Axiom ax-hvdistr2 27866

Description: Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-hvdistr2	|- ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( ( A + B ) .h C ) = ( ( A .h C ) +h ( B .h C ) ) )

Detailed syntax breakdown of Axiom ax-hvdistr2

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		chil 27776	. . . 4 class ~H
8	6, 7	wcel 1990	. . 3 wff C e. ~H
9	3, 5, 8	w3a 1037	. 2 wff ( A e. CC /\ B e. CC /\ C e. ~H )
10		caddc 9939	. . . . 5 class +
11	1, 4, 10	co 6650	. . . 4 class ( A + B )
12		csm 27778	. . . 4 class .h
13	11, 6, 12	co 6650	. . 3 class ( ( A + B ) .h C )
14	1, 6, 12	co 6650	. . . 4 class ( A .h C )
15	4, 6, 12	co 6650	. . . 4 class ( B .h C )
16		cva 27777	. . . 4 class +h
17	14, 15, 16	co 6650	. . 3 class ( ( A .h C ) +h ( B .h C ) )
18	13, 17	wceq 1483	. 2 wff ( ( A + B ) .h C ) = ( ( A .h C ) +h ( B .h C ) )
19	9, 18	wi 4	1 wff ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( ( A + B ) .h C ) = ( ( A .h C ) +h ( B .h C ) ) )

This axiom is referenced by:  hvsubid  27883  hvsubdistr2  27907  hv2times  27918  hilvc  28019  hhssnv  28121  hoadddir  28663  superpos  29213