Theorem mul32d 8224

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 8201	. 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 1249	1 |- ( ph -> ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) )

Syntax hints:   -> wi 4   = wceq 1372   e. wcel 2175  (class class class)co 5943   CCcc 7922   x. cmul 7929

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186  ax-mulcom 8025  ax-mulass 8027

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-iota 5231  df-fv 5278  df-ov 5946

This theorem is referenced by:  conjmulap  8801  modqmul1  10520  binom3  10800  bernneq  10803  bcm1k  10903  bcp1n  10904  resqrexlemcalc1  11296  resqrexlemnm  11300  reccn2ap  11595  binomlem  11765  tanaddap  12021  eirraplem  12059  dvds2ln  12106  divgcdcoprm0  12394  modprm0  12548  binom4  15422  gausslemma2d  15517  lgsquadlem1  15525  2lgslem3b  15542  2lgslem3c  15543  2lgslem3d  15544