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Theorem mulsub 7572
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 7423 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
2 negsub 7423 . . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C + -u D ) = ( C - D ) )
31, 2oveqan12d 5562 . 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 7375 . . . 4 |- ( B e. CC -> -u B e. CC )
5 negcl 7375 . . . . 5 |- ( D e. CC -> -u D e. CC )
6 muladd 7555 . . . . 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 397 . . . 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 395 . . 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 7569 . . . . . . 7 |- ( ( D e. CC /\ B e. CC ) -> ( -u D x. -u B ) = ( D x. B ) )
109ancoms 264 . . . . . 6 |- ( ( B e. CC /\ D e. CC ) -> ( -u D x. -u B ) = ( D x. B ) )
1110oveq2d 5559 . . . . 5 |- ( ( B e. CC /\ D e. CC ) -> ( ( A x. C ) + ( -u D x. -u B ) ) = ( ( A x. C ) + ( D x. B ) ) )
1211ad2ant2l 492 . . . 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 7567 . . . . . . . 8 |- ( ( A e. CC /\ D e. CC ) -> ( A x. -u D ) = -u ( A x. D ) )
14 mulneg2 7567 . . . . . . . 8 |- ( ( C e. CC /\ B e. CC ) -> ( C x. -u B ) = -u ( C x. B ) )
1513, 14oveqan12d 5562 . . . . . . 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 7162 . . . . . . . 8 |- ( ( A e. CC /\ D e. CC ) -> ( A x. D ) e. CC )
17 mulcl 7162 . . . . . . . 8 |- ( ( C e. CC /\ B e. CC ) -> ( C x. B ) e. CC )
18 negdi 7432 . . . . . . . 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 283 . . . . . . 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 2117 . . . . . 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 531 . . . . 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 554 . . . 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 5561 . . 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 7162 . . . . . 6 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) e. CC )
25 mulcl 7162 . . . . . . 7 |- ( ( D e. CC /\ B e. CC ) -> ( D x. B ) e. CC )
2625ancoms 264 . . . . . 6 |- ( ( B e. CC /\ D e. CC ) -> ( D x. B ) e. CC )
27 addcl 7160 . . . . . 6 |- ( ( ( A x. C ) e. CC /\ ( D x. B ) e. CC ) -> ( ( A x. C ) + ( D x. B ) ) e. CC )
2824, 26, 27syl2an 283 . . . . 5 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B e. CC /\ D e. CC ) ) -> ( ( A x. C ) + ( D x. B ) ) e. CC )
2928an4s 553 . . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. C ) + ( D x. B ) ) e. CC )
3017ancoms 264 . . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( C x. B ) e. CC )
31 addcl 7160 . . . . . 6 |- ( ( ( A x. D ) e. CC /\ ( C x. B ) e. CC ) -> ( ( A x. D ) + ( C x. B ) ) e. CC )
3216, 30, 31syl2an 283 . . . . 5 |- ( ( ( A e. CC /\ D e. CC ) /\ ( B e. CC /\ C e. CC ) ) -> ( ( A x. D ) + ( C x. B ) ) e. CC )
3332an42s 554 . . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. D ) + ( C x. B ) ) e. CC )
3429, 33negsubd 7492 . . 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 2118 . 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 2116 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 102   = wceq 1285   e. wcel 1434  (class class class)co 5543   CCcc 7041   + caddc 7046   x. cmul 7048   - cmin 7346   -ucneg 7347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288  ax-resscn 7130  ax-1cn 7131  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-sub 7348  df-neg 7349
This theorem is referenced by:  mulsubd  7588  muleqadd  7825  addltmul  8334  sqabssub  10080
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