MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-mulass Structured version   Unicode version

Axiom ax-mulass 9058
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 9034. Proofs should normally use mulass 9080 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 8990 . . . 4  class  CC
31, 2wcel 1726 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1726 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1726 . . 3  wff  C  e.  CC
83, 5, 7w3a 937 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 cmul 8997 . . . . 5  class  x.
101, 4, 9co 6083 . . . 4  class  ( A  x.  B )
1110, 6, 9co 6083 . . 3  class  ( ( A  x.  B )  x.  C )
124, 6, 9co 6083 . . . 4  class  ( B  x.  C )
131, 12, 9co 6083 . . 3  class  ( A  x.  ( B  x.  C ) )
1411, 13wceq 1653 . 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  9080
  Copyright terms: Public domain W3C validator