MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mul4d Structured version   Unicode version

Theorem mul4d 9280
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 9237 . 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 1186 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 1653    e. wcel 1726  (class class class)co 6083   CCcc 8990    x. cmul 8997
This theorem is referenced by:  remullem  11935  absmul  12101  cosadd  12768  tanadd  12770  eulerthlem2  13173  mul4sqlem  13323  odadd2  15466  itgmulc2  19727  plymullem1  20135  chordthmlem4  20678  quartlem1  20699  dchrmulcl  21035  bposlem9  21078  lgsdir  21116  lgsdi  21118  lgsquad2lem1  21144  chtppilimlem1  21169  rplogsumlem1  21180  dchrvmasumlem1  21191  dchrvmasum2lem  21192  chpdifbndlem1  21249  pntlemf  21301  circum  25113  binomrisefac  25360  brbtwn2  25846  colinearalglem4  25850  itgmulc2nc  26275  pellexlem6  26899  pell1234qrmulcl  26920  rmxyadd  26986  wallispi2lem2  27799  cevathlem1  27835
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-mulcl 9054  ax-mulcom 9056  ax-mulass 9058
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086
  Copyright terms: Public domain W3C validator