Theorem hvdistr1i 29702

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

Hypotheses

Ref	Expression
hvdistr1.1	⊢ 𝐴 ∈ ℂ
hvdistr1.2	⊢ 𝐵 ∈ ℋ
hvdistr1.3	⊢ 𝐶 ∈ ℋ

Assertion

Ref	Expression
hvdistr1i	⊢ (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶))

Proof of Theorem hvdistr1i

Step	Hyp	Ref	Expression
1		hvdistr1.1	. 2 ⊢ 𝐴 ∈ ℂ
2		hvdistr1.2	. 2 ⊢ 𝐵 ∈ ℋ
3		hvdistr1.3	. 2 ⊢ 𝐶 ∈ ℋ
4		ax-hvdistr1 29659	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶)))
5	1, 2, 3, 4	mp3an 1460	1 ⊢ (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶))

Syntax hints:   = wceq 1540   ∈ wcel 2105  (class class class)co 7338  ℂcc 10971   ℋchba 29570   +_ℎ cva 29571   ·_ℎ csm 29572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvdistr1 29659

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088

This theorem is referenced by:  hvsubsub4i  29710  hvnegdii  29713  pjmulii  30328  lnophmlem2  30668