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 ) )