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Theorem mul4d 9024
Description: Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
muld.1  |-  ( ph  ->  A  e.  CC )
addcomd.2  |-  ( ph  ->  B  e.  CC )
addcand.3  |-  ( ph  ->  C  e.  CC )
mul4d.4  |-  ( ph  ->  D  e.  CC )
Assertion
Ref Expression
mul4d  |-  ( ph  ->  ( ( A  x.  B )  x.  ( C  x.  D )
)  =  ( ( A  x.  C )  x.  ( B  x.  D ) ) )

Proof of Theorem mul4d
StepHypRef Expression
1 muld.1 . 2  |-  ( ph  ->  A  e.  CC )
2 addcomd.2 . 2  |-  ( ph  ->  B  e.  CC )
3 addcand.3 . 2  |-  ( ph  ->  C  e.  CC )
4 mul4d.4 . 2  |-  ( ph  ->  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, 5syl22anc 1183 1  |-  ( ph  ->  ( ( A  x.  B )  x.  ( C  x.  D )
)  =  ( ( A  x.  C )  x.  ( B  x.  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735    x. cmul 8742
This theorem is referenced by:  remullem  11613  absmul  11779  cosadd  12445  tanadd  12447  eulerthlem2  12850  mul4sqlem  13000  odadd2  15141  itgmulc2  19188  plymullem1  19596  chordthmlem4  20132  quartlem1  20153  dchrmulcl  20488  bposlem9  20531  lgsdir  20569  lgsdi  20571  lgsquad2lem1  20597  chtppilimlem1  20622  rplogsumlem1  20633  dchrvmasumlem1  20644  dchrvmasum2lem  20645  chpdifbndlem1  20702  pntlemf  20754  circum  23418  brbtwn2  23944  colinearalglem4  23948  pellexlem6  26331  pell1234qrmulcl  26352  rmxyadd  26418  wallispi2lem2  27233  cevathlem1  27269
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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