Theorem mul32d 8480

Description: Commutative/associative law that swaps the last 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
mul32d	|- ( ph -> ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) )

Proof of Theorem mul32d

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		mul32 8457	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) )
5	1, 2, 3, 4	syl3anc 1278	1 |- ( ph -> ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209  (class class class)co 6085   CCcc 8177   x. cmul 8184

This proof depends on 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 8280  ax-mulass 8282

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088

This theorem is used by:  conjmulap  9061  modqmul1  10827  binom3  11107  bernneq  11111  bcm1k  11212  bcp1n  11213  resqrexlemcalc1  11794  resqrexlemnm  11798  reccn2ap  12095  binomlem  12266  tanaddap  12522  eirraplem  12560  dvds2ln  12607  divgcdcoprm0  12895  modprm0  13053  binom4  16138  bcmono  16202  gausslemma2d  16286  lgsquadlem1  16294  2lgslem3b  16311  2lgslem3c  16312  2lgslem3d  16313