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Metamath Proof Explorer

Theorem hocoi

Description: Composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000) (New usage is discouraged.)

Ref Expression
Hypotheses hoeq.1 ⊢ 𝑆 : ℋ ⟶ ℋ
hoeq.2 ⊢ 𝑇 : ℋ ⟶ ℋ
Assertion hocoi ( 𝐴 ∈ ℋ → ( ( 𝑆 ∘ 𝑇 ) ‘ 𝐴 ) = ( 𝑆 ‘ ( 𝑇 ‘ 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 hoeq.1 ⊢ 𝑆 : ℋ ⟶ ℋ
2 hoeq.2 ⊢ 𝑇 : ℋ ⟶ ℋ
3 fvco3 ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝑆 ∘ 𝑇 ) ‘ 𝐴 ) = ( 𝑆 ‘ ( 𝑇 ‘ 𝐴 ) ) )
4 2 3 mpan ⊢ ( 𝐴 ∈ ℋ → ( ( 𝑆 ∘ 𝑇 ) ‘ 𝐴 ) = ( 𝑆 ‘ ( 𝑇 ‘ 𝐴 ) ) )