Theorem mul12d 8321

Description: Commutative/associative law that swaps the first 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
mul12d	|- ( ph -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) )

Proof of Theorem mul12d

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		mul12 8298	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) )
5	1, 2, 3, 4	syl3anc 1271	1 |- ( ph -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200  (class class class)co 6013   CCcc 8020   x. cmul 8027

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-mulcom 8123  ax-mulass 8125

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-iota 5284  df-fv 5332  df-ov 6016

This theorem is referenced by:  mulreim  8774  divrecap  8858  remullem  11422  cvgratnnlemnexp  12075  cvgratnnlemmn  12076  tanval3ap  12265  sinadd  12287  dvdscmulr  12371  bezoutlemnewy  12557  dvdsmulgcd  12586  lcmgcdlem  12639  cncongr1  12665  prmdiv  12797  tangtx  15552  gausslemma2dlem6  15786  lgseisenlem2  15790  lgseisenlem4  15792  lgsquadlem1  15796  2sqlem4  15837