Theorem mul32d 8331

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 8308	. 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 1273	1 |- ( ph -> ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202  (class class class)co 6017   CCcc 8029   x. cmul 8036

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-mulcom 8132  ax-mulass 8134

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6020

This theorem is referenced by:  conjmulap  8908  modqmul1  10638  binom3  10918  bernneq  10921  bcm1k  11021  bcp1n  11022  resqrexlemcalc1  11574  resqrexlemnm  11578  reccn2ap  11873  binomlem  12043  tanaddap  12299  eirraplem  12337  dvds2ln  12384  divgcdcoprm0  12672  modprm0  12826  binom4  15702  gausslemma2d  15797  lgsquadlem1  15805  2lgslem3b  15822  2lgslem3c  15823  2lgslem3d  15824