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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