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Metamath Proof Explorer

Theorem hvdistr1i

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 hvdistr1i ( 𝐴 ·ℎ ( 𝐵 +ℎ 𝐶 ) ) = ( ( 𝐴 ·ℎ 𝐵 ) +ℎ ( 𝐴 ·ℎ 𝐶 ) )

Proof

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