ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mul12 Unicode version

Theorem mul12 7293
Description: Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
Assertion
Ref Expression
mul12  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C )
) )

Proof of Theorem mul12
StepHypRef Expression
1 mulcom 7153 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 5552 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  x.  C
)  =  ( ( B  x.  A )  x.  C ) )
323adant3 959 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( B  x.  A )  x.  C ) )
4 mulass 7155 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( A  x.  ( B  x.  C
) ) )
5 mulass 7155 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  CC )  ->  (
( B  x.  A
)  x.  C )  =  ( B  x.  ( A  x.  C
) ) )
653com12 1143 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  x.  A
)  x.  C )  =  ( B  x.  ( A  x.  C
) ) )
73, 4, 63eqtr3d 2122 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    = wceq 1285    e. wcel 1434  (class class class)co 5537   CCcc 7030    x. cmul 7037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-mulcom 7128  ax-mulass 7130
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-iota 4891  df-fv 4934  df-ov 5540
This theorem is referenced by:  mul12i  7310  mul12d  7316  mulreap  9878  dvdscmul  10356  dvdstr  10366
  Copyright terms: Public domain W3C validator