Theorem mul31d 7634

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 7611	. 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 1174	1 |- ( ph -> ( ( A x. B ) x. C ) = ( ( C x. B ) x. A ) )

Syntax hints:   -> wi 4   = wceq 1289   e. wcel 1438  (class class class)co 5652   CCcc 7346   x. cmul 7353

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-mulcl 7441  ax-mulcom 7444  ax-mulass 7446

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655

This theorem is referenced by: (None)