Theorem homul12

Description: Swap first and second factors in a nested operator scalar product. (Contributed by NM, 12-Aug-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	homul12	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝐴 ·op ( 𝐵 ·op 𝑇 ) ) = ( 𝐵 ·op ( 𝐴 ·op 𝑇 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mulcom	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
2	1	oveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ·op 𝑇 ) = ( ( 𝐵 · 𝐴 ) ·op 𝑇 ) )
3	2	3adant3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ( 𝐴 · 𝐵 ) ·op 𝑇 ) = ( ( 𝐵 · 𝐴 ) ·op 𝑇 ) )
4		homulass	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ( 𝐴 · 𝐵 ) ·op 𝑇 ) = ( 𝐴 ·op ( 𝐵 ·op 𝑇 ) ) )
5		homulass	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ( 𝐵 · 𝐴 ) ·op 𝑇 ) = ( 𝐵 ·op ( 𝐴 ·op 𝑇 ) ) )
6	5	3com12	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ( 𝐵 · 𝐴 ) ·op 𝑇 ) = ( 𝐵 ·op ( 𝐴 ·op 𝑇 ) ) )
7	3 4 6	3eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝐴 ·op ( 𝐵 ·op 𝑇 ) ) = ( 𝐵 ·op ( 𝐴 ·op 𝑇 ) ) )