Axiom ax-hvdistr1 4993

Description: Scalar multiplication distributive law

Assertion

Ref	Expression
ax-hvdistr1	|- ((A e. CC /\ B e. H~ /\ C e. H~) -> (A .s (B +v C)) = ((A .s B) +v (A .s C)))

Detailed syntax breakdown of Axiom ax-hvdistr1

Step	Hyp	Ref	Expression
1		cA	. . . 4 class A
2		cc 4026	. . . 4 class CC
3	1, 2	wcel 1092	. . 3 wff A e. CC
4		cB	. . . 4 class B
5		chil 4958	. . . 4 class H~
6	4, 5	wcel 1092	. . 3 wff B e. H~
7		cC	. . . 4 class C
8	7, 5	wcel 1092	. . 3 wff C e. H~
9	3, 6, 8	w3a 581	. 2 wff (A e. CC /\ B e. H~ /\ C e. H~)
10		cva 4959	. . . . 5 class +v
11	4, 7, 10	co 3001	. . . 4 class (B +v C)
12		csm 4960	. . . 4 class .s
13	1, 11, 12	co 3001	. . 3 class (A .s (B +v C))
14	1, 4, 12	co 3001	. . . 4 class (A .s B)
15	1, 7, 12	co 3001	. . . 4 class (A .s C)
16	14, 15, 10	co 3001	. . 3 class ((A .s B) +v (A .s C))
17	13, 16	wceq 1091	. 2 wff (A .s (B +v C)) = ((A .s B) +v (A .s C))
18	9, 17	wi 2	1 wff ((A e. CC /\ B e. H~ /\ C e. H~) -> (A .s (B +v C)) = ((A .s B) +v (A .s C)))

This axiom is referenced by:  hvsub4t 5014  hvdistr1 5023  shscl 5282  spanunsn 5482

metamath.org