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Theorem mul4i 9813
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 9785 . 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 673 1  |-  ( ( A  x.  B )  x.  ( C  x.  D ) )  =  ( ( A  x.  C )  x.  ( B  x.  D )
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1407    e. wcel 1844  (class class class)co 6280   CCcc 9522    x. cmul 9529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-mulcl 9586  ax-mulcom 9588  ax-mulass 9590
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-iota 5535  df-fv 5579  df-ov 6283
This theorem is referenced by:  faclbnd4lem1  12417  bposlem8  23949  normlem1  26454
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