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Metamath Proof Explorer

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 ) ) )