Theorem mul4i 7161

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 7145	. 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 403	1 |- ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) )

Syntax hints:   = wceq 1243   e. wcel 1393  (class class class)co 5512   CCcc 6887   x. cmul 6894

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-mulcl 6982  ax-mulcom 6985  ax-mulass 6987

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by: (None)