Theorem mul4d 10248

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

Step	Hyp	Ref	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 10205	. 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 ) ) )
6	1, 2, 3, 4, 5	syl22anc 1327	1 |- ( ph -> ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   x. cmul 9941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-mulcl 9998  ax-mulcom 10000  ax-mulass 10002

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  remullem  13868  absmul  14034  binomrisefac  14773  cosadd  14895  tanadd  14897  eulerthlem2  15487  mul4sqlem  15657  odadd2  18252  itgmulc2  23600  plymullem1  23970  chordthmlem4  24562  heron  24565  quartlem1  24584  dchrmulcl  24974  bposlem9  25017  lgsdir  25057  lgsdi  25059  lgsquad2lem1  25109  chtppilimlem1  25162  rplogsumlem1  25173  dchrvmasumlem1  25184  dchrvmasum2lem  25185  chpdifbndlem1  25242  pntlemf  25294  brbtwn2  25785  colinearalglem4  25789  madjusmdetlem4  29896  hgt750lemf  30731  hgt750leme  30736  circum  31568  itgmulc2nc  33478  pellexlem6  37398  pell1234qrmulcl  37419  rmxyadd  37486  wallispi2lem2  40289  dirkertrigeqlem3  40317  cevathlem1  41056