. E ) ) )
- 12bc:12bb:oveq2d |- ( x e. RR
- -> ( E + ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) )
- = ( E + ( ( ( ( x - A ) / ( B - A ) ) x. F ) - ( ( ( x - A ) / ( B - A ) ) x. E ) ) ) )
- 12bd:11,12bc:syl5eq |- ( x e. RR -> V
- = ( E + ( ( ( ( x - A ) / ( B - A ) ) x. F ) - ( ( ( x - A ) / ( B - A ) ) x. E ) ) ) )
- 12be::addsub12 |- ( ( E e. CC /\ ( ( ( x - A ) / ( B - A ) ) x. F ) e. CC /\ ( ( ( x - A ) / ( B - A ) ) x. E ) e. CC )
- -> ( E + ( ( ( ( x - A ) / ( B - A ) ) x. F ) - ( ( ( x - A ) / ( B - A ) ) x. E ) ) )
- = ( ( ( ( x - A ) / ( B - A ) ) x. F ) + ( E - ( ( ( x - A ) / ( B - A ) ) x. E ) ) ) )
- 12bf:27ab,10fe:mulcld |- ( x e. RR -> ( ( ( x - A ) / ( B - A ) ) x. F ) e. CC )
- 12bg:27ab,10ee:mulcld |- ( x e. RR -> ( ( ( x - A ) / ( B - A ) ) x. E ) e. CC )
- 12bh:10ee,12bf,12bg,12be:syl3anc |- ( x e. RR
- -> ( E + ( ( ( ( x - A ) / ( B - A ) ) x. F ) - ( ( ( x - A ) / ( B - A ) ) x. E ) ) )
- = ( ( ( ( x - A ) / ( B - A ) ) x. F ) + ( E - ( ( ( x - A ) / ( B - A ) ) x. E ) ) ) )
- 12bi:12bd,12bh:eqtrd |- ( x e. RR -> V = ( ( ( ( x - A ) / ( B - A ) ) x. F ) + ( E - ( ( ( x - A ) / ( B - A ) ) x. E ) ) ) )
- 12bk:10ec:mulid2i |- ( 1 x. E ) = E
- 12bl::oveq1 |- ( ( 1 x. E ) = E -> ( ( 1 x. E ) - ( ( ( x - A ) / ( B - A ) ) x. E ) )
- = ( E - ( ( ( x - A ) / ( B - A ) ) x. E ) ) )
- 12bm:12bk,12bl:ax-mp |- ( ( 1 x. E ) - ( ( ( x - A ) / ( B - A ) ) x. E ) ) = ( E - ( ( ( x - A ) / ( B - A ) ) x. E ) )
- 12bn::subdir |- ( ( 1 e. CC /\ ( ( x - A ) / ( B - A ) ) e. CC /\ E e. CC )
- -> ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. E ) = ( ( 1 x. E ) - ( ( ( x - A ) / ( B - A ) ) x. E ) ) )
- 12bo:28ab,27ab,10ee,12bn:syl3anc |- ( x e. RR -> ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. E ) = ( ( 1 x. E ) - ( ( ( x - A ) / ( B - A ) ) x. E ) ) )
- 12bp:12bo,12bm:syl6eq |- ( x e. RR -> ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. E ) = ( E - ( ( ( x - A ) / ( B - A ) ) x. E ) ) )
- 12bq:12bp:oveq2d |- ( x e. RR -> ( ( ( ( x - A ) / ( B - A ) ) x. F ) + ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. E ) )
- = ( ( ( ( x - A ) / ( B - A ) ) x. F ) + ( E - ( ( ( x - A ) / ( B - A ) ) x. E ) ) ) )
- 12br:12bi,12bq:eqtr4d |- ( x e. RR -> V = ( ( ( ( x - A ) / ( B - A ) ) x. F ) + ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. E ) ) )
- 12bs::addcomgi |- ( ( ( ( x - A ) / ( B - A ) ) x. F ) + ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. E ) )
- = ( ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. E ) + ( ( ( x - A ) / ( B - A ) ) x. F ) )
- 12bu:12br,12bs:syl6eq |- ( x e. RR -> V =
- ( ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. E ) + ( ( ( x - A ) / ( B - A ) ) x. F ) ) )
- 12bv:29c,10ee:mulcomd |- ( x e. RR
- -> ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. E ) = ( E x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) )
- 12bw:27ab,10fe:mulcomd |- ( x e. RR -> ( ( ( x - A ) / ( B - A ) ) x. F ) = ( F x. ( ( x - A ) / ( B - A ) ) ) )
- 12bx:12bv,12bw:oveq12d |- ( x e. RR -> ( ( ( 1 - ( ( x - A ) / ( B - A ) ) ) x. E ) + ( ( ( x - A ) / ( B - A ) ) x. F ) )
- = ( ( E x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) + ( F x. ( ( x - A ) / ( B - A ) ) ) ) )
- 12by:12bu,12bx:eqtrd |- ( x e. RR -> V = ( ( E x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) + ( F x. ( ( x - A ) / ( B - A ) ) ) ) )
- 12bz:15f,12by:syl |- ( x e. ( A [,] B ) -> V = ( ( E x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) + ( F x. ( ( x - A ) / ( B - A ) ) ) ) )
- *now get U,V in RR
- 31v:30e,30f:readdcld |- ( x e. RR -> ( ( E x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) + ( F x. ( ( x - A ) / ( B - A ) ) ) ) e. RR )
- 31u:30c,30d:readdcld |- ( x e. RR -> ( ( C x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) + ( D x. ( ( x - A ) / ( B - A ) ) ) ) e. RR )
- 36:12ay,31u:eqeltrd |- ( x e. RR -> U e. RR )
- 37:12by,31v:eqeltrd |- ( x e. RR -> V e. RR )
- 36b:15f,36:syl |- ( x e. ( A [,] B ) -> U e. RR )
- 36c:15f,37:syl |- ( x e. ( A [,] B ) -> V e. RR )
- 36d:36b,36c:jca |- ( x e. ( A [,] B ) -> ( U e. RR /\ V e. RR ) )
- 37a:36,37:jca |- ( x e. RR -> ( U e. RR /\ V e. RR ) )
- 37aa:37,36:jca |- ( x e. RR -> ( V e. RR /\ U e. RR ) )
- 37e::resubcl |- ( ( V e. RR /\ U e. RR ) -> ( V - U ) e. RR )
- 37f:37aa,37e:syl |- ( x e. RR -> ( V - U ) e. RR )
- 37fb:36c,36b:resubcld |- ( x e. ( A [,] B ) -> ( V - U ) e. RR )
- 37g:37f:recnd |- ( x e. RR -> ( V - U ) e. CC )
- 38::iccssre |- ( ( U e. RR /\ V e. RR ) -> ( U [,] V ) C_ RR )
- 39:36,37,38:syl2anc |- ( x e. RR -> ( U [,] V ) C_ RR )
- *prove U <_V in ( A [,] B )
- *first get both 1 - ( ( x - A ) / ( B - A ) ) >_ 0, then 0 <_ ( ( x - A ) / ( B - A ) )
- 40a::iccleub |- ( ( A e. RR* /\ B e. RR* /\ x e. ( A [,] B ) ) -> x <_ B )
- 40b:10ae,10be,40a:mp3an12 |- ( x e. ( A [,] B ) -> x <_ B )
- d11:2:a1i |- ( x e. ( A [,] B ) -> B e. RR )
- d12:1:a1i |- ( x e. ( A [,] B ) -> A e. RR )
- 40c:15f,d11,d12:lesub1d |- ( x e. ( A [,] B ) -> ( x <_ B <-> ( x - A ) <_ ( B - A ) ) )
- 40e:40b,40c:mpbid |- ( x e. ( A [,] B ) -> ( x - A ) <_ ( B - A ) )
- 40f:d12,d11,d12:ltsub1d |- ( x e. ( A [,] B ) -> ( A < B <-> ( A - A ) < ( B - A ) ) )
- 40g:7,40f:mpbii |- ( x e. ( A [,] B ) -> ( A - A ) < ( B - A ) )
- 40h::subid |- ( A e. CC -> ( A - A ) = 0 )
- 40i:10ac,40h:ax-mp |- ( A - A ) = 0
- 40j:40i,40g:syl5eqbrr |- ( x e. ( A [,] B ) -> 0 < ( B - A ) )
- 40k:26aa,40j:jca |- ( x e. ( A [,] B ) -> ( ( B - A ) e. RR /\ 0 < ( B - A ) ) )
- 40l::lediv1 |- ( ( ( x - A ) e. RR /\ ( B - A ) e. RR /\ ( ( B - A ) e. RR /\ 0 < ( B - A ) ) )
- -> ( ( x - A ) <_ ( B - A ) <-> ( ( x - A ) / ( B - A ) ) <_ ( ( B - A ) / ( B - A ) ) ) )
- 40m:16cb,26aa,40k,40l:syl3anc |- ( x e. ( A [,] B )
- -> ( ( x - A ) <_ ( B - A ) <-> ( ( x - A ) / ( B - A ) ) <_ ( ( B - A ) / ( B - A ) ) ) )
- 40n:40e,40m:mpbid |- ( x e. ( A [,] B )
- -> ( ( x - A ) / ( B - A ) ) <_ ( ( B - A ) / ( B - A ) ) )
- 40o:16a,24:dividi |- ( ( B - A ) / ( B - A ) ) = 1
- 40p:40n,40o:syl6breq |- ( x e. ( A [,] B ) -> ( ( x - A ) / ( B - A ) ) <_ 1 )
- d13:15f,27:syl |- ( x e. ( A [,] B ) -> ( ( x - A ) / ( B - A ) ) e. RR )
- d14:15f,28a:syl |- ( x e. ( A [,] B ) -> 1 e. RR )
- 40q:d13,d14,d13:lesub1d |- ( x e. ( A [,] B ) -> ( ( ( x - A ) / ( B - A ) ) <_ 1
- <-> ( ( ( x - A ) / ( B - A ) ) - ( ( x - A ) / ( B - A ) ) ) <_ ( 1 - ( ( x - A ) / ( B - A ) ) ) ) )
- 40r:40p,40q:mpbid |- ( x e. ( A [,] B ) -> ( ( ( x - A ) / ( B - A ) ) - ( ( x - A ) / ( B - A ) ) )
- <_ ( 1 - ( ( x - A ) / ( B - A ) ) ) )
- 40s::subid |- ( ( ( x - A ) / ( B - A ) ) e. CC -> ( ( ( x - A ) / ( B - A ) ) - ( ( x - A ) / ( B - A ) ) ) = 0 )
- 40t:27b,40s:syl |- ( x e. ( A [,] B ) -> ( ( ( x - A ) / ( B - A ) ) - ( ( x - A ) / ( B - A ) ) ) = 0 )
- 40u:40t,40r:eqbrtrrd |- ( x e. ( A [,] B ) -> 0 <_ ( 1 - ( ( x - A ) / ( B - A ) ) ) )
- *not get 0 <_ ( ( x - A ) / ( B - A ) )
- 41::elicc1 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,] B ) <-> ( x e. RR* /\ A <_ x /\ x <_ B ) ) )
- 41a::simp2 |- ( ( x e. RR* /\ A <_ x /\ x <_ B ) -> A <_ x )
- 41b:41,41a:syl6bi |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,] B ) -> A <_ x ) )
- 41c:41b:3impia |- ( ( A e. RR* /\ B e. RR* /\ x e. ( A [,] B ) ) -> A <_ x )
- 41d:10ae,10be,41c:mp3an12 |- ( x e. ( A [,] B ) -> A <_ x )
- 41e:d12,15f,d12:lesub1d |- ( x e. ( A [,] B ) -> ( A <_ x <-> ( A - A ) <_ ( x - A ) ) )
- 41f::subid |- ( A e. CC -> ( A - A ) = 0 )
- 41g:10ac,41f:ax-mp |- ( A - A ) = 0
- 41h:41d,41e:mpbid |- ( x e. ( A [,] B ) -> ( A - A ) <_ ( x - A ) )
- 41i:41g,41h:syl5eqbrr |- ( x e. ( A [,] B ) -> 0 <_ ( x - A ) )
- 41j::lediv1 |- ( ( 0 e. RR /\ ( x - A ) e. RR /\ ( ( B - A ) e. RR /\ 0 < ( B - A ) ) )
- -> ( 0 <_ ( x - A ) <-> ( 0 / ( B - A ) ) <_ ( ( x - A ) / ( B - A ) ) ) )
- 41k:206,16cb,40k,41j:eel011 |- ( x e. ( A [,] B ) -> ( 0 <_ ( x - A ) <-> ( 0 / ( B - A ) ) <_ ( ( x - A ) / ( B - A ) ) ) )
- 41l:41i,41k:mpbid |- ( x e. ( A [,] B ) -> ( 0 / ( B - A ) ) <_ ( ( x - A ) / ( B - A ) ) )
- 41m::diveq0 |- ( ( 0 e. CC /\ ( B - A ) e. CC /\ ( B - A ) =/= 0 ) -> ( ( 0 / ( B - A ) ) = 0 <-> 0 = 0 ) )
- 41n:206ccc,16a,24,41m:mp3an |- ( ( 0 / ( B - A ) ) = 0 <-> 0 = 0 )
- 41o::eqid |- 0 = 0
- 41p:41o,41n:mpbir |- ( 0 / ( B - A ) ) = 0
- 41q:41p,41l:syl5eqbrr |- ( x e. ( A [,] B ) -> 0 <_ ( ( x - A ) / ( B - A ) ) )
- *now establish the inequalities
- d5:3:a1i |- ( x e. ( A [,] B ) -> C e. RR )
- d6:5:a1i |- ( x e. ( A [,] B ) -> E e. RR )
- d7:15f,29:syl |- ( x e. ( A [,] B ) -> ( 1 - ( ( x - A ) / ( B - A ) ) ) e. RR )
- d9:11b:a1i |- ( x e. ( A [,] B ) -> C <_ E )
- 42:d5,d6,d7,40u,d9:lemul1ad
- |- ( x e. ( A [,] B ) -> ( C x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) <_ ( E x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) )
- d15:4:a1i |- ( x e. ( A [,] B ) -> D e. RR )
- d16:6:a1i |- ( x e. ( A [,] B ) -> F e. RR )
- d19:11c:a1i |- ( x e. ( A [,] B ) -> D <_ F )
- 43:d15,d16,27a,41q,d19:lemul1ad
- |- ( x e. ( A [,] B ) -> ( D x. ( ( x - A ) / ( B - A ) ) ) <_ ( F x. ( ( x - A ) / ( B - A ) ) ) )
- d20:15f,30c:syl |- ( x e. ( A [,] B ) -> ( C x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) e. RR )
- d21:15f,30d:syl |- ( x e. ( A [,] B ) -> ( D x. ( ( x - A ) / ( B - A ) ) ) e. RR )
- d22:15f,30e:syl |- ( x e. ( A [,] B ) -> ( E x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) e. RR )
- d23:15f,30f:syl |- ( x e. ( A [,] B ) -> ( F x. ( ( x - A ) / ( B - A ) ) ) e. RR )
- 44:d20,d21,d22,d23,42,43:le2addd
- |- ( x e. ( A [,] B ) -> ( ( C x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) + ( D x. ( ( x - A ) / ( B - A ) ) ) )
- <_ ( ( E x. ( 1 - ( ( x - A ) / ( B - A ) ) ) ) + ( F x. ( ( x - A ) / ( B - A ) ) ) ) )
- 45:44,12az,12bz:3brtr4d |- ( x e. ( A [,] B ) -> U <_ V )
- 46:36b,36c,45:3jca |- ( x e. ( A [,] B ) -> ( U e. RR /\ V e. RR /\ U <_ V ) )
- *you did it :) **
- 50a::iftrue |- ( x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( U [,] V ) , (/) ) = ( U [,] V ) )
- 50b::iffalse |- ( -. x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( U [,] V ) , (/) ) = (/) )
- *compute the image of S
- 51::vex |- x e. _V
- 52::vex |- y e. _V
- 53:51,52:elimasn |- ( y e. ( S " { x } ) <-> <. x , y >. e. S )
- 54::nfopab2 |- F/_ y { <. x , y >. | ( x e. ( A [,] B ) /\ y e. ( U [,] V ) ) }
- 55:12,54:nfcxfr |- F/_ y S
- 56::nfcv |- F/_ y { x }
- 57:55,56:nfima |- F/_ y ( S " { x } )
- 58::nfcv |- F/_ y ( U [,] V )
- 59::nfv |- F/ y x e. ( A [,] B )
- 60a:12:eleq2i |- ( <. x , y >. e. S <-> <. x , y >. e. { <. x , y >. | ( x e. ( A [,] B ) /\ y e. ( U [,] V ) ) } )
- 60b::opabid |- ( <. x , y >. e. { <. x , y >. | ( x e. ( A [,] B ) /\ y e. ( U [,] V ) ) } <-> ( x e. ( A [,] B ) /\ y e. ( U [,] V ) ) )
- 60ba:60a,60b:bitri |- ( <. x , y >. e. S <-> ( x e. ( A [,] B ) /\ y e. ( U [,] V ) ) )
- 60c:53,60a,60b:3bitri |- ( y e. ( S " { x } ) <-> ( x e. ( A [,] B ) /\ y e. ( U [,] V ) ) )
- 60d:60c:baib |- ( x e. ( A [,] B ) -> ( y e. ( S " { x } ) <-> y e. ( U [,] V ) ) )
- 60e:59,57,58,60d:eqrd |- ( x e. ( A [,] B ) -> ( S " { x } ) = ( U [,] V ) )
- 60f:59:nfn |- F/ y -. x e. ( A [,] B )
- 60g::nfcv |- F/_ y (/)
- 60h:60c:simplbi |- ( y e. ( S " { x } ) -> x e. ( A [,] B ) )
- 60i::noel |- -. y e. (/)
- 60j:60i:pm2.21i |- ( y e. (/) -> x e. ( A [,] B ) )
- 60k:60h,60j:pm5.21ni |- ( -. x e. ( A [,] B ) -> ( y e. ( S " { x } ) <-> y e. (/) ) )
- 60l:60f,57,60g,60k:eqrd |- ( -. x e. ( A [,] B ) -> ( S " { x } ) = (/) )
- 60m:60e,50a:eqtr4d |- ( x e. ( A [,] B ) -> ( S " { x } ) = if ( x e. ( A [,] B ) , ( U [,] V ) , (/) ) )
- 60n:60l,50b:eqtr4d |- ( -. x e. ( A [,] B ) -> ( S " { x } ) = if ( x e. ( A [,] B ) , ( U [,] V ) , (/) ) )
- *work with the image of S at x and find it's integral
- 60:60m,60n:pm2.61i |- ( S " { x } ) = if ( x e. ( A [,] B ) , ( U [,] V ) , (/) )
- 61:60:fveq2i |- ( vol ` ( S " { x } ) ) = ( vol ` if ( x e. ( A [,] B ) , ( U [,] V ) , (/) ) )
- 62:61:a1i |- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = ( vol ` if ( x e. ( A [,] B ) , ( U [,] V ) , (/) ) ) )
- 63:61:a1i |- ( -. x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = ( vol ` if ( x e. ( A [,] B ) , ( U [,] V ) , (/) ) ) )
- 64:50a:fveq2d |- ( x e. ( A [,] B )
- -> ( vol ` if ( x e. ( A [,] B ) , ( U [,] V ) , (/) ) ) = ( vol ` ( U [,] V ) ) )
- 65:50b:fveq2d |- ( -. x e. ( A [,] B )
- -> ( vol ` if ( x e. ( A [,] B ) , ( U [,] V ) , (/) ) ) = ( vol ` (/) ) )
- 66:62,64:eqtrd |- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = ( vol ` ( U [,] V ) ) )
- 67:63,65:eqtrd |- ( -. x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = ( vol ` (/) ) )
- *part 1 of the conditions of dmarea prove S subset R2
- 70c:15f,39:syl |- ( x e. ( A [,] B ) -> ( U [,] V ) C_ RR )
- 71::dfss3 |- ( ( U [,] V ) C_ RR <-> A. y e. ( U [,] V ) y e. RR )
- 71a:70c,71:sylib |- ( x e. ( A [,] B ) -> A. y e. ( U [,] V ) y e. RR )
- 71b::rsp |- ( A. y e. ( U [,] V ) y e. RR -> ( y e. ( U [,] V ) -> y e. RR ) )
- 71c:71a,71b:syl |- ( x e. ( A [,] B ) -> ( y e. ( U [,] V ) -> y e. RR ) )
- 72:15f:adantr |- ( ( x e. ( A [,] B ) /\ y e. ( U [,] V ) ) -> x e. RR )
- 72a:71c:imp |- ( ( x e. ( A [,] B ) /\ y e. ( U [,] V ) ) -> y e. RR )
- 72b:72,72a:jca |- ( ( x e. ( A [,] B ) /\ y e. ( U [,] V ) ) -> ( x e. RR /\ y e. RR ) )
- 75e::df-xp |- ( RR X. RR ) = { <. x , y >. | ( x e. RR /\ y e. RR ) }
- 75f:72b:ssopab2i |- { <. x , y >. | ( x e. ( A [,] B ) /\ y e. ( U [,] V ) ) } C_ { <. x , y >. | ( x e. RR /\ y e. RR ) }
- 76:75f,12,75e:3sstr4i |- S C_ ( RR X. RR )
- *part 2 of the conditions of dmarea prove A. x e. RR ( S " { x } ) e. ( `' vol " RR )
- 100::iccmbl |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) e. dom vol )
- 101:1,2,100:mp2an |- ( A [,] B ) e. dom vol
- 102::iccmbl |- ( ( U e. RR /\ V e. RR ) -> ( U [,] V ) e. dom vol )
- 104:36d,102:syl |- ( x e. ( A [,] B ) -> ( U [,] V ) e. dom vol )
- 104a:37a,102:syl |- ( x e. RR -> ( U [,] V ) e. dom vol )
- 105::0mbl |- (/) e. dom vol
- 106:105:a1i |- ( x e. RR -> (/) e. dom vol )
- 107:104a,106:jca |- ( x e. RR -> ( ( U [,] V ) e. dom vol /\ (/) e. dom vol ) )
- 108::ifcl |- ( ( ( U [,] V ) e. dom vol /\ (/) e. dom vol )
- -> if ( x e. ( A [,] B ) , ( U [,] V ) , (/) ) e. dom vol )
- 109:107,108:syl |- ( x e. RR -> if ( x e. ( A [,] B ) , ( U [,] V ) , (/) ) e. dom vol )
- 110:60,109:syl5eqel |- ( x e. RR -> ( S " { x } ) e. dom vol )
- *jump to 200 for the completion of this part
- *compute vols
- 168::ovolicc |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( vol* ` ( A [,] B ) ) = ( B - A ) )
- 168a::ovolicc |- ( ( U e. RR /\ V e. RR /\ U <_ V ) -> ( vol* ` ( U [,] V ) ) = ( V - U ) )
- 169:1,2,11ad,168:mp3an |- ( vol* ` ( A [,] B ) ) = ( B - A )
- 169a:46,168a:syl |- ( x e. ( A [,] B ) -> ( vol* ` ( U [,] V ) ) = ( V - U ) )
- 170::mblvol |- ( ( A [,] B ) e. dom vol -> ( vol ` ( A [,] B ) ) = ( vol* ` ( A [,] B ) ) )
- 170a::mblvol |- ( ( U [,] V ) e. dom vol -> ( vol ` ( U [,] V ) ) = ( vol* ` ( U [,] V ) ) )
- 171:101,170:ax-mp |- ( vol ` ( A [,] B ) ) = ( vol* ` ( A [,] B ) )
- 171a:104,170a:syl |- ( x e. ( A [,] B ) -> ( vol ` ( U [,] V ) ) = ( vol* ` ( U [,] V ) ) )
- 172:171,169:eqtri |- ( vol ` ( A [,] B ) ) = ( B - A )
- 172aa:172,16:eqeltri |- ( vol ` ( A [,] B ) ) e. RR
- 172a:171a,169a:eqtrd |- ( x e. ( A [,] B ) -> ( vol ` ( U [,] V ) ) = ( V - U ) )
- 178::ssel2 |- ( ( ( A [,] B ) C_ RR /\ x e. ( A [,] B ) ) -> x e. RR )
- 181::ovol0 |- ( vol* ` (/) ) = 0
- 182::mblvol |- ( (/) e. dom vol -> ( vol ` (/) ) = ( vol* ` (/) ) )
- 183:105,182:ax-mp |- ( vol ` (/) ) = ( vol* ` (/) )
- 184:183,181:eqtri |- ( vol ` (/) ) = 0
- 185:184:a1i |- ( -. x e. ( A [,] B ) -> ( vol ` (/) ) = 0 )
- *apply these to S
- 186:66,172a:eqtrd |- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = ( V - U ) )
- 188:186,37fb:eqeltrd |- ( x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) e. RR )
- 189:67,184:syl6eq |- ( -. x e. ( A [,] B ) -> ( vol ` ( S " { x } ) ) = 0 )
- 190::iftrue |- ( x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( V - U ) , 0 ) = ( V - U ) )
- 191::iffalse |- ( -. x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( V - U ) , 0 ) = 0 )
- 192:190,186:eqtr4d |- ( x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( V - U ) , 0 ) = ( vol ` ( S " { x } ) ) )
- 192a:191,189:eqtr4d |- ( -. x e. ( A [,] B ) -> if ( x e. ( A [,] B ) , ( V - U ) , 0 ) = ( vol ` ( S " { x } ) ) )
- 193:192,192a:pm2.61i |- if ( x e. ( A [,] B ) , ( V - U ) , 0 ) = ( vol ` ( S " { x } ) )
- 194:193:eqcomi |- ( vol ` ( S " { x } ) ) = if ( x e. ( A [,] B ) , ( V - U ) , 0 )
- 195:194:a1i |- ( x e. RR -> ( vol ` ( S " { x } ) ) = if ( x e. ( A [,] B ) , ( V - U ) , 0 ) )
- 196:195:rgen |- A. x e. RR ( vol ` ( S " { x } ) ) = if ( x e. ( A [,] B ) , ( V - U ) , 0 )
- 197::itgeq2 |- ( A. x e. RR ( vol ` ( S " { x } ) ) = if ( x e. ( A [,] B ) , ( V - U ) , 0 )
- -> S. RR ( vol ` ( S " { x } ) ) _d x = S. RR if ( x e. ( A [,] B ) , ( V - U ) , 0 ) _d x )
- 198:196,197:ax-mp |- S. RR ( vol ` ( S " { x } ) ) _d x = S. RR if ( x e. ( A [,] B ) , ( V - U ) , 0 ) _d x
- *continue with part 2 of conditions of dmarea
- 200::fvimacnv |- ( ( Fun vol /\ ( S " { x } ) e. dom vol ) -> ( ( vol ` ( S " { x } ) ) e. RR <-> ( S " { x } ) e. ( `' vol " RR ) ) )
- 201::volf |- vol : dom vol --> ( 0 [,] +oo )
- 202::ffun |- ( vol : dom vol --> ( 0 [,] +oo ) -> Fun vol )
- 203:201,202:ax-mp |- Fun vol
- 203a:110,203:jctil |- ( x e. RR -> ( Fun vol /\ ( S " { x } ) e. dom vol ) )
- 204:203a,200:syl |- ( x e. RR -> ( ( vol ` ( S " { x } ) ) e. RR <-> ( S " { x } ) e. ( `' vol " RR ) ) )
- 205::ifcl |- ( ( ( V - U ) e. RR /\ 0 e. RR ) -> if ( x e. ( A [,] B ) , ( V - U ) , 0 ) e. RR )
- 206a:37f,206:jctir |- ( x e. RR -> ( ( V - U ) e. RR /\ 0 e. RR ) )
- 207:206a,205:syl |- ( x e. RR -> if ( x e. ( A [,] B ) , ( V - U ) , 0 ) e. RR )
- 208:195,207:eqeltrd |- ( x e. RR -> ( vol ` ( S " { x } ) ) e. RR )
- 209:208,204:mpbid |- ( x e. RR -> ( S " { x } ) e. ( `' vol " RR ) )
- 210:209:rgen |- A. x e. RR ( S " { x } ) e. ( `' vol " RR )
- *part 3 of the conditions of dmarea, prove L1
- 220:15:a1i |- ( 0 e. RR -> ( A [,] B ) C_ RR )
- 221::rembl |- RR e. dom vol
- 222:221:a1i |- ( 0 e. RR -> RR e. dom vol )
- 223:188:adantl |- ( ( 0 e. RR /\ x e. ( A [,] B ) ) -> ( vol ` ( S " { x } ) ) e. RR )
- 224::eldifn |- ( x e. ( RR \ ( A [,] B ) ) -> -. x e. ( A [,] B ) )
- 225:224,189:syl |- ( x e. ( RR \ ( A [,] B ) ) -> ( vol ` ( S " { x } ) ) = 0 )
- 226:225:adantl |- ( ( 0 e. RR /\ x e. ( RR \ ( A [,] B ) ) ) -> ( vol ` ( S " { x } ) ) = 0 )
- 227::cncfmptc |- ( ( ( V - U ) e. CC /\ ( A [,] B ) C_ CC /\ CC C_ CC )
- -> ( x e. ( A [,] B ) |-> ( V - U ) ) e. ( ( A [,] B ) -cn-> CC ) )
- 228::ssid |- CC C_ CC
- 228a:228:a1i |- ( x e. RR -> CC C_ CC )
- 228b:37g,15c,228a:3jca |- ( x e. RR -> ( ( V - U ) e. CC /\ ( A [,] B ) C_ CC /\ CC C_ CC ) )
- 229:228b,227:syl |- ( x e. RR -> ( x e. ( A [,] B ) |-> ( V - U ) ) e. ( ( A [,] B ) -cn-> CC ) )
- 230::cniccibl |- ( ( A e. RR /\ B e. RR /\ ( x e. ( A [,] B ) |-> ( V - U ) ) e. ( ( A [,] B ) -cn-> CC ) )
- -> ( x e. ( A [,] B ) |-> ( V - U ) ) e. L^1 )
- 231:10ad,10bd,229:3jca |- ( x e. RR -> ( A e. RR /\ B e. RR /\ ( x e. ( A [,] B ) |-> ( V - U ) ) e. ( ( A [,] B ) -cn-> CC ) ) )
- 232:231,230:syl |- ( x e. RR -> ( x e. ( A [,] B ) |-> ( V - U ) ) e. L^1 )
- 233:186:mpteq2ia |- ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) = ( x e. ( A [,] B ) |-> ( V - U ) )
- 234:233,232:syl5eqel |- ( x e. RR -> ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) e. L^1 )
- 234a:15f,234:syl |- ( x e. ( A [,] B ) -> ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) e. L^1 )
- 234ab:234:rgen |- A. x e. RR ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) e. L^1
- 234b:234ab: |- ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) e. L^1
- 235:234b:a1i |- ( 0 e. RR -> ( x e. ( A [,] B ) |-> ( vol ` ( S " { x } ) ) ) e. L^1 )
- 236:220,222,223,226,235:iblss2
- |- ( 0 e. RR -> ( x e. RR |-> ( vol ` ( S " { x } ) ) ) e. L^1 )
- 237:206,236:ax-mp |- ( x e. RR |-> ( vol ` ( S " { x } ) ) ) e. L^1
- *value of the integral of x
- 250:: |- S. ( A [,] B ) x _d x = ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 )
- *evaluate the integral
- d24:186:adantl |- ( ( T. /\ x e. ( A [,] B ) ) -> ( vol ` ( S " { x } ) ) = ( V - U ) )
- 300:d24:itgeq2dv |- ( T. -> S. ( A [,] B ) ( vol ` ( S " { x } ) ) _d x = S. ( A [,] B ) ( V - U ) _d x )
- 300a:11e,300:ax-mp |- S. ( A [,] B ) ( vol ` ( S " { x } ) ) _d x = S. ( A [,] B ) ( V - U ) _d x
- d25:36c:adantl |- ( ( T. /\ x e. ( A [,] B ) ) -> V e. RR )
- !d26:: |- ( T. -> ( x e. ( A [,] B ) |-> V ) e. L^1 )
- d27:36b:adantl |- ( ( T. /\ x e. ( A [,] B ) ) -> U e. RR )
- !d28:: |- ( T. -> ( x e. ( A [,] B ) |-> U ) e. L^1 )
- *break up int S = int V - int U
- 301:d25,d26,d27,d28:itgsub |- ( T. -> S. ( A [,] B ) ( V - U ) _d x = ( S. ( A [,] B ) V _d x - S. ( A [,] B ) U _d x ) )
- 301a:11e,301:ax-mp |- S. ( A [,] B ) ( V - U ) _d x = ( S. ( A [,] B ) V _d x - S. ( A [,] B ) U _d x )
- 302:10:a1i |- ( ( T. /\ x e. ( A [,] B ) ) -> U = ( C + ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) ) )
- 303:302:itgeq2dv |- ( T. -> S. ( A [,] B ) U _d x
- = S. ( A [,] B ) ( C + ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) ) _d x )
- 303a:11e,303:ax-mp |- S. ( A [,] B ) U _d x
- = S. ( A [,] B ) ( C + ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) ) _d x
- d29:3:a1i |- ( ( T. /\ x e. ( A [,] B ) ) -> C e. RR )
- d29a::cncfmptc |- ( ( C e. CC /\ ( A [,] B ) C_ CC /\ CC C_ CC )
- -> ( x e. ( A [,] B ) |-> C ) e. ( ( A [,] B ) -cn-> CC ) )
- d29b:10cc,15b,228,d29a:mp3an |- ( x e. ( A [,] B ) |-> C ) e. ( ( A [,] B ) -cn-> CC )
- d29c::cniccibl |- ( ( A e. RR /\ B e. RR /\ ( x e. ( A [,] B ) |-> C ) e. ( ( A [,] B ) -cn-> CC ) )
- -> ( x e. ( A [,] B ) |-> C ) e. L^1 )
- d29d:1,2,d29b,d29c:mp3an |- ( x e. ( A [,] B ) |-> C ) e. L^1
- d30:d29d:a1i |- ( T. -> ( x e. ( A [,] B ) |-> C ) e. L^1 )
- d50:27a:adantl |- ( ( T. /\ x e. ( A [,] B ) ) -> ( ( x - A ) / ( B - A ) ) e. RR )
- d51:4:a1i |- ( ( T. /\ x e. ( A [,] B ) ) -> D e. RR )
- d52:d51,d29:resubcld |- ( ( T. /\ x e. ( A [,] B ) ) -> ( D - C ) e. RR )
- d31:d50,d52:remulcld |- ( ( T. /\ x e. ( A [,] B ) ) -> ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) e. RR )
- !d32:: |- ( T. -> ( x e. ( A [,] B ) |-> ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) ) e. L^1 )
- *int U = int C + int I (D - C)
- 304:d29,d30,d31,d32:itgadd |- ( T. -> S. ( A [,] B ) ( C + ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) ) _d x
- = ( S. ( A [,] B ) C _d x + S. ( A [,] B ) ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) _d x ) )
- 304a:11e,304:ax-mp |- S. ( A [,] B ) ( C + ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) ) _d x
- = ( S. ( A [,] B ) C _d x + S. ( A [,] B ) ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) _d x )
- 305::itgconst |- ( ( ( A [,] B ) e. dom vol /\ ( vol ` ( A [,] B ) ) e. RR /\ C e. CC )
- -> S. ( A [,] B ) C _d x = ( C x. ( vol ` ( A [,] B ) ) ) )
- 305a:101,172aa,10cc,305:mp3an |- S. ( A [,] B ) C _d x = ( C x. ( vol ` ( A [,] B ) ) )
- 305b:172:oveq2i |- ( C x. ( vol ` ( A [,] B ) ) ) = ( C x. ( B - A ) )
- 305c:305a,305b:eqtri |- S. ( A [,] B ) C _d x = ( C x. ( B - A ) )
- d43:10dc:a1i |- ( T. -> D e. CC )
- d44:10cc:a1i |- ( T. -> C e. CC )
- d33:d43,d44:subcld |- ( T. -> ( D - C ) e. CC )
- !d35:: |- ( T. -> ( x e. ( A [,] B ) |-> ( ( x - A ) / ( B - A ) ) ) e. L^1 )
- *int I ( D - C ) = ( D - C ) int I
- 306:d33,d50,d35:itgmulc2 |- ( T. -> ( ( D - C ) x. S. ( A [,] B ) ( ( x - A ) / ( B - A ) ) _d x )
- = S. ( A [,] B ) ( ( D - C ) x. ( ( x - A ) / ( B - A ) ) ) _d x )
- 306a:11e,306:ax-mp |- ( ( D - C ) x. S. ( A [,] B ) ( ( x - A ) / ( B - A ) ) _d x )
- = S. ( A [,] B ) ( ( D - C ) x. ( ( x - A ) / ( B - A ) ) ) _d x
- d36:15a,16cb:sseldi |- ( x e. ( A [,] B ) -> ( x - A ) e. CC )
- d37:16a:a1i |- ( x e. ( A [,] B ) -> ( B - A ) e. CC )
- d38:24:a1i |- ( x e. ( A [,] B ) -> ( B - A ) =/= 0 )
- 307:d36,d37,d38:divrecd |- ( x e. ( A [,] B ) -> ( ( x - A ) / ( B - A ) ) = ( ( x - A ) x. ( 1 / ( B - A ) ) ) )
- d39:d37,d38:reccld |- ( x e. ( A [,] B ) -> ( 1 / ( B - A ) ) e. CC )
- 308:d36,d39:mulcomd |- ( x e. ( A [,] B ) -> ( ( x - A ) x. ( 1 / ( B - A ) ) ) = ( ( 1 / ( B - A ) ) x. ( x - A ) ) )
- 309:307,308:eqtrd |- ( x e. ( A [,] B ) -> ( ( x - A ) / ( B - A ) ) = ( ( 1 / ( B - A ) ) x. ( x - A ) ) )
- 309a:309:rgen |- A. x e. ( A [,] B ) ( ( x - A ) / ( B - A ) ) = ( ( 1 / ( B - A ) ) x. ( x - A ) )
- 309c::itgeq2 |- ( A. x e. ( A [,] B ) ( ( x - A ) / ( B - A ) ) = ( ( 1 / ( B - A ) ) x. ( x - A ) )
- -> S. ( A [,] B ) ( ( x - A ) / ( B - A ) ) _d x = S. ( A [,] B ) ( ( 1 / ( B - A ) ) x. ( x - A ) ) _d x )
- 309d:309a,309c:ax-mp |- S. ( A [,] B ) ( ( x - A ) / ( B - A ) ) _d x = S. ( A [,] B ) ( ( 1 / ( B - A ) ) x. ( x - A ) ) _d x
- d45:10bc:a1i |- ( T. -> B e. CC )
- d46:10ac:a1i |- ( T. -> A e. CC )
- d47:d45,d46:subcld |- ( T. -> ( B - A ) e. CC )
- d48:18:a1i |- ( T. -> B =/= A )
- d49:d45,d46,d48:subne0d |- ( T. -> ( B - A ) =/= 0 )
- d40:d47,d49:reccld |- ( T. -> ( 1 / ( B - A ) ) e. CC )
- d40a:11e,d40:ax-mp |- ( 1 / ( B - A ) ) e. CC
- d41:16cb:adantl |- ( ( T. /\ x e. ( A [,] B ) ) -> ( x - A ) e. RR )
- !d42:: |- ( T. -> ( x e. ( A [,] B ) |-> ( x - A ) ) e. L^1 )
- *int I = int ( x - A ) / B - A = 1 / B - A int ( x - A )
- 310:d40,d41,d42:itgmulc2 |- ( T. -> ( ( 1 / ( B - A ) ) x. S. ( A [,] B ) ( x - A ) _d x )
- = S. ( A [,] B ) ( ( 1 / ( B - A ) ) x. ( x - A ) ) _d x )
- 310a:11e,310:ax-mp |- ( ( 1 / ( B - A ) ) x. S. ( A [,] B ) ( x - A ) _d x )
- = S. ( A [,] B ) ( ( 1 / ( B - A ) ) x. ( x - A ) ) _d x
- d53:15f:adantl |- ( ( T. /\ x e. ( A [,] B ) ) -> x e. RR )
- !d54:: |- ( T. -> ( x e. ( A [,] B ) |-> x ) e. L^1 )
- d55:1:a1i |- ( ( T. /\ x e. ( A [,] B ) ) -> A e. RR )
- d56a::cncfmptc |- ( ( A e. CC /\ ( A [,] B ) C_ CC /\ CC C_ CC )
- -> ( x e. ( A [,] B ) |-> A ) e. ( ( A [,] B ) -cn-> CC ) )
- d56b:10ac,15b,228,d56a:mp3an |- ( x e. ( A [,] B ) |-> A ) e. ( ( A [,] B ) -cn-> CC )
- d56c::cniccibl |- ( ( A e. RR /\ B e. RR /\ ( x e. ( A [,] B ) |-> A ) e. ( ( A [,] B ) -cn-> CC ) )
- -> ( x e. ( A [,] B ) |-> A ) e. L^1 )
- d56:1,2,d56b,d56c:eel000cT |- ( T. -> ( x e. ( A [,] B ) |-> A ) e. L^1 )
- * int x - A = int X - int A
- 315:d53,d54,d55,d56:itgsub
- |- ( T. -> S. ( A [,] B ) ( x - A ) _d x = ( S. ( A [,] B ) x _d x - S. ( A [,] B ) A _d x ) )
- 315a:11e,315:ax-mp |- S. ( A [,] B ) ( x - A ) _d x = ( S. ( A [,] B ) x _d x - S. ( A [,] B ) A _d x )
- 316::itgconst |- ( ( ( A [,] B ) e. dom vol /\ ( vol ` ( A [,] B ) ) e. RR /\ A e. CC )
- -> S. ( A [,] B ) A _d x = ( A x. ( vol ` ( A [,] B ) ) ) )
- 317:101,172aa,10ac,316:mp3an |- S. ( A [,] B ) A _d x = ( A x. ( vol ` ( A [,] B ) ) )
- 318:172:oveq2i |- ( A x. ( vol ` ( A [,] B ) ) ) = ( A x. ( B - A ) )
- 319:317,318:eqtri |- S. ( A [,] B ) A _d x = ( A x. ( B - A ) )
- *now rebuild :)
- 320:250,319:oveq12i |- ( S. ( A [,] B ) x _d x - S. ( A [,] B ) A _d x ) = ( ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) - ( A x. ( B - A ) ) )
- 321:315a,320:eqtri |- S. ( A [,] B ) ( x - A ) _d x = ( ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) - ( A x. ( B - A ) ) )
- 322:321:oveq2i |- ( ( 1 / ( B - A ) ) x. S. ( A [,] B ) ( x - A ) _d x )
- = ( ( 1 / ( B - A ) ) x. ( ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) - ( A x. ( B - A ) ) ) )
- 323::subdi |- ( ( ( 1 / ( B - A ) ) e. CC /\ ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) e. CC /\ ( A x. ( B - A ) ) e. CC )
- -> ( ( 1 / ( B - A ) ) x. ( ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) - ( A x. ( B - A ) ) ) )
- = ( ( ( 1 / ( B - A ) ) x. ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) ) - ( ( 1 / ( B - A ) ) x. ( A x. ( B - A ) ) ) ) )
- 324:d40a,16af,16ab,323:mp3an |- ( ( 1 / ( B - A ) ) x. ( ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) - ( A x. ( B - A ) ) ) )
- = ( ( ( 1 / ( B - A ) ) x. ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) ) - ( ( 1 / ( B - A ) ) x. ( A x. ( B - A ) ) ) )
- 345:d40a,16ab:mulcomi |- ( ( 1 / ( B - A ) ) x. ( A x. ( B - A ) ) ) = ( ( A x. ( B - A ) ) x. ( 1 / ( B - A ) ) )
- 346:16ab,16a,24:divreci |- ( ( A x. ( B - A ) ) / ( B - A ) ) = ( ( A x. ( B - A ) ) x. ( 1 / ( B - A ) ) )
- 347:345,346:eqtr4i |- ( ( 1 / ( B - A ) ) x. ( A x. ( B - A ) ) ) = ( ( A x. ( B - A ) ) / ( B - A ) )
- 348:10ac,16a,24:divcan4i |- ( ( A x. ( B - A ) ) / ( B - A ) ) = A
- 349:347,348:eqtri |- ( ( 1 / ( B - A ) ) x. ( A x. ( B - A ) ) ) = A
- 350:322,324:eqtri |- ( ( 1 / ( B - A ) ) x. S. ( A [,] B ) ( x - A ) _d x )
- = ( ( ( 1 / ( B - A ) ) x. ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) ) - ( ( 1 / ( B - A ) ) x. ( A x. ( B - A ) ) ) )
- 351:349:oveq2i |- ( ( ( 1 / ( B - A ) ) x. ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) ) - ( ( 1 / ( B - A ) ) x. ( A x. ( B - A ) ) ) )
- = ( ( ( 1 / ( B - A ) ) x. ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) ) - A )
- 352:350,351:eqtri |- ( ( 1 / ( B - A ) ) x. S. ( A [,] B ) ( x - A ) _d x )
- = ( ( ( 1 / ( B - A ) ) x. ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) ) - A )
- 353:310a,352:eqtr3i |- S. ( A [,] B ) ( ( 1 / ( B - A ) ) x. ( x - A ) ) _d x
- = ( ( ( 1 / ( B - A ) ) x. ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) ) - A )
- *this is to be simplified
- 354:309d,353:eqtri |- S. ( A [,] B ) ( ( x - A ) / ( B - A ) ) _d x
- = ( ( ( 1 / ( B - A ) ) x. ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) ) - A )
- 355:10bc,10ac:subsqi |- ( ( B ^ 2 ) - ( A ^ 2 ) ) = ( ( B + A ) x. ( B - A ) )
- 355a:355:oveq1i |- ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) = ( ( ( B + A ) x. ( B - A ) ) / 2 )
- 355b:355a:oveq2i |- ( ( 1 / ( B - A ) ) x. ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) )
- = ( ( 1 / ( B - A ) ) x. ( ( ( B + A ) x. ( B - A ) ) / 2 ) )
- d58:355,16ae:eqeltrri |- ( ( B + A ) x. ( B - A ) ) e. CC
- 356:d40a,d58,11f,11g:divassi
- |- ( ( ( 1 / ( B - A ) ) x. ( ( B + A ) x. ( B - A ) ) ) / 2 ) = ( ( 1 / ( B - A ) ) x. ( ( ( B + A ) x. ( B - A ) ) / 2 ) )
- 357:355b,356:eqtr4i |- ( ( 1 / ( B - A ) ) x. ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) )
- = ( ( ( 1 / ( B - A ) ) x. ( ( B + A ) x. ( B - A ) ) ) / 2 )
- 358:d40a,d58:mulcomi
- |- ( ( 1 / ( B - A ) ) x. ( ( B + A ) x. ( B - A ) ) ) = ( ( ( B + A ) x. ( B - A ) ) x. ( 1 / ( B - A ) ) )
- 359:d58,16a,24:divreci
- |- ( ( ( B + A ) x. ( B - A ) ) / ( B - A ) ) = ( ( ( B + A ) x. ( B - A ) ) x. ( 1 / ( B - A ) ) )
- 360:358,359:eqtr4i |- ( ( 1 / ( B - A ) ) x. ( ( B + A ) x. ( B - A ) ) )
- = ( ( ( B + A ) x. ( B - A ) ) / ( B - A ) )
- d59:10bc,10ac:addcli |- ( B + A ) e. CC
- 361:d59,16a,24:divcan4i |- ( ( ( B + A ) x. ( B - A ) ) / ( B - A ) ) = ( B + A )
- 362:360,361:eqtri |- ( ( 1 / ( B - A ) ) x. ( ( B + A ) x. ( B - A ) ) ) = ( B + A )
- 363:362:oveq1i |- ( ( ( 1 / ( B - A ) ) x. ( ( B + A ) x. ( B - A ) ) ) / 2 ) = ( ( B + A ) / 2 )
- 364:357,363:eqtri |- ( ( 1 / ( B - A ) ) x. ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) ) = ( ( B + A ) / 2 )
- 365:364:oveq1i |- ( ( ( 1 / ( B - A ) ) x. ( ( ( B ^ 2 ) - ( A ^ 2 ) ) / 2 ) ) - A ) = ( ( ( B + A ) / 2 ) - A )
- *this is to be simplified further
- 366:354,365:eqtri |- S. ( A [,] B ) ( ( x - A ) / ( B - A ) ) _d x = ( ( ( B + A ) / 2 ) - A )
- 367:10ac,11f,11g:divcan4i |- ( ( A x. 2 ) / 2 ) = A
- 368:367:oveq2i |- ( ( ( B + A ) / 2 ) - ( ( A x. 2 ) / 2 ) ) = ( ( ( B + A ) / 2 ) - A )
- 369::divsubdir |- ( ( ( B + A ) e. CC /\ ( A x. 2 ) e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) )
- -> ( ( ( B + A ) - ( A x. 2 ) ) / 2 ) = ( ( ( B + A ) / 2 ) - ( ( A x. 2 ) / 2 ) ) )
- 370:d59,11h,11i,369:mp3an |- ( ( ( B + A ) - ( A x. 2 ) ) / 2 ) = ( ( ( B + A ) / 2 ) - ( ( A x. 2 ) / 2 ) )
- 371:370,368:eqtri |- ( ( ( B + A ) - ( A x. 2 ) ) / 2 ) = ( ( ( B + A ) / 2 ) - A )
- 372:10bc,10ac,11h:addsubassi |- ( ( B + A ) - ( A x. 2 ) ) = ( B + ( A - ( A x. 2 ) ) )
- 373::subsub2 |- ( ( B e. CC /\ ( A x. 2 ) e. CC /\ A e. CC )
- -> ( B - ( ( A x. 2 ) - A ) ) = ( B + ( A - ( A x. 2 ) ) ) )
- 374:10bc,11h,10ac,373:mp3an |- ( B - ( ( A x. 2 ) - A ) ) = ( B + ( A - ( A x. 2 ) ) )
- *375::pncan |- ( ( A e. CC /\ A e. CC ) -> ( ( A + A ) - A ) = A )
- 376:10ac,10ac:pncan3oi |- ( ( A + A ) - A ) = A
- 377:10ac:times2i |- ( A x. 2 ) = ( A + A )
- 378:377:oveq1i |- ( ( A x. 2 ) - A ) = ( ( A + A ) - A )
- 379:378,376:eqtri |- ( ( A x. 2 ) - A ) = A
- 380:379:oveq2i |- ( B - ( ( A x. 2 ) - A ) ) = ( B - A )
- 381:374,380:eqtr3i |- ( B + ( A - ( A x. 2 ) ) ) = ( B - A )
- 382:372,381:eqtri |- ( ( B + A ) - ( A x. 2 ) ) = ( B - A )
- 383:382:oveq1i |- ( ( ( B + A ) - ( A x. 2 ) ) / 2 ) = ( ( B - A ) / 2 )
- 384:370,383:eqtr3i |- ( ( ( B + A ) / 2 ) - ( ( A x. 2 ) / 2 ) ) = ( ( B - A ) / 2 )
- 385:368,384:eqtr3i |- ( ( ( B + A ) / 2 ) - A ) = ( ( B - A ) / 2 )
- *this is the integral of I :) :)
- 386:366,385:eqtri |- S. ( A [,] B ) ( ( x - A ) / ( B - A ) ) _d x = ( ( B - A ) / 2 )
- *build up to the integral of U
- 387:386:oveq2i |- ( ( D - C ) x. S. ( A [,] B ) ( ( x - A ) / ( B - A ) ) _d x )
- = ( ( D - C ) x. ( ( B - A ) / 2 ) )
- 388:387,306a:eqtr3i |- ( ( D - C ) x. ( ( B - A ) / 2 ) )
- = S. ( A [,] B ) ( ( D - C ) x. ( ( x - A ) / ( B - A ) ) ) _d x
- d60:11e,d33:ax-mp |- ( D - C ) e. CC
- d60a:d60:a1i |- ( x e. ( A [,] B ) -> ( D - C ) e. CC )
- 389::mulcom |- ( ( ( D - C ) e. CC /\ ( ( x - A ) / ( B - A ) ) e. CC )
- -> ( ( D - C ) x. ( ( x - A ) / ( B - A ) ) ) = ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) )
- 390:d60a,27b,389:syl2anc |- ( x e. ( A [,] B )
- -> ( ( D - C ) x. ( ( x - A ) / ( B - A ) ) ) = ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) )
- 391:390:rgen |- A. x e. ( A [,] B ) ( ( D - C ) x. ( ( x - A ) / ( B - A ) ) ) = ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) )
- 392::itgeq2 |- ( A. x e. ( A [,] B ) ( ( D - C ) x. ( ( x - A ) / ( B - A ) ) )
- = ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) )
- -> S. ( A [,] B ) ( ( D - C ) x. ( ( x - A ) / ( B - A ) ) ) _d x
- = S. ( A [,] B ) ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) _d x )
- 393:391,392:ax-mp |- S. ( A [,] B ) ( ( D - C ) x. ( ( x - A ) / ( B - A ) ) ) _d x
- = S. ( A [,] B ) ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) _d x
- 394:388,393:eqtr2i |- S. ( A [,] B ) ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) _d x
- = ( ( D - C ) x. ( ( B - A ) / 2 ) )
- 395:305c,394:oveq12i |- ( S. ( A [,] B ) C _d x + S. ( A [,] B ) ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) _d x )
- = ( ( C x. ( B - A ) ) + ( ( D - C ) x. ( ( B - A ) / 2 ) ) )
- 396:304a,395:eqtri |- S. ( A [,] B ) ( C + ( ( ( x - A ) / ( B - A ) ) x. ( D - C ) ) ) _d x
- = ( ( C x. ( B - A ) ) + ( ( D - C ) x. ( ( B - A ) / 2 ) ) )
- *now simplify this
- 397:303a,396:eqtri |- S. ( A [,] B ) U _d x = ( ( C x. ( B - A ) ) + ( ( D - C ) x. ( ( B - A ) / 2 ) ) )
- 398:10cc,11f,11g:divcan4i |- ( ( C x. 2 ) / 2 ) = C
- 399:398:oveq1i |- ( ( ( C x. 2 ) / 2 ) x. ( B - A ) ) = ( C x. ( B - A ) )
- 400::div32 |- ( ( ( C x. 2 ) e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) /\ ( B - A ) e. CC )
- -> ( ( ( C x. 2 ) / 2 ) x. ( B - A ) ) = ( ( C x. 2 ) x. ( ( B - A ) / 2 ) ) )
- 401:11hc,11i,16a,400:mp3an |- ( ( ( C x. 2 ) / 2 ) x. ( B - A ) ) = ( ( C x. 2 ) x. ( ( B - A ) / 2 ) )
- 402:399,401:eqtr3i |- ( C x. ( B - A ) ) = ( ( C x. 2 ) x. ( ( B - A ) / 2 ) )
- 403:402:oveq1i |- ( ( C x. ( B - A ) ) + ( ( D - C ) x. ( ( B - A ) / 2 ) ) )
- = ( ( ( C x. 2 ) x. ( ( B - A ) / 2 ) ) + ( ( D - C ) x. ( ( B - A ) / 2 ) ) )
- 404:397,403:eqtri |- S. ( A [,] B ) U _d x
- = ( ( ( C x. 2 ) x. ( ( B - A ) / 2 ) ) + ( ( D - C ) x. ( ( B - A ) / 2 ) ) )
- d63:16a,11f,11g:divcli |- ( ( B - A ) / 2 ) e. CC
- 405:11hc,d60,d63:adddiri |- ( ( ( C x. 2 ) + ( D - C ) ) x. ( ( B - A ) / 2 ) )
- = ( ( ( C x. 2 ) x. ( ( B - A ) / 2 ) ) + ( ( D - C ) x. ( ( B - A ) / 2 ) ) )
- 406:404,405:eqtr4i |- S. ( A [,] B ) U _d x
- = ( ( ( C x. 2 ) + ( D - C ) ) x. ( ( B - A ) / 2 ) )
- 407::addsub12 |- ( ( D e. CC /\ ( C x. 2 ) e. CC /\ C e. CC )
- -> ( D + ( ( C x. 2 ) - C ) ) = ( ( C x. 2 ) + ( D - C ) ) )
- 408:10dc,11hc,10cc,407:mp3an |- ( D + ( ( C x. 2 ) - C ) ) = ( ( C x. 2 ) + ( D - C ) )
- 409:10cc,10cc:pncan3oi |- ( ( C + C ) - C ) = C
- 410:10cc:times2i |- ( C x. 2 ) = ( C + C )
- 411:410:oveq1i |- ( ( C x. 2 ) - C ) = ( ( C + C ) - C )
- 412:411,409:eqtri |- ( ( C x. 2 ) - C ) = C
- 413:412:oveq2i |- ( D + ( ( C x. 2 ) - C ) ) = ( D + C )
- 414:408,413:eqtr3i |- ( ( C x. 2 ) + ( D - C ) ) = ( D + C )
- 415:414:oveq1i |- ( ( ( C x. 2 ) + ( D - C ) ) x. ( ( B - A ) / 2 ) ) = ( ( D + C ) x. ( ( B - A ) / 2 ) )
- *we have the integral for U :) :) :)
- 416:406,415:eqtri |- S. ( A [,] B ) U _d x = ( ( D + C ) x. ( ( B - A ) / 2 ) )
- 417::itgeq2 |- ( A. x e. ( A [,] B ) V = ( E + ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) )
- -> S. ( A [,] B ) V _d x = S. ( A [,] B ) ( E + ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) ) _d x )
- 418:11:a1i |- ( x e. ( A [,] B ) -> V = ( E + ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) ) )
- 419:418:rgen |- A. x e. ( A [,] B ) V = ( E + ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) )
- *this is the integral for V to be proven
- 420:419,417:ax-mp |- S. ( A [,] B ) V _d x = S. ( A [,] B ) ( E + ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) ) _d x
- d64:5:a1i |- ( ( T. /\ x e. ( A [,] B ) ) -> E e. RR )
- d65a:5:a1i |- ( ( T. /\ x e. ( A [,] B ) ) -> E e. RR )
- d66a::cncfmptc |- ( ( E e. CC /\ ( A [,] B ) C_ CC /\ CC C_ CC )
- -> ( x e. ( A [,] B ) |-> E ) e. ( ( A [,] B ) -cn-> CC ) )
- d66b:10ec,15b,228,d66a:mp3an |- ( x e. ( A [,] B ) |-> E ) e. ( ( A [,] B ) -cn-> CC )
- d66c::cniccibl |- ( ( A e. RR /\ B e. RR /\ ( x e. ( A [,] B ) |-> E ) e. ( ( A [,] B ) -cn-> CC ) )
- -> ( x e. ( A [,] B ) |-> E ) e. L^1 )
- d66ca:1,2,d66b,d66c:mp3an |- ( x e. ( A [,] B ) |-> E ) e. L^1
- d66d:d66ca:a1i |- ( T. -> ( x e. ( A [,] B ) |-> E ) e. L^1 )
- d68:6:a1i |- ( ( T. /\ x e. ( A [,] B ) ) -> F e. RR )
- d69:d68,d64:resubcld |- ( ( T. /\ x e. ( A [,] B ) ) -> ( F - E ) e. RR )
- d66:d50,d69:remulcld |- ( ( T. /\ x e. ( A [,] B ) ) -> ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) e. RR )
- !d67:: |- ( T. -> ( x e. ( A [,] B ) |-> ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) ) e. L^1 )
- 421:d64,d66d,d66,d67:itgadd |- ( T. -> S. ( A [,] B ) ( E + ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) ) _d x
- = ( S. ( A [,] B ) E _d x + S. ( A [,] B ) ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) _d x ) )
- 421a:11e,421:ax-mp |- S. ( A [,] B ) ( E + ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) ) _d x
- = ( S. ( A [,] B ) E _d x + S. ( A [,] B ) ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) _d x )
- 422::itgconst |- ( ( ( A [,] B ) e. dom vol /\ ( vol ` ( A [,] B ) ) e. RR /\ E e. CC )
- -> S. ( A [,] B ) E _d x = ( E x. ( vol ` ( A [,] B ) ) ) )
- 422a:101,172aa,10ec,422:mp3an |- S. ( A [,] B ) E _d x = ( E x. ( vol ` ( A [,] B ) ) )
- 422b:172:oveq2i |- ( E x. ( vol ` ( A [,] B ) ) ) = ( E x. ( B - A ) )
- 422c:422a,422b:eqtri |- S. ( A [,] B ) E _d x = ( E x. ( B - A ) )
- *get int I ( F - E )
- d73:10fc:a1i |- ( T. -> F e. CC )
- d74:10ec:a1i |- ( T. -> E e. CC )
- d70:d73,d74:subcld |- ( T. -> ( F - E ) e. CC )
- 423:d70,d50,d35:itgmulc2 |- ( T. -> ( ( F - E ) x. S. ( A [,] B ) ( ( x - A ) / ( B - A ) ) _d x )
- = S. ( A [,] B ) ( ( F - E ) x. ( ( x - A ) / ( B - A ) ) ) _d x )
- 423a:11e,423:ax-mp |- ( ( F - E ) x. S. ( A [,] B ) ( ( x - A ) / ( B - A ) ) _d x )
- = S. ( A [,] B ) ( ( F - E ) x. ( ( x - A ) / ( B - A ) ) ) _d x
- 424:386:oveq2i |- ( ( F - E ) x. S. ( A [,] B ) ( ( x - A ) / ( B - A ) ) _d x )
- = ( ( F - E ) x. ( ( B - A ) / 2 ) )
- 425:423a,424:eqtr3i |- S. ( A [,] B ) ( ( F - E ) x. ( ( x - A ) / ( B - A ) ) ) _d x = ( ( F - E ) x. ( ( B - A ) / 2 ) )
- 426::itgeq2 |- ( A. x e. ( A [,] B ) ( ( F - E ) x. ( ( x - A ) / ( B - A ) ) )
- = ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) )
- -> S. ( A [,] B ) ( ( F - E ) x. ( ( x - A ) / ( B - A ) ) ) _d x
- = S. ( A [,] B ) ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) _d x )
- 427::mulcom |- ( ( ( F - E ) e. CC /\ ( ( x - A ) / ( B - A ) ) e. CC )
- -> ( ( F - E ) x. ( ( x - A ) / ( B - A ) ) ) = ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) )
- 428:11k,27b,427:syl2anc |- ( x e. ( A [,] B )
- -> ( ( F - E ) x. ( ( x - A ) / ( B - A ) ) ) = ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) )
- 429:428:rgen |- A. x e. ( A [,] B ) ( ( F - E ) x. ( ( x - A ) / ( B - A ) ) )
- = ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) )
- 430:429,426:ax-mp |- S. ( A [,] B ) ( ( F - E ) x. ( ( x - A ) / ( B - A ) ) ) _d x
- = S. ( A [,] B ) ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) _d x
- 431:430,425:eqtr3i |- S. ( A [,] B ) ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) _d x
- = ( ( F - E ) x. ( ( B - A ) / 2 ) )
- 432:422c,431:oveq12i |- ( S. ( A [,] B ) E _d x + S. ( A [,] B ) ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) _d x )
- = ( ( E x. ( B - A ) ) + ( ( F - E ) x. ( ( B - A ) / 2 ) ) )
- 433:421a,432:eqtri |- S. ( A [,] B ) ( E + ( ( ( x - A ) / ( B - A ) ) x. ( F - E ) ) ) _d x
- = ( ( E x. ( B - A ) ) + ( ( F - E ) x. ( ( B - A ) / 2 ) ) )
- 434:420,433:eqtri |- S. ( A [,] B ) V _d x = ( ( E x. ( B - A ) ) + ( ( F - E ) x. ( ( B - A ) / 2 ) ) )
- *now simplify this
- 498:10ec,11f,11g:divcan4i |- ( ( E x. 2 ) / 2 ) = E
- 499:498:oveq1i |- ( ( ( E x. 2 ) / 2 ) x. ( B - A ) ) = ( E x. ( B - A ) )
- 500::div32 |- ( ( ( E x. 2 ) e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) /\ ( B - A ) e. CC )
- -> ( ( ( E x. 2 ) / 2 ) x. ( B - A ) ) = ( ( E x. 2 ) x. ( ( B - A ) / 2 ) ) )
- 501:11he,11i,16a,500:mp3an |- ( ( ( E x. 2 ) / 2 ) x. ( B - A ) ) = ( ( E x. 2 ) x. ( ( B - A ) / 2 ) )
- 502:499,501:eqtr3i |- ( E x. ( B - A ) ) = ( ( E x. 2 ) x. ( ( B - A ) / 2 ) )
- 503:502:oveq1i |- ( ( E x. ( B - A ) ) + ( ( F - E ) x. ( ( B - A ) / 2 ) ) )
- = ( ( ( E x. 2 ) x. ( ( B - A ) / 2 ) ) + ( ( F - E ) x. ( ( B - A ) / 2 ) ) )
- 504:434,503:eqtri |- S. ( A [,] B ) V _d x
- = ( ( ( E x. 2 ) x. ( ( B - A ) / 2 ) ) + ( ( F - E ) x. ( ( B - A ) / 2 ) ) )
- 505:11he,11j,d63:adddiri |- ( ( ( E x. 2 ) + ( F - E ) ) x. ( ( B - A ) / 2 ) )
- = ( ( ( E x. 2 ) x. ( ( B - A ) / 2 ) ) + ( ( F - E ) x. ( ( B - A ) / 2 ) ) )
- 506:504,505:eqtr4i |- S. ( A [,] B ) V _d x
- = ( ( ( E x. 2 ) + ( F - E ) ) x. ( ( B - A ) / 2 ) )
- 507::addsub12 |- ( ( F e. CC /\ ( E x. 2 ) e. CC /\ E e. CC )
- -> ( F + ( ( E x. 2 ) - E ) ) = ( ( E x. 2 ) + ( F - E ) ) )
- 508:10fc,11he,10ec,507:mp3an |- ( F + ( ( E x. 2 ) - E ) ) = ( ( E x. 2 ) + ( F - E ) )
- 509:10ec,10ec:pncan3oi |- ( ( E + E ) - E ) = E
- 510:10ec:times2i |- ( E x. 2 ) = ( E + E )
- 511:510:oveq1i |- ( ( E x. 2 ) - E ) = ( ( E + E ) - E )
- 512:511,509:eqtri |- ( ( E x. 2 ) - E ) = E
- 513:512:oveq2i |- ( F + ( ( E x. 2 ) - E ) ) = ( F + E )
- 514:508,513:eqtr3i |- ( ( E x. 2 ) + ( F - E ) ) = ( F + E )
- 515:514:oveq1i |- ( ( ( E x. 2 ) + ( F - E ) ) x. ( ( B - A ) / 2 ) ) = ( ( F + E ) x. ( ( B - A ) / 2 ) )
- *we have the integral for V :) :) :)
- 516:506,515:eqtri |- S. ( A [,] B ) V _d x = ( ( F + E ) x. ( ( B - A ) / 2 ) )
- 517:516,416:oveq12i |- ( S. ( A [,] B ) V _d x - S. ( A [,] B ) U _d x )
- = ( ( ( F + E ) x. ( ( B - A ) / 2 ) ) - ( ( D + C ) x. ( ( B - A ) / 2 ) ) )
- 518:301a,517:eqtri |- S. ( A [,] B ) ( V - U ) _d x
- = ( ( ( F + E ) x. ( ( B - A ) / 2 ) ) - ( ( D + C ) x. ( ( B - A ) / 2 ) ) )
- 519:300a,518:eqtri |- S. ( A [,] B ) ( vol ` ( S " { x } ) ) _d x
- = ( ( ( F + E ) x. ( ( B - A ) / 2 ) ) - ( ( D + C ) x. ( ( B - A ) / 2 ) ) )
- *simplify
- 520::div32 |- ( ( ( F + E ) e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) /\ ( B - A ) e. CC )
- -> ( ( ( F + E ) / 2 ) x. ( B - A ) ) = ( ( F + E ) x. ( ( B - A ) / 2 ) ) )
- 521:11fe,11i,16a,520:mp3an |- ( ( ( F + E ) / 2 ) x. ( B - A ) ) = ( ( F + E ) x. ( ( B - A ) / 2 ) )
- 522::div32 |- ( ( ( D + C ) e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) /\ ( B - A ) e. CC )
- -> ( ( ( D + C ) / 2 ) x. ( B - A ) ) = ( ( D + C ) x. ( ( B - A ) / 2 ) ) )
- 523:11dc,11i,16a,522:mp3an |- ( ( ( D + C ) / 2 ) x. ( B - A ) ) = ( ( D + C ) x. ( ( B - A ) / 2 ) )
- 524:521,523:oveq12i |- ( ( ( ( F + E ) / 2 ) x. ( B - A ) ) - ( ( ( D + C ) / 2 ) x. ( B - A ) ) )
- = ( ( ( F + E ) x. ( ( B - A ) / 2 ) ) - ( ( D + C ) x. ( ( B - A ) / 2 ) ) )
- 525:519,524:eqtr4i |- S. ( A [,] B ) ( vol ` ( S " { x } ) ) _d x
- = ( ( ( ( F + E ) / 2 ) x. ( B - A ) ) - ( ( ( D + C ) / 2 ) x. ( B - A ) ) )
- d75:11fe,11f,11g:divcli |- ( ( F + E ) / 2 ) e. CC
- d76:11dc,11f,11g:divcli |- ( ( D + C ) / 2 ) e. CC
- 526:d75,d76,16a:subdiri |- ( ( ( ( F + E ) / 2 ) - ( ( D + C ) / 2 ) ) x. ( B - A ) )
- = ( ( ( ( F + E ) / 2 ) x. ( B - A ) ) - ( ( ( D + C ) / 2 ) x. ( B - A ) ) )
- *this is the integral on A,B !! :)
- 527:525,526:eqtr4i |- S. ( A [,] B ) ( vol ` ( S " { x } ) ) _d x
- = ( ( ( ( F + E ) / 2 ) - ( ( D + C ) / 2 ) ) x. ( B - A ) )
- 528:300a,527:eqtr3i |- S. ( A [,] B ) ( V - U ) _d x
- = ( ( ( ( F + E ) / 2 ) - ( ( D + C ) / 2 ) ) x. ( B - A ) )
- *now get the integral on RR
- 600::itgss2 |- ( ( A [,] B ) C_ RR -> S. ( A [,] B ) ( V - U ) _d x
- = S. RR if ( x e. ( A [,] B ) , ( V - U ) , 0 ) _d x )
- 600a:15,600:ax-mp |- S. ( A [,] B ) ( V - U ) _d x = S. RR if ( x e. ( A [,] B ) , ( V - U ) , 0 ) _d x
- *41::itgconst |- ( ( ( A [,] B ) e. dom vol /\ ( vol ` ( A [,] B ) ) e. RR /\ ( D - C ) e. CC ) -> S. ( A [,] B ) ( D - C ) _d x = ( ( D - C ) x. ( vol ` ( A [,] B ) ) ) )
- *42:16a,25d,25ac,41:mp3an |- S. ( A [,] B ) ( D - C ) _d x = ( ( D - C ) x. ( vol ` ( A [,] B ) ) )
- 601:600a,528:eqtr3i |- S. RR if ( x e. ( A [,] B ) , ( V - U ) , 0 ) _d x
- = ( ( ( ( F + E ) / 2 ) - ( ( D + C ) / 2 ) ) x. ( B - A ) )
- *44:43,25aaa:eqtri |- S. RR if ( x e. ( A [,] B ) , ( D - C ) , 0 ) _d x = ( ( D - C ) x. ( B - A ) )
- 602:198,601:eqtri |- S. RR ( vol ` ( S " { x } ) ) _d x
- = ( ( ( ( F + E ) / 2 ) - ( ( D + C ) / 2 ) ) x. ( B - A ) )
- *46:45,25acb:eqtri |- S. RR ( vol ` ( S " { x } ) ) _d x = ( ( B - A ) x. ( D - C ) )
- *combine facts to get the result
- 997:76,210,237:3pm3.2i |- ( S C_ ( RR X. RR ) /\ A. x e. RR ( S " { x } ) e. ( `' vol " RR )
- /\ ( x e. RR |-> ( vol ` ( S " { x } ) ) ) e. L^1 )
- 998:997,dmarea:mpbir |- S e. dom area
- 999:998,areaval:ax-mp |- ( area ` S ) = S. RR ( vol ` ( S " { x } ) ) _d x
- qed:999,602:eqtri |- ( area ` S ) = ( ( ( ( F + E ) / 2 ) - ( ( D + C ) / 2 ) ) x. ( B - A ) )
- $)
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