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Theorem mulsub 9238
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 9111 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
2 negsub 9111 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  -u D )  =  ( C  -  D ) )
31, 2oveqan12d 5893 . 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 9068 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
5 negcl 9068 . . . . 5  |-  ( D  e.  CC  ->  -u D  e.  CC )
6 muladd 9228 . . . . 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 634 . . . 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 632 . . 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 9235 . . . . . . 7  |-  ( ( D  e.  CC  /\  B  e.  CC )  ->  ( -u D  x.  -u B )  =  ( D  x.  B ) )
109ancoms 439 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC )  ->  ( -u D  x.  -u B )  =  ( D  x.  B ) )
1110oveq2d 5890 . . . . 5  |-  ( ( B  e.  CC  /\  D  e.  CC )  ->  ( ( A  x.  C )  +  (
-u D  x.  -u B
) )  =  ( ( A  x.  C
)  +  ( D  x.  B ) ) )
1211ad2ant2l 726 . . . 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 9233 . . . . . . . 8  |-  ( ( A  e.  CC  /\  D  e.  CC )  ->  ( A  x.  -u D
)  =  -u ( A  x.  D )
)
14 mulneg2 9233 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  x.  -u B
)  =  -u ( C  x.  B )
)
1513, 14oveqan12d 5893 . . . . . . 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 8837 . . . . . . . 8  |-  ( ( A  e.  CC  /\  D  e.  CC )  ->  ( A  x.  D
)  e.  CC )
17 mulcl 8837 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  x.  B
)  e.  CC )
18 negdi 9120 . . . . . . . 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 463 . . . . . . 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 2331 . . . . . 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 777 . . . . 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 800 . . . 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 5892 . . 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 8837 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
25 mulcl 8837 . . . . . . 7  |-  ( ( D  e.  CC  /\  B  e.  CC )  ->  ( D  x.  B
)  e.  CC )
2625ancoms 439 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC )  ->  ( D  x.  B
)  e.  CC )
27 addcl 8835 . . . . . 6  |-  ( ( ( A  x.  C
)  e.  CC  /\  ( D  x.  B
)  e.  CC )  ->  ( ( A  x.  C )  +  ( D  x.  B
) )  e.  CC )
2824, 26, 27syl2an 463 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  C )  +  ( D  x.  B ) )  e.  CC )
2928an4s 799 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  C )  +  ( D  x.  B ) )  e.  CC )
3017ancoms 439 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( C  x.  B
)  e.  CC )
31 addcl 8835 . . . . . 6  |-  ( ( ( A  x.  D
)  e.  CC  /\  ( C  x.  B
)  e.  CC )  ->  ( ( A  x.  D )  +  ( C  x.  B
) )  e.  CC )
3216, 30, 31syl2an 463 . . . . 5  |-  ( ( ( A  e.  CC  /\  D  e.  CC )  /\  ( B  e.  CC  /\  C  e.  CC ) )  -> 
( ( A  x.  D )  +  ( C  x.  B ) )  e.  CC )
3332an42s 800 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  D )  +  ( C  x.  B ) )  e.  CC )
3429, 33negsubd 9179 . . 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 2332 . 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 2330 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 358    = wceq 1632    e. wcel 1696  (class class class)co 5874   CCcc 8751    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054
This theorem is referenced by:  mulsubd  9254  muleqadd  9428  addltmul  9963  sqabssub  11784  mod2xnegi  13102  addltmulALT  23042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056
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