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

Axiom ax-mulass 7351
Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 7311. Proofs should normally use mulass 7376 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-mulass  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( A  x.  ( B  x.  C
) ) )

Detailed syntax breakdown of Axiom ax-mulass
StepHypRef Expression
1 cA . . . 4  class  A
2 cc 7251 . . . 4  class  CC
31, 2wcel 1434 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1434 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1434 . . 3  wff  C  e.  CC
83, 5, 7w3a 920 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 cmul 7258 . . . . 5  class  x.
101, 4, 9co 5591 . . . 4  class  ( A  x.  B )
1110, 6, 9co 5591 . . 3  class  ( ( A  x.  B )  x.  C )
124, 6, 9co 5591 . . . 4  class  ( B  x.  C )
131, 12, 9co 5591 . . 3  class  ( A  x.  ( B  x.  C ) )
1411, 13wceq 1285 . 2  wff  ( ( A  x.  B )  x.  C )  =  ( A  x.  ( B  x.  C )
)
158, 14wi 4 1  wff  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( A  x.  ( B  x.  C
) ) )
Colors of variables: wff set class
This axiom is referenced by:  mulass  7376
  Copyright terms: Public domain W3C validator