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Theorem mul12d 7878
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 7855 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C )
) )
51, 2, 3, 4syl3anc 1199 1  |-  ( ph  ->  ( A  x.  ( B  x.  C )
)  =  ( B  x.  ( A  x.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463  (class class class)co 5740   CCcc 7582    x. cmul 7589
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-mulcom 7685  ax-mulass 7687
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-iota 5056  df-fv 5099  df-ov 5743
This theorem is referenced by:  mulreim  8329  divrecap  8408  remullem  10583  cvgratnnlemnexp  11233  cvgratnnlemmn  11234  tanval3ap  11320  sinadd  11342  dvdscmulr  11418  bezoutlemnewy  11580  dvdsmulgcd  11609  lcmgcdlem  11654  cncongr1  11680
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