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Theorem mulsub2 8184
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 7981 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
2 subcl 7981 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  -  D
)  e.  CC )
3 mul2neg 8180 . . 3  |-  ( ( ( A  -  B
)  e.  CC  /\  ( C  -  D
)  e.  CC )  ->  ( -u ( A  -  B )  x.  -u ( C  -  D ) )  =  ( ( A  -  B )  x.  ( C  -  D )
) )
41, 2, 3syl2an 287 . 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 8041 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( A  -  B )  =  ( B  -  A ) )
6 negsubdi2 8041 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC )  -> 
-u ( C  -  D )  =  ( D  -  C ) )
75, 6oveqan12d 5797 . 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 2175 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 set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481  (class class class)co 5778   CCcc 7638    x. cmul 7645    - cmin 7953   -ucneg 7954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-setind 4456  ax-resscn 7732  ax-1cn 7733  ax-icn 7735  ax-addcl 7736  ax-addrcl 7737  ax-mulcl 7738  ax-addcom 7740  ax-mulcom 7741  ax-addass 7742  ax-distr 7744  ax-i2m1 7745  ax-0id 7748  ax-rnegex 7749  ax-cnre 7751
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-br 3934  df-opab 3994  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-iota 5092  df-fun 5129  df-fv 5135  df-riota 5734  df-ov 5781  df-oprab 5782  df-mpo 5783  df-sub 7955  df-neg 7956
This theorem is referenced by: (None)
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