Axiom ax-distr 10003

Description: Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 9979. Proofs should normally use adddi 10025 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

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

Detailed syntax breakdown of Axiom ax-distr

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	4, 6, 9	co 6650	. . . 4 class ( B + C )
11		cmul 9941	. . . 4 class x.
12	1, 10, 11	co 6650	. . 3 class ( A x. ( B + C ) )
13	1, 4, 11	co 6650	. . . 4 class ( A x. B )
14	1, 6, 11	co 6650	. . . 4 class ( A x. C )
15	13, 14, 9	co 6650	. . 3 class ( ( A x. B ) + ( A x. C ) )
16	12, 15	wceq 1483	. 2 wff ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) )
17	8, 16	wi 4	1 wff ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) )

This axiom is referenced by:  adddi  10025