Theorem mul32d 8255

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 8232	. 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 2177  (class class class)co 5962   CCcc 7953   x. cmul 7960

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-mulcom 8056  ax-mulass 8058

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-iota 5246  df-fv 5293  df-ov 5965

This theorem is referenced by:  conjmulap  8832  modqmul1  10554  binom3  10834  bernneq  10837  bcm1k  10937  bcp1n  10938  resqrexlemcalc1  11410  resqrexlemnm  11414  reccn2ap  11709  binomlem  11879  tanaddap  12135  eirraplem  12173  dvds2ln  12220  divgcdcoprm0  12508  modprm0  12662  binom4  15536  gausslemma2d  15631  lgsquadlem1  15639  2lgslem3b  15656  2lgslem3c  15657  2lgslem3d  15658