Axiom ax-hvdistr2 31490

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

Assertion

Ref	Expression
ax-hvdistr2	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 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 11122	. . . 4 class ℂ
3	1, 2	wcel 2145	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2145	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7		chba 31400	. . . 4 class ℋ
8	6, 7	wcel 2145	. . 3 wff 𝐶 ∈ ℋ
9	3, 5, 8	w3a 1103	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ)
10		caddc 11127	. . . . 5 class +
11	1, 4, 10	co 7413	. . . 4 class (𝐴 + 𝐵)
12		csm 31402	. . . 4 class ·ℎ
13	11, 6, 12	co 7413	. . 3 class ((𝐴 + 𝐵) ·ℎ 𝐶)
14	1, 6, 12	co 7413	. . . 4 class (𝐴 ·ℎ 𝐶)
15	4, 6, 12	co 7413	. . . 4 class (𝐵 ·ℎ 𝐶)
16		cva 31401	. . . 4 class +ℎ
17	14, 15, 16	co 7413	. . 3 class ((𝐴 ·ℎ 𝐶) +ℎ (𝐵 ·ℎ 𝐶))
18	13, 17	wceq 1570	. 2 wff ((𝐴 + 𝐵) ·ℎ 𝐶) = ((𝐴 ·ℎ 𝐶) +ℎ (𝐵 ·ℎ 𝐶))
19	9, 18	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ℎ 𝐶) = ((𝐴 ·ℎ 𝐶) +ℎ (𝐵 ·ℎ 𝐶)))

This axiom is used by:  hvsubid  31507  hvsubdistr2  31531  hv2times  31542  hilvc  31643  hhssnv  31745  hoadddir  32285  superpos  32835