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 ) ) )