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Theorem mulsub 7858
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 7709 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
2 negsub 7709 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  -u D )  =  ( C  -  D ) )
31, 2oveqan12d 5653 . 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 7661 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
5 negcl 7661 . . . . 5  |-  ( D  e.  CC  ->  -u D  e.  CC )
6 muladd 7841 . . . . 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 7855 . . . . . . 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 5650 . . . . 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 7853 . . . . . . . 8  |-  ( ( A  e.  CC  /\  D  e.  CC )  ->  ( A  x.  -u D
)  =  -u ( A  x.  D )
)
14 mulneg2 7853 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  x.  -u B
)  =  -u ( C  x.  B )
)
1513, 14oveqan12d 5653 . . . . . . 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 7448 . . . . . . . 8  |-  ( ( A  e.  CC  /\  D  e.  CC )  ->  ( A  x.  D
)  e.  CC )
17 mulcl 7448 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  x.  B
)  e.  CC )
18 negdi 7718 . . . . . . . 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 2123 . . . . . 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 533 . . . . 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 556 . . . 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 5652 . . 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 7448 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
25 mulcl 7448 . . . . . . 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 7446 . . . . . 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 555 . . . 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 7446 . . . . . 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 556 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  D )  +  ( C  x.  B ) )  e.  CC )
3429, 33negsubd 7778 . . 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 2124 . 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 2122 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 1289    e. wcel 1438  (class class class)co 5634   CCcc 7327    + caddc 7332    x. cmul 7334    - cmin 7632   -ucneg 7633
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343  ax-resscn 7416  ax-1cn 7417  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-sub 7634  df-neg 7635
This theorem is referenced by:  mulsubd  7874  muleqadd  8111  addltmul  8622  sqabssub  10454
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