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Theorem mul4 7206
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 7204 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( A  x.  C )  x.  B ) )
21oveq1d 5555 . . . 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 1115 . . 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 456 . 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 7066 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
6 mulass 7070 . . . 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 1116 . . 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 271 . 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 7066 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
10 mulass 7070 . . . . 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 1116 . . . 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 271 . . 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 530 . 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 2096 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 101    /\ w3a 896    = wceq 1259    e. wcel 1409  (class class class)co 5540   CCcc 6945    x. cmul 6952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-mulcl 7040  ax-mulcom 7043  ax-mulass 7045
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543
This theorem is referenced by:  mul4i  7222  mul4d  7229  recextlem1  7706  divmuldivap  7763  mulexp  9459
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