Theorem mul12d 8468

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

Step	Hyp	Ref	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 8445	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) )
5	1, 2, 3, 4	syl3anc 1278	1 |- ( ph -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209  (class class class)co 6075   CCcc 8167   x. cmul 8174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-mulcom 8270  ax-mulass 8272

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380  df-ov 6078

This theorem is referenced by:  mulreim  8922  divrecap  9008  remullem  11614  sq01  11638  cvgratnnlemnexp  12269  cvgratnnlemmn  12270  tanval3ap  12459  sinadd  12481  dvdscmulr  12565  bezoutlemnewy  12751  dvdsmulgcd  12780  lcmgcdlem  12833  cncongr1  12859  prmdiv  12991  tangtx  15862  gausslemma2dlem6  16100  lgseisenlem2  16104  lgseisenlem4  16106  lgsquadlem1  16110  2sqlem4  16151