Theorem mul4i 8174

Description: Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)

Hypotheses

Ref	Expression
mul.1	|- A e. CC
mul.2	|- B e. CC
mul.3	|- C e. CC
mul4.4	|- D e. CC

Assertion

Ref	Expression
mul4i	|- ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) )

Proof of Theorem mul4i

Step	Hyp	Ref	Expression
1		mul.1	. 2 |- A e. CC
2		mul.2	. 2 |- B e. CC
3		mul.3	. 2 |- C e. CC
4		mul4.4	. 2 |- D e. CC
5		mul4 8158	. 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	mp4an 427	1 |- ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) )

Syntax hints:   = wceq 1364   e. wcel 2167  (class class class)co 5922   CCcc 7877   x. cmul 7884

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-mulcl 7977  ax-mulcom 7980  ax-mulass 7982

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925

This theorem is referenced by: (None)