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Theorem mulsub 8691
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 8537 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
2 negsub 8537 . . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C + -u D ) = ( C - D ) )
31, 2oveqan12d 6077 . 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 8489 . . . 4 |- ( B e. CC -> -u B e. CC )
5 negcl 8489 . . . . 5 |- ( D e. CC -> -u D e. CC )
6 muladd 8674 . . . . 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 405 . . . 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 403 . . 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 8688 . . . . . . 7 |- ( ( D e. CC /\ B e. CC ) -> ( -u D x. -u B ) = ( D x. B ) )
109ancoms 268 . . . . . 6 |- ( ( B e. CC /\ D e. CC ) -> ( -u D x. -u B ) = ( D x. B ) )
1110oveq2d 6074 . . . . 5 |- ( ( B e. CC /\ D e. CC ) -> ( ( A x. C ) + ( -u D x. -u B ) ) = ( ( A x. C ) + ( D x. B ) ) )
1211ad2ant2l 508 . . . 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 8686 . . . . . . . 8 |- ( ( A e. CC /\ D e. CC ) -> ( A x. -u D ) = -u ( A x. D ) )
14 mulneg2 8686 . . . . . . . 8 |- ( ( C e. CC /\ B e. CC ) -> ( C x. -u B ) = -u ( C x. B ) )
1513, 14oveqan12d 6077 . . . . . . 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 8270 . . . . . . . 8 |- ( ( A e. CC /\ D e. CC ) -> ( A x. D ) e. CC )
17 mulcl 8270 . . . . . . . 8 |- ( ( C e. CC /\ B e. CC ) -> ( C x. B ) e. CC )
18 negdi 8546 . . . . . . . 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 289 . . . . . . 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 2270 . . . . . 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 568 . . . . 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 593 . . . 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 6076 . . 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 8270 . . . . . 6 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) e. CC )
25 mulcl 8270 . . . . . . 7 |- ( ( D e. CC /\ B e. CC ) -> ( D x. B ) e. CC )
2625ancoms 268 . . . . . 6 |- ( ( B e. CC /\ D e. CC ) -> ( D x. B ) e. CC )
27 addcl 8268 . . . . . 6 |- ( ( ( A x. C ) e. CC /\ ( D x. B ) e. CC ) -> ( ( A x. C ) + ( D x. B ) ) e. CC )
2824, 26, 27syl2an 289 . . . . 5 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B e. CC /\ D e. CC ) ) -> ( ( A x. C ) + ( D x. B ) ) e. CC )
2928an4s 592 . . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. C ) + ( D x. B ) ) e. CC )
3017ancoms 268 . . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( C x. B ) e. CC )
31 addcl 8268 . . . . . 6 |- ( ( ( A x. D ) e. CC /\ ( C x. B ) e. CC ) -> ( ( A x. D ) + ( C x. B ) ) e. CC )
3216, 30, 31syl2an 289 . . . . 5 |- ( ( ( A e. CC /\ D e. CC ) /\ ( B e. CC /\ C e. CC ) ) -> ( ( A x. D ) + ( C x. B ) ) e. CC )
3332an42s 593 . . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. D ) + ( C x. B ) ) e. CC )
3429, 33negsubd 8606 . . 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 2271 . 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 2269 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 104   = wceq 1398   e. wcel 2205  (class class class)co 6058   CCcc 8141   + caddc 8146   x. cmul 8148   - cmin 8460   -ucneg 8461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-setind 4664  ax-resscn 8235  ax-1cn 8236  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-sub 8462  df-neg 8463
This theorem is referenced by:  mulsubd  8707  muleqadd  8959  addltmul  9492  sqabssub  11766
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