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Metamath Proof Explorer

Theorem hvmulcom

Description: Scalar multiplication commutative law. (Contributed by NM, 19-May-2005) (New usage is discouraged.)

Ref Expression
Assertion hvmulcom ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 ·ℎ ( 𝐵 ·ℎ 𝐶 ) ) = ( 𝐵 ·ℎ ( 𝐴 ·ℎ 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 mulcom ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
2 1 oveq1d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ·ℎ 𝐶 ) = ( ( 𝐵 · 𝐴 ) ·ℎ 𝐶 ) )
3 2 3adant3 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 · 𝐵 ) ·ℎ 𝐶 ) = ( ( 𝐵 · 𝐴 ) ·ℎ 𝐶 ) )
4 ax-hvmulass ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 · 𝐵 ) ·ℎ 𝐶 ) = ( 𝐴 ·ℎ ( 𝐵 ·ℎ 𝐶 ) ) )
5 ax-hvmulass ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐵 · 𝐴 ) ·ℎ 𝐶 ) = ( 𝐵 ·ℎ ( 𝐴 ·ℎ 𝐶 ) ) )
6 5 3com12 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐵 · 𝐴 ) ·ℎ 𝐶 ) = ( 𝐵 ·ℎ ( 𝐴 ·ℎ 𝐶 ) ) )
7 3 4 6 3eqtr3d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 ·ℎ ( 𝐵 ·ℎ 𝐶 ) ) = ( 𝐵 ·ℎ ( 𝐴 ·ℎ 𝐶 ) ) )