Axiom ax-hvdistr1 30013

Description: Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-hvdistr1	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶)))

Detailed syntax breakdown of Axiom ax-hvdistr1

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 11058	. . . 4 class ℂ
3	1, 2	wcel 2106	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5		chba 29924	. . . 4 class ℋ
6	4, 5	wcel 2106	. . 3 wff 𝐵 ∈ ℋ
7		cC	. . . 4 class 𝐶
8	7, 5	wcel 2106	. . 3 wff 𝐶 ∈ ℋ
9	3, 6, 8	w3a 1087	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)
10		cva 29925	. . . . 5 class +ℎ
11	4, 7, 10	co 7362	. . . 4 class (𝐵 +ℎ 𝐶)
12		csm 29926	. . . 4 class ·ℎ
13	1, 11, 12	co 7362	. . 3 class (𝐴 ·ℎ (𝐵 +ℎ 𝐶))
14	1, 4, 12	co 7362	. . . 4 class (𝐴 ·ℎ 𝐵)
15	1, 7, 12	co 7362	. . . 4 class (𝐴 ·ℎ 𝐶)
16	14, 15, 10	co 7362	. . 3 class ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶))
17	13, 16	wceq 1541	. 2 wff (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶))
18	9, 17	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶)))

This axiom is referenced by:  hvsub4  30042  hvsubass  30049  hvsubdistr1  30054  hvdistr1i  30056  hv2times  30066  hilvc  30167  hhssnv  30269  shscli  30322  spanunsni  30584  hoadddi  30808  lnopmi  31005  lnophsi  31006