Axiom ax-hvdistr2 9027

Description: Scalar multiplication distributive law

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 5155	. . . 4 class CC
3	1, 2	wcel 1105	. . 3 wff A e. CC
4		cB	. . . 4 class B
5	4, 2	wcel 1105	. . 3 wff B e. CC
6		cC	. . . 4 class C
7		chil 8968	. . . 4 class H~
8	6, 7	wcel 1105	. . 3 wff C e. H~
9	3, 5, 8	w3a 772	. 2 wff (A e. CC /\ B e. CC /\ C e. H~)
10		caddc 5160	. . . . 5 class +
11	1, 4, 10	co 3902	. . . 4 class (A + B)
12		csm 8970	. . . 4 class .h
13	11, 6, 12	co 3902	. . 3 class ((A + B) .h C)
14	1, 6, 12	co 3902	. . . 4 class (A .h C)
15	4, 6, 12	co 3902	. . . 4 class (B .h C)
16		cva 8969	. . . 4 class +h
17	14, 15, 16	co 3902	. . 3 class ((A .h C) +h (B .h C))
18	13, 17	wceq 1099	. 2 wff ((A + B) .h C) = ((A .h C) +h (B .h C))
19	9, 18	wi 3	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:  hvsubidt 9044  hvsubdistr2t 9066  hv2timest 9077  hilvc 9178  hoadddirt 9861  superpos 10403

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