Axiom ax-addass 10001

Description: Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 9977. Proofs should normally use addass 10023 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-addass	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )

Detailed syntax breakdown of Axiom ax-addass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class A
2		cc 9934	. . . 4 class CC
3	1, 2	wcel 1990	. . 3 wff A e. CC
4		cB	. . . 4 class B
5	4, 2	wcel 1990	. . 3 wff B e. CC
6		cC	. . . 4 class C
7	6, 2	wcel 1990	. . 3 wff C e. CC
8	3, 5, 7	w3a 1037	. 2 wff ( A e. CC /\ B e. CC /\ C e. CC )
9		caddc 9939	. . . . 5 class +
10	1, 4, 9	co 6650	. . . 4 class ( A + B )
11	10, 6, 9	co 6650	. . 3 class ( ( A + B ) + C )
12	4, 6, 9	co 6650	. . . 4 class ( B + C )
13	1, 12, 9	co 6650	. . 3 class ( A + ( B + C ) )
14	11, 13	wceq 1483	. 2 wff ( ( A + B ) + C ) = ( A + ( B + C ) )
15	8, 14	wi 4	1 wff ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )

This axiom is referenced by:  addass  10023