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Theorem mulsub 9468
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 9341 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
2 negsub 9341 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  -u D )  =  ( C  -  D ) )
31, 2oveqan12d 6092 . 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 9298 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
5 negcl 9298 . . . . 5  |-  ( D  e.  CC  ->  -u D  e.  CC )
6 muladd 9458 . . . . 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 635 . . . 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 633 . . 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 9465 . . . . . . 7  |-  ( ( D  e.  CC  /\  B  e.  CC )  ->  ( -u D  x.  -u B )  =  ( D  x.  B ) )
109ancoms 440 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC )  ->  ( -u D  x.  -u B )  =  ( D  x.  B ) )
1110oveq2d 6089 . . . . 5  |-  ( ( B  e.  CC  /\  D  e.  CC )  ->  ( ( A  x.  C )  +  (
-u D  x.  -u B
) )  =  ( ( A  x.  C
)  +  ( D  x.  B ) ) )
1211ad2ant2l 727 . . . 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 9463 . . . . . . . 8  |-  ( ( A  e.  CC  /\  D  e.  CC )  ->  ( A  x.  -u D
)  =  -u ( A  x.  D )
)
14 mulneg2 9463 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  x.  -u B
)  =  -u ( C  x.  B )
)
1513, 14oveqan12d 6092 . . . . . . 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 9066 . . . . . . . 8  |-  ( ( A  e.  CC  /\  D  e.  CC )  ->  ( A  x.  D
)  e.  CC )
17 mulcl 9066 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  x.  B
)  e.  CC )
18 negdi 9350 . . . . . . . 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 464 . . . . . . 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 2470 . . . . . 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 778 . . . . 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 801 . . . 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 6091 . . 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 9066 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
25 mulcl 9066 . . . . . . 7  |-  ( ( D  e.  CC  /\  B  e.  CC )  ->  ( D  x.  B
)  e.  CC )
2625ancoms 440 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC )  ->  ( D  x.  B
)  e.  CC )
27 addcl 9064 . . . . . 6  |-  ( ( ( A  x.  C
)  e.  CC  /\  ( D  x.  B
)  e.  CC )  ->  ( ( A  x.  C )  +  ( D  x.  B
) )  e.  CC )
2824, 26, 27syl2an 464 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  C )  +  ( D  x.  B ) )  e.  CC )
2928an4s 800 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  C )  +  ( D  x.  B ) )  e.  CC )
3017ancoms 440 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( C  x.  B
)  e.  CC )
31 addcl 9064 . . . . . 6  |-  ( ( ( A  x.  D
)  e.  CC  /\  ( C  x.  B
)  e.  CC )  ->  ( ( A  x.  D )  +  ( C  x.  B
) )  e.  CC )
3216, 30, 31syl2an 464 . . . . 5  |-  ( ( ( A  e.  CC  /\  D  e.  CC )  /\  ( B  e.  CC  /\  C  e.  CC ) )  -> 
( ( A  x.  D )  +  ( C  x.  B ) )  e.  CC )
3332an42s 801 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  D )  +  ( C  x.  B ) )  e.  CC )
3429, 33negsubd 9409 . . 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 2471 . 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 2469 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 359    = wceq 1652    e. wcel 1725  (class class class)co 6073   CCcc 8980    + caddc 8985    x. cmul 8987    - cmin 9283   -ucneg 9284
This theorem is referenced by:  mulsubd  9484  muleqadd  9658  addltmul  10195  sqabssub  12080  mod2xnegi  13399  addltmulALT  23941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-sub 9285  df-neg 9286
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