Metamath Proof Explorer


Axiom ax-mulass

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass . Proofs should normally use mulass instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994)

Ref Expression
Assertion ax-mulass
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA
 |-  A
1 cc
 |-  CC
2 0 1 wcel
 |-  A e. CC
3 cB
 |-  B
4 3 1 wcel
 |-  B e. CC
5 cC
 |-  C
6 5 1 wcel
 |-  C e. CC
7 2 4 6 w3a
 |-  ( A e. CC /\ B e. CC /\ C e. CC )
8 cmul
 |-  x.
9 0 3 8 co
 |-  ( A x. B )
10 9 5 8 co
 |-  ( ( A x. B ) x. C )
11 3 5 8 co
 |-  ( B x. C )
12 0 11 8 co
 |-  ( A x. ( B x. C ) )
13 10 12 wceq
 |-  ( ( A x. B ) x. C ) = ( A x. ( B x. C ) )
14 7 13 wi
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )