Metamath Proof Explorer


Axiom ax-addass

Description: Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by Theorem axaddass . Proofs should normally use addass instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994)

Ref Expression
Assertion ax-addass
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + 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 caddc
 |-  +
9 0 3 8 co
 |-  ( A + B )
10 9 5 8 co
 |-  ( ( A + B ) + C )
11 3 5 8 co
 |-  ( B + C )
12 0 11 8 co
 |-  ( A + ( B + C ) )
13 10 12 wceq
 |-  ( ( A + B ) + C ) = ( A + ( B + C ) )
14 7 13 wi
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )