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Theorem mulsub 8475
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 8322 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
2 negsub 8322 . . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C + -u D ) = ( C - D ) )
31, 2oveqan12d 5965 . 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 8274 . . . 4 |- ( B e. CC -> -u B e. CC )
5 negcl 8274 . . . . 5 |- ( D e. CC -> -u D e. CC )
6 muladd 8458 . . . . 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 8472 . . . . . . 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 5962 . . . . 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 8470 . . . . . . . 8 |- ( ( A e. CC /\ D e. CC ) -> ( A x. -u D ) = -u ( A x. D ) )
14 mulneg2 8470 . . . . . . . 8 |- ( ( C e. CC /\ B e. CC ) -> ( C x. -u B ) = -u ( C x. B ) )
1513, 14oveqan12d 5965 . . . . . . 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 8054 . . . . . . . 8 |- ( ( A e. CC /\ D e. CC ) -> ( A x. D ) e. CC )
17 mulcl 8054 . . . . . . . 8 |- ( ( C e. CC /\ B e. CC ) -> ( C x. B ) e. CC )
18 negdi 8331 . . . . . . . 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 2241 . . . . . 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 566 . . . . 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 589 . . . 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 5964 . . 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 8054 . . . . . 6 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) e. CC )
25 mulcl 8054 . . . . . . 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 8052 . . . . . 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 588 . . . 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 8052 . . . . . 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 589 . . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. D ) + ( C x. B ) ) e. CC )
3429, 33negsubd 8391 . . 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 2242 . 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 2240 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 1373   e. wcel 2176  (class class class)co 5946   CCcc 7925   + caddc 7930   x. cmul 7932   - cmin 8245   -ucneg 8246
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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-setind 4586  ax-resscn 8019  ax-1cn 8020  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-distr 8031  ax-i2m1 8032  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-sub 8247  df-neg 8248
This theorem is referenced by:  mulsubd  8491  muleqadd  8743  addltmul  9276  sqabssub  11400
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