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Theorem mul4d 9040
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 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, 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 1632    e. wcel 1696  (class class class)co 5874   CCcc 8751    x. cmul 8758
This theorem is referenced by:  remullem  11629  absmul  11795  cosadd  12461  tanadd  12463  eulerthlem2  12866  mul4sqlem  13016  odadd2  15157  itgmulc2  19204  plymullem1  19612  chordthmlem4  20148  quartlem1  20169  dchrmulcl  20504  bposlem9  20547  lgsdir  20585  lgsdi  20587  lgsquad2lem1  20613  chtppilimlem1  20638  rplogsumlem1  20649  dchrvmasumlem1  20660  dchrvmasum2lem  20661  chpdifbndlem1  20718  pntlemf  20770  circum  24022  brbtwn2  24605  colinearalglem4  24609  itgmulc2nc  25019  pellexlem6  27022  pell1234qrmulcl  27043  rmxyadd  27109  wallispi2lem2  27924  cevathlem1  27960
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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