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Axiom ax-mulass 8803
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 8779. Proofs should normally use mulass 8825 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 8735 . . . 4  class  CC
31, 2wcel 1684 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1684 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1684 . . 3  wff  C  e.  CC
83, 5, 7w3a 934 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 cmul 8742 . . . . 5  class  x.
101, 4, 9co 5858 . . . 4  class  ( A  x.  B )
1110, 6, 9co 5858 . . 3  class  ( ( A  x.  B )  x.  C )
124, 6, 9co 5858 . . . 4  class  ( B  x.  C )
131, 12, 9co 5858 . . 3  class  ( A  x.  ( B  x.  C ) )
1411, 13wceq 1623 . 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  8825
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