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Axiom ax-mulass 9048
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 9024. Proofs should normally use mulass 9070 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 8980 . . . 4  class  CC
31, 2wcel 1725 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1725 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1725 . . 3  wff  C  e.  CC
83, 5, 7w3a 936 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 cmul 8987 . . . . 5  class  x.
101, 4, 9co 6073 . . . 4  class  ( A  x.  B )
1110, 6, 9co 6073 . . 3  class  ( ( A  x.  B )  x.  C )
124, 6, 9co 6073 . . . 4  class  ( B  x.  C )
131, 12, 9co 6073 . . 3  class  ( A  x.  ( B  x.  C ) )
1411, 13wceq 1652 . 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  9070
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