ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mul4 Unicode version

Theorem mul4 8158
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 8156 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( A  x.  C )  x.  B ) )
21oveq1d 5937 . . . 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 1205 . . 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 479 . 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 8006 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
6 mulass 8010 . . . 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 1206 . . 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 283 . 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 8006 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
10 mulass 8010 . . . . 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 1206 . . . 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 283 . . 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 588 . 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 2237 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 104    /\ w3a 980    = wceq 1364    e. wcel 2167  (class class class)co 5922   CCcc 7877    x. cmul 7884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-mulcl 7977  ax-mulcom 7980  ax-mulass 7982
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925
This theorem is referenced by:  mul4i  8174  mul4d  8181  recextlem1  8678  divmuldivap  8739  mulexp  10670  demoivreALT  11939
  Copyright terms: Public domain W3C validator