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Metamath Proof Explorer

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 ) )