Theorem mul4d 8257

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 8234	. 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 1251	1 |- ( ph -> ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177  (class class class)co 5962   CCcc 7953   x. cmul 7960

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-mulcl 8053  ax-mulcom 8056  ax-mulass 8058

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-iota 5246  df-fv 5293  df-ov 5965

This theorem is referenced by:  mulreim  8707  remullem  11267  absmul  11465  cosadd  12133  tanaddap  12135  eulerthlema  12637  mul4sqlem  12801  plymullem1  15305  lgsdir  15597  lgsdi  15599  lgsquad2lem1  15643