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Theorem mul4i 9009
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
StepHypRef 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 8981 . 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 ) ) )
61, 2, 3, 4, 5mp4an 654 1  |-  ( ( A  x.  B )  x.  ( C  x.  D ) )  =  ( ( A  x.  C )  x.  ( B  x.  D )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735    x. cmul 8742
This theorem is referenced by:  faclbnd4lem1  11306  bposlem8  20530  normlem1  21689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-mulcl 8799  ax-mulcom 8801  ax-mulass 8803
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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