Theorem mul12d 8485

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 8462	. 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 1278	1 |- ( ph -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209  (class class class)co 6088   CCcc 8183   x. cmul 8190

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-mulcom 8286  ax-mulass 8288

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-iota 5340  df-fv 5388  df-ov 6091

This theorem is used by:  mulreim  8940  divrecap  9026  remullem  11657  sq01  11681  cvgratnnlemnexp  12315  cvgratnnlemmn  12316  tanval3ap  12505  sinadd  12527  dvdscmulr  12611  bezoutlemnewy  12797  dvdsmulgcd  12826  lcmgcdlem  12879  cncongr1  12905  prmdiv  13041  tangtx  16082  bcp1ctr  16323  gausslemma2dlem6  16408  lgseisenlem2  16412  lgseisenlem4  16414  lgsquadlem1  16418  2sqlem4  16459