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Theorem mulsub2 9902
Description: Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)
Assertion
Ref Expression
mulsub2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  -  B )  x.  ( C  -  D )
)  =  ( ( B  -  A )  x.  ( D  -  C ) ) )

Proof of Theorem mulsub2
StepHypRef Expression
1 subcl 9723 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
2 subcl 9723 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  -  D
)  e.  CC )
3 mul2neg 9898 . . 3  |-  ( ( ( A  -  B
)  e.  CC  /\  ( C  -  D
)  e.  CC )  ->  ( -u ( A  -  B )  x.  -u ( C  -  D ) )  =  ( ( A  -  B )  x.  ( C  -  D )
) )
41, 2, 3syl2an 477 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( -u ( A  -  B )  x.  -u ( C  -  D )
)  =  ( ( A  -  B )  x.  ( C  -  D ) ) )
5 negsubdi2 9782 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( A  -  B )  =  ( B  -  A ) )
6 negsubdi2 9782 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  -> 
-u ( C  -  D )  =  ( D  -  C ) )
75, 6oveqan12d 6222 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( -u ( A  -  B )  x.  -u ( C  -  D )
)  =  ( ( B  -  A )  x.  ( D  -  C ) ) )
84, 7eqtr3d 2497 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  -  B )  x.  ( C  -  D )
)  =  ( ( B  -  A )  x.  ( D  -  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758  (class class class)co 6203   CCcc 9394    x. cmul 9401    - cmin 9709   -ucneg 9710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-ltxr 9537  df-sub 9711  df-neg 9712
This theorem is referenced by:  brbtwn2  23323
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