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Theorem submul2 8279
Description: Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.)
Assertion
Ref Expression
submul2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  x.  C ) )  =  ( A  +  ( B  x.  -u C
) ) )

Proof of Theorem submul2
StepHypRef Expression
1 mulneg2 8276 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  -u C
)  =  -u ( B  x.  C )
)
21adantl 275 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( B  x.  -u C )  = 
-u ( B  x.  C ) )
32oveq2d 5843 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( A  +  ( B  x.  -u C ) )  =  ( A  +  -u ( B  x.  C
) ) )
4 mulcl 7862 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  e.  CC )
5 negsub 8128 . . . 4  |-  ( ( A  e.  CC  /\  ( B  x.  C
)  e.  CC )  ->  ( A  +  -u ( B  x.  C
) )  =  ( A  -  ( B  x.  C ) ) )
64, 5sylan2 284 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( A  +  -u ( B  x.  C ) )  =  ( A  -  ( B  x.  C )
) )
73, 6eqtr2d 2191 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( A  -  ( B  x.  C ) )  =  ( A  +  ( B  x.  -u C
) ) )
873impb 1181 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  x.  C ) )  =  ( A  +  ( B  x.  -u C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    = wceq 1335    e. wcel 2128  (class class class)co 5827   CCcc 7733    + caddc 7738    x. cmul 7740    - cmin 8051   -ucneg 8052
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172  ax-setind 4499  ax-resscn 7827  ax-1cn 7828  ax-icn 7830  ax-addcl 7831  ax-addrcl 7832  ax-mulcl 7833  ax-addcom 7835  ax-mulcom 7836  ax-addass 7837  ax-distr 7839  ax-i2m1 7840  ax-0id 7843  ax-rnegex 7844  ax-cnre 7846
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4029  df-id 4256  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-iota 5138  df-fun 5175  df-fv 5181  df-riota 5783  df-ov 5830  df-oprab 5831  df-mpo 5832  df-sub 8053  df-neg 8054
This theorem is referenced by:  cjap  10818
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