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Theorem mulsub 9958
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 9821 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
2 negsub 9821 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  -u D )  =  ( C  -  D ) )
31, 2oveqan12d 6251 . 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 9774 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
5 negcl 9774 . . . . 5  |-  ( D  e.  CC  ->  -u D  e.  CC )
6 muladd 9948 . . . . 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 651 . . . 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 649 . . 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 9955 . . . . . . 7  |-  ( ( D  e.  CC  /\  B  e.  CC )  ->  ( -u D  x.  -u B )  =  ( D  x.  B ) )
109ancoms 451 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC )  ->  ( -u D  x.  -u B )  =  ( D  x.  B ) )
1110oveq2d 6248 . . . . 5  |-  ( ( B  e.  CC  /\  D  e.  CC )  ->  ( ( A  x.  C )  +  (
-u D  x.  -u B
) )  =  ( ( A  x.  C
)  +  ( D  x.  B ) ) )
1211ad2ant2l 744 . . . 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 9953 . . . . . . . 8  |-  ( ( A  e.  CC  /\  D  e.  CC )  ->  ( A  x.  -u D
)  =  -u ( A  x.  D )
)
14 mulneg2 9953 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  x.  -u B
)  =  -u ( C  x.  B )
)
1513, 14oveqan12d 6251 . . . . . . 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 9524 . . . . . . . 8  |-  ( ( A  e.  CC  /\  D  e.  CC )  ->  ( A  x.  D
)  e.  CC )
17 mulcl 9524 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  x.  B
)  e.  CC )
18 negdi 9830 . . . . . . . 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 475 . . . . . . 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 2444 . . . . . 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 801 . . . . 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 826 . . . 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 6250 . . 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 9524 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
25 mulcl 9524 . . . . . . 7  |-  ( ( D  e.  CC  /\  B  e.  CC )  ->  ( D  x.  B
)  e.  CC )
2625ancoms 451 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC )  ->  ( D  x.  B
)  e.  CC )
27 addcl 9522 . . . . . 6  |-  ( ( ( A  x.  C
)  e.  CC  /\  ( D  x.  B
)  e.  CC )  ->  ( ( A  x.  C )  +  ( D  x.  B
) )  e.  CC )
2824, 26, 27syl2an 475 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  C )  +  ( D  x.  B ) )  e.  CC )
2928an4s 825 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  C )  +  ( D  x.  B ) )  e.  CC )
3017ancoms 451 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( C  x.  B
)  e.  CC )
31 addcl 9522 . . . . . 6  |-  ( ( ( A  x.  D
)  e.  CC  /\  ( C  x.  B
)  e.  CC )  ->  ( ( A  x.  D )  +  ( C  x.  B
) )  e.  CC )
3216, 30, 31syl2an 475 . . . . 5  |-  ( ( ( A  e.  CC  /\  D  e.  CC )  /\  ( B  e.  CC  /\  C  e.  CC ) )  -> 
( ( A  x.  D )  +  ( C  x.  B ) )  e.  CC )
3332an42s 826 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  D )  +  ( C  x.  B ) )  e.  CC )
3429, 33negsubd 9891 . . 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 2445 . 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 2443 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840  (class class class)co 6232   CCcc 9438    + caddc 9443    x. cmul 9445    - cmin 9759   -ucneg 9760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-pnf 9578  df-mnf 9579  df-ltxr 9581  df-sub 9761  df-neg 9762
This theorem is referenced by:  mulsubd  9974  muleqadd  10152  addltmul  10733  sqabssub  13170  mod2xnegi  14656  addltmulALT  27659
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