Theorem mul4i 7910

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 7894	. 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 423	1 |- ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) )

Syntax hints:   = wceq 1331   e. wcel 1480  (class class class)co 5774   CCcc 7618   x. cmul 7625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-mulcl 7718  ax-mulcom 7721  ax-mulass 7723

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777

This theorem is referenced by: (None)