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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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