Theorem mul4d 8297

Description: Rearrangement of 4 factors. (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 )
mul4d.4	|- ( ph -> D e. CC )

Assertion

Ref	Expression
mul4d	|- ( ph -> ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) ) )

Proof of Theorem mul4d

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		mul4d.4	. 2 |- ( ph -> D e. CC )
5		mul4 8274	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) ) )
6	1, 2, 3, 4, 5	syl22anc 1272	1 |- ( ph -> ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200  (class class class)co 6000   CCcc 7993   x. cmul 8000

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-mulcl 8093  ax-mulcom 8096  ax-mulass 8098

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 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325  df-ov 6003

This theorem is referenced by:  mulreim  8747  remullem  11377  absmul  11575  cosadd  12243  tanaddap  12245  eulerthlema  12747  mul4sqlem  12911  plymullem1  15416  lgsdir  15708  lgsdi  15710  lgsquad2lem1  15754