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Theorem mul4i 9554
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 9526 . 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 666 1  |-  ( ( A  x.  B )  x.  ( C  x.  D ) )  =  ( ( A  x.  C )  x.  ( B  x.  D )
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1362    e. wcel 1755  (class class class)co 6080   CCcc 9268    x. cmul 9275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-mulcl 9332  ax-mulcom 9334  ax-mulass 9336
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-iota 5369  df-fv 5414  df-ov 6083
This theorem is referenced by:  faclbnd4lem1  12053  bposlem8  22515  normlem1  24335
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