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Theorem mulsub 8299
Description: Product of two differences. (Contributed by NM, 14-Jan-2006.)
Assertion
Ref Expression
mulsub |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A - B ) x. ( C - D ) ) = ( ( ( A x. C ) + ( D x. B ) ) - ( ( A x. D ) + ( C x. B ) ) ) )
Proof of Theorem mulsub
StepHypRef Expression
1 negsub 8146 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
2 negsub 8146 . . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C + -u D ) = ( C - D ) )
31, 2oveqan12d 5861 . 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + -u B ) x. ( C + -u D ) ) = ( ( A - B ) x. ( C - D ) ) )
4 negcl 8098 . . . 4 |- ( B e. CC -> -u B e. CC )
5 negcl 8098 . . . . 5 |- ( D e. CC -> -u D e. CC )
6 muladd 8282 . . . . 5 |- ( ( ( A e. CC /\ -u B e. CC ) /\ ( C e. CC /\ -u D e. CC ) ) -> ( ( A + -u B ) x. ( C + -u D ) ) = ( ( ( A x. C ) + ( -u D x. -u B ) ) + ( ( A x. -u D ) + ( C x. -u B ) ) ) )
75, 6sylanr2 403 . . . 4 |- ( ( ( A e. CC /\ -u B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + -u B ) x. ( C + -u D ) ) = ( ( ( A x. C ) + ( -u D x. -u B ) ) + ( ( A x. -u D ) + ( C x. -u B ) ) ) )
84, 7sylanl2 401 . . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + -u B ) x. ( C + -u D ) ) = ( ( ( A x. C ) + ( -u D x. -u B ) ) + ( ( A x. -u D ) + ( C x. -u B ) ) ) )
9 mul2neg 8296 . . . . . . 7 |- ( ( D e. CC /\ B e. CC ) -> ( -u D x. -u B ) = ( D x. B ) )
109ancoms 266 . . . . . 6 |- ( ( B e. CC /\ D e. CC ) -> ( -u D x. -u B ) = ( D x. B ) )
1110oveq2d 5858 . . . . 5 |- ( ( B e. CC /\ D e. CC ) -> ( ( A x. C ) + ( -u D x. -u B ) ) = ( ( A x. C ) + ( D x. B ) ) )
1211ad2ant2l 500 . . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. C ) + ( -u D x. -u B ) ) = ( ( A x. C ) + ( D x. B ) ) )
13 mulneg2 8294 . . . . . . . 8 |- ( ( A e. CC /\ D e. CC ) -> ( A x. -u D ) = -u ( A x. D ) )
14 mulneg2 8294 . . . . . . . 8 |- ( ( C e. CC /\ B e. CC ) -> ( C x. -u B ) = -u ( C x. B ) )
1513, 14oveqan12d 5861 . . . . . . 7 |- ( ( ( A e. CC /\ D e. CC ) /\ ( C e. CC /\ B e. CC ) ) -> ( ( A x. -u D ) + ( C x. -u B ) ) = ( -u ( A x. D ) + -u ( C x. B ) ) )
16 mulcl 7880 . . . . . . . 8 |- ( ( A e. CC /\ D e. CC ) -> ( A x. D ) e. CC )
17 mulcl 7880 . . . . . . . 8 |- ( ( C e. CC /\ B e. CC ) -> ( C x. B ) e. CC )
18 negdi 8155 . . . . . . . 8 |- ( ( ( A x. D ) e. CC /\ ( C x. B ) e. CC ) -> -u ( ( A x. D ) + ( C x. B ) ) = ( -u ( A x. D ) + -u ( C x. B ) ) )
1916, 17, 18syl2an 287 . . . . . . 7 |- ( ( ( A e. CC /\ D e. CC ) /\ ( C e. CC /\ B e. CC ) ) -> -u ( ( A x. D ) + ( C x. B ) ) = ( -u ( A x. D ) + -u ( C x. B ) ) )
2015, 19eqtr4d 2201 . . . . . 6 |- ( ( ( A e. CC /\ D e. CC ) /\ ( C e. CC /\ B e. CC ) ) -> ( ( A x. -u D ) + ( C x. -u B ) ) = -u ( ( A x. D ) + ( C x. B ) ) )
2120ancom2s 556 . . . . 5 |- ( ( ( A e. CC /\ D e. CC ) /\ ( B e. CC /\ C e. CC ) ) -> ( ( A x. -u D ) + ( C x. -u B ) ) = -u ( ( A x. D ) + ( C x. B ) ) )
2221an42s 579 . . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. -u D ) + ( C x. -u B ) ) = -u ( ( A x. D ) + ( C x. B ) ) )
2312, 22oveq12d 5860 . . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( A x. C ) + ( -u D x. -u B ) ) + ( ( A x. -u D ) + ( C x. -u B ) ) ) = ( ( ( A x. C ) + ( D x. B ) ) + -u ( ( A x. D ) + ( C x. B ) ) ) )
24 mulcl 7880 . . . . . 6 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) e. CC )
25 mulcl 7880 . . . . . . 7 |- ( ( D e. CC /\ B e. CC ) -> ( D x. B ) e. CC )
2625ancoms 266 . . . . . 6 |- ( ( B e. CC /\ D e. CC ) -> ( D x. B ) e. CC )
27 addcl 7878 . . . . . 6 |- ( ( ( A x. C ) e. CC /\ ( D x. B ) e. CC ) -> ( ( A x. C ) + ( D x. B ) ) e. CC )
2824, 26, 27syl2an 287 . . . . 5 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B e. CC /\ D e. CC ) ) -> ( ( A x. C ) + ( D x. B ) ) e. CC )
2928an4s 578 . . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. C ) + ( D x. B ) ) e. CC )
3017ancoms 266 . . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( C x. B ) e. CC )
31 addcl 7878 . . . . . 6 |- ( ( ( A x. D ) e. CC /\ ( C x. B ) e. CC ) -> ( ( A x. D ) + ( C x. B ) ) e. CC )
3216, 30, 31syl2an 287 . . . . 5 |- ( ( ( A e. CC /\ D e. CC ) /\ ( B e. CC /\ C e. CC ) ) -> ( ( A x. D ) + ( C x. B ) ) e. CC )
3332an42s 579 . . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. D ) + ( C x. B ) ) e. CC )
3429, 33negsubd 8215 . . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( A x. C ) + ( D x. B ) ) + -u ( ( A x. D ) + ( C x. B ) ) ) = ( ( ( A x. C ) + ( D x. B ) ) - ( ( A x. D ) + ( C x. B ) ) ) )
358, 23, 343eqtrd 2202 . 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + -u B ) x. ( C + -u D ) ) = ( ( ( A x. C ) + ( D x. B ) ) - ( ( A x. D ) + ( C x. B ) ) ) )
363, 35eqtr3d 2200 1 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A - B ) x. ( C - D ) ) = ( ( ( A x. C ) + ( D x. B ) ) - ( ( A x. D ) + ( C x. B ) ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136  (class class class)co 5842   CCcc 7751   + caddc 7756   x. cmul 7758   - cmin 8069   -ucneg 8070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-sub 8071  df-neg 8072
This theorem is referenced by:  mulsubd  8315  muleqadd  8565  addltmul  9093  sqabssub  10998
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