Theorem hvmulassi

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

	Ref	Expression
Hypotheses	hvmulcom.1	⊢ 𝐴 ∈ ℂ
	hvmulcom.2	⊢ 𝐵 ∈ ℂ
	hvmulcom.3	⊢ 𝐶 ∈ ℋ
Assertion	hvmulassi	⊢ ( ( 𝐴 · 𝐵 ) ·ℎ 𝐶 ) = ( 𝐴 ·ℎ ( 𝐵 ·ℎ 𝐶 ) )

Proof

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