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Theorem mul4 7675
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 7673 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( A  x.  C )  x.  B ) )
21oveq1d 5681 . . . 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 1144 . . 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 464 . 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 7530 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
6 mulass 7534 . . . 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 1145 . . 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 278 . 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 7530 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
10 mulass 7534 . . . . 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 1145 . . . 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 278 . . 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 556 . 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 2129 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 103    /\ w3a 925    = wceq 1290    e. wcel 1439  (class class class)co 5666   CCcc 7409    x. cmul 7416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-mulcl 7504  ax-mulcom 7507  ax-mulass 7509
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-iota 4993  df-fv 5036  df-ov 5669
This theorem is referenced by:  mul4i  7691  mul4d  7698  recextlem1  8181  divmuldivap  8240  mulexp  10055  demoivreALT  11124
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