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

Axiom ax-mulass 8848
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 8824. Proofs should normally use mulass 8870 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 8780 . . . 4  class  CC
31, 2wcel 1701 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1701 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1701 . . 3  wff  C  e.  CC
83, 5, 7w3a 934 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 cmul 8787 . . . . 5  class  x.
101, 4, 9co 5900 . . . 4  class  ( A  x.  B )
1110, 6, 9co 5900 . . 3  class  ( ( A  x.  B )  x.  C )
124, 6, 9co 5900 . . . 4  class  ( B  x.  C )
131, 12, 9co 5900 . . 3  class  ( A  x.  ( B  x.  C ) )
1411, 13wceq 1633 . 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  8870
  Copyright terms: Public domain W3C validator