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Theorem hvdistr1i 28832
 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 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))

Proof of Theorem hvdistr1i
StepHypRef Expression
1 hvdistr1.1 . 2 𝐴 ∈ ℂ
2 hvdistr1.2 . 2 𝐵 ∈ ℋ
3 hvdistr1.3 . 2 𝐶 ∈ ℋ
4 ax-hvdistr1 28789 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
51, 2, 3, 4mp3an 1458 1 (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2114  (class class class)co 7140  ℂcc 10524   ℋchba 28700   +ℎ cva 28701   ·ℎ csm 28702 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvdistr1 28789 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086 This theorem is referenced by:  hvsubsub4i  28840  hvnegdii  28843  pjmulii  29458  lnophmlem2  29798
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