Theorem mul31d 8142

Description: Commutative/associative law. (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
mul31d	|- ( ph -> ( ( A x. B ) x. C ) = ( ( C x. B ) x. A ) )

Proof of Theorem mul31d

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		mul31 8119	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( ( C x. B ) x. A ) )
5	1, 2, 3, 4	syl3anc 1249	1 |- ( ph -> ( ( A x. B ) x. C ) = ( ( C x. B ) x. A ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2160  (class class class)co 5897   CCcc 7840   x. cmul 7847

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-mulcl 7940  ax-mulcom 7943  ax-mulass 7945

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5900

This theorem is referenced by: (None)