Metamath Proof Explorer


Theorem hvsubdistr1i

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

Ref Expression
Hypotheses hvdistr1.1 𝐴 ∈ ℂ
hvdistr1.2 𝐵 ∈ ℋ
hvdistr1.3 𝐶 ∈ ℋ
Assertion hvsubdistr1i ( 𝐴 · ( 𝐵 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) )

Proof

Step Hyp Ref Expression
1 hvdistr1.1 𝐴 ∈ ℂ
2 hvdistr1.2 𝐵 ∈ ℋ
3 hvdistr1.3 𝐶 ∈ ℋ
4 hvsubdistr1 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 · ( 𝐵 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) ) )
5 1 2 3 4 mp3an ( 𝐴 · ( 𝐵 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) )