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Theorem mul4i 9025
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 8997 . 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 1632    e. wcel 1696  (class class class)co 5874   CCcc 8751    x. cmul 8758
This theorem is referenced by:  faclbnd4lem1  11322  bposlem8  20546  normlem1  21705
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-mulcl 8815  ax-mulcom 8817  ax-mulass 8819
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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