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

Axiom ax-mulass 9012
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 8988. Proofs should normally use mulass 9034 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 8944 . . . 4  class  CC
31, 2wcel 1721 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1721 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1721 . . 3  wff  C  e.  CC
83, 5, 7w3a 936 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 cmul 8951 . . . . 5  class  x.
101, 4, 9co 6040 . . . 4  class  ( A  x.  B )
1110, 6, 9co 6040 . . 3  class  ( ( A  x.  B )  x.  C )
124, 6, 9co 6040 . . . 4  class  ( B  x.  C )
131, 12, 9co 6040 . . 3  class  ( A  x.  ( B  x.  C ) )
1411, 13wceq 1649 . 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  9034
  Copyright terms: Public domain W3C validator