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Axiom ax-mulass 9087
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 9063. Proofs should normally use mulass 9109 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 9019 . . . 4  class  CC
31, 2wcel 1727 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1727 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1727 . . 3  wff  C  e.  CC
83, 5, 7w3a 937 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 cmul 9026 . . . . 5  class  x.
101, 4, 9co 6110 . . . 4  class  ( A  x.  B )
1110, 6, 9co 6110 . . 3  class  ( ( A  x.  B )  x.  C )
124, 6, 9co 6110 . . . 4  class  ( B  x.  C )
131, 12, 9co 6110 . . 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  9109
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