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

Axiom ax-mulass 7735
Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 7693. Proofs should normally use mulass 7763 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 7630 . . . 4  class  CC
31, 2wcel 1480 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1480 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1480 . . 3  wff  C  e.  CC
83, 5, 7w3a 962 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 cmul 7637 . . . . 5  class  x.
101, 4, 9co 5774 . . . 4  class  ( A  x.  B )
1110, 6, 9co 5774 . . 3  class  ( ( A  x.  B )  x.  C )
124, 6, 9co 5774 . . . 4  class  ( B  x.  C )
131, 12, 9co 5774 . . 3  class  ( A  x.  ( B  x.  C ) )
1411, 13wceq 1331 . 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  7763
  Copyright terms: Public domain W3C validator