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Theorem mul12d 7536
Description: Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
muld.1  |-  ( ph  ->  A  e.  CC )
addcomd.2  |-  ( ph  ->  B  e.  CC )
mul12d.3  |-  ( ph  ->  C  e.  CC )
Assertion
Ref Expression
mul12d  |-  ( ph  ->  ( A  x.  ( B  x.  C )
)  =  ( B  x.  ( A  x.  C ) ) )

Proof of Theorem mul12d
StepHypRef Expression
1 muld.1 . 2  |-  ( ph  ->  A  e.  CC )
2 addcomd.2 . 2  |-  ( ph  ->  B  e.  CC )
3 mul12d.3 . 2  |-  ( ph  ->  C  e.  CC )
4 mul12 7513 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C )
) )
51, 2, 3, 4syl3anc 1170 1  |-  ( ph  ->  ( A  x.  ( B  x.  C )
)  =  ( B  x.  ( A  x.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434  (class class class)co 5590   CCcc 7250    x. cmul 7257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-mulcom 7348  ax-mulass 7350
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-iota 4933  df-fv 4976  df-ov 5593
This theorem is referenced by:  mulreim  7980  divrecap  8052  remullem  10131  dvdscmulr  10604  bezoutlemnewy  10764  dvdsmulgcd  10793  lcmgcdlem  10838  cncongr1  10864
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