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Theorem mul4 8981
Description: Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)
Assertion
Ref Expression
mul4  |-  ( ( ( 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 ) ) )

Proof of Theorem mul4
StepHypRef Expression
1 mul32 8979 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( A  x.  C )  x.  B ) )
21oveq1d 5873 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  x.  B )  x.  C
)  x.  D )  =  ( ( ( A  x.  C )  x.  B )  x.  D ) )
323expa 1151 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  C  e.  CC )  ->  ( ( ( A  x.  B )  x.  C )  x.  D )  =  ( ( ( A  x.  C )  x.  B
)  x.  D ) )
43adantrr 697 . 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 ) )
5 mulcl 8821 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
6 mulass 8825 . . . 4  |-  ( ( ( A  x.  B
)  e.  CC  /\  C  e.  CC  /\  D  e.  CC )  ->  (
( ( A  x.  B )  x.  C
)  x.  D )  =  ( ( A  x.  B )  x.  ( C  x.  D
) ) )
763expb 1152 . . 3  |-  ( ( ( A  x.  B
)  e.  CC  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( (
( A  x.  B
)  x.  C )  x.  D )  =  ( ( A  x.  B )  x.  ( C  x.  D )
) )
85, 7sylan 457 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( A  x.  B )  x.  C )  x.  D
)  =  ( ( A  x.  B )  x.  ( C  x.  D ) ) )
9 mulcl 8821 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
10 mulass 8825 . . . . 5  |-  ( ( ( A  x.  C
)  e.  CC  /\  B  e.  CC  /\  D  e.  CC )  ->  (
( ( A  x.  C )  x.  B
)  x.  D )  =  ( ( A  x.  C )  x.  ( B  x.  D
) ) )
11103expb 1152 . . . 4  |-  ( ( ( A  x.  C
)  e.  CC  /\  ( B  e.  CC  /\  D  e.  CC ) )  ->  ( (
( A  x.  C
)  x.  B )  x.  D )  =  ( ( A  x.  C )  x.  ( B  x.  D )
) )
129, 11sylan 457 . . 3  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( A  x.  C )  x.  B )  x.  D
)  =  ( ( A  x.  C )  x.  ( B  x.  D ) ) )
1312an4s 799 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( A  x.  C )  x.  B )  x.  D
)  =  ( ( A  x.  C )  x.  ( B  x.  D ) ) )
144, 8, 133eqtr3d 2323 1  |-  ( ( ( 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 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735    x. cmul 8742
This theorem is referenced by:  mul4i  9009  mul4d  9024  recextlem1  9398  divmuldiv  9460  mulexp  11141  demoivreALT  12481  bposlem9  20531
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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