Theorem mul4d 8227

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 8204	. 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 2176  (class class class)co 5944   CCcc 7923   x. cmul 7930

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-mulcl 8023  ax-mulcom 8026  ax-mulass 8028

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947

This theorem is referenced by:  mulreim  8677  remullem  11182  absmul  11380  cosadd  12048  tanaddap  12050  eulerthlema  12552  mul4sqlem  12716  plymullem1  15220  lgsdir  15512  lgsdi  15514  lgsquad2lem1  15558