Theorem hvdistr1 5023

Description: Scalar multiplication distributive law.

Hypotheses

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

Assertion

Ref	Expression
hvdistr1	⊢ (A ·s (B +v C)) = ((A ·s B) +v (A ·s C))

Proof of Theorem hvdistr1

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 4993	. 2 ⊢ ((A ∈ ℂ ∧ B ∈ ℋ ∧ C ∈ ℋ ) → (A ·s (B +v C)) = ((A ·s B) +v (A ·s C)))
5	1, 2, 3, 4	mp3an 642	1 ⊢ (A ·s (B +v C)) = ((A ·s B) +v (A ·s C))

Syntax hints:   = wceq 1091   ∈ wcel 1092  (class class class)co 3001  ℂcc 4026   ℋ chil 4958   +_v cva 4959   ·_s csm 4960

This theorem is referenced by:  hvsubdistr1 5024  hvsubass 5027  hvsubsub4 5031  hv2times 5033  hvnegdi 5034  pjmul 5568

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-hvdistr1 4993

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583

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