Theorem mul32d 8225

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 8202	. 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 1250	1 |- ( ph -> ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176  (class class class)co 5944   CCcc 7923   x. cmul 7930

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-mulcom 8026  ax-mulass 8028

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947

This theorem is referenced by:  conjmulap  8802  modqmul1  10522  binom3  10802  bernneq  10805  bcm1k  10905  bcp1n  10906  resqrexlemcalc1  11325  resqrexlemnm  11329  reccn2ap  11624  binomlem  11794  tanaddap  12050  eirraplem  12088  dvds2ln  12135  divgcdcoprm0  12423  modprm0  12577  binom4  15451  gausslemma2d  15546  lgsquadlem1  15554  2lgslem3b  15571  2lgslem3c  15572  2lgslem3d  15573