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Theorem mul4 9559
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 9557 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( A  x.  C )  x.  B ) )
21oveq1d 6127 . . . 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 1187 . . 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 716 . 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 9387 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
6 mulass 9391 . . . 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 1188 . . 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 471 . 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 9387 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
10 mulass 9391 . . . . 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 1188 . . . 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 471 . . 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 822 . 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 2483 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756  (class class class)co 6112   CCcc 9301    x. cmul 9308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-mulcl 9365  ax-mulcom 9367  ax-mulass 9369
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-iota 5402  df-fv 5447  df-ov 6115
This theorem is referenced by:  mul4i  9587  mul4d  9602  recextlem1  9987  divmuldiv  10052  mulexp  11924  demoivreALT  13506  bposlem9  22653
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