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Axiom ax-mulass 8799
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 8775. Proofs should normally use mulass 8821 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 8731 . . . 4  class  CC
31, 2wcel 1685 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1685 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
76, 2wcel 1685 . . 3  wff  C  e.  CC
83, 5, 7w3a 936 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
9 cmul 8738 . . . . 5  class  x.
101, 4, 9co 5820 . . . 4  class  ( A  x.  B )
1110, 6, 9co 5820 . . 3  class  ( ( A  x.  B )  x.  C )
124, 6, 9co 5820 . . . 4  class  ( B  x.  C )
131, 12, 9co 5820 . . 3  class  ( A  x.  ( B  x.  C ) )
1411, 13wceq 1624 . 2  wff  ( ( A  x.  B )  x.  C )  =  ( A  x.  ( B  x.  C )
)
158, 14wi 6 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  8821
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