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Metamath Proof Explorer

Theorem hoeqi

Description: Equality of Hilbert space operators. (Contributed by NM, 14-Nov-2000) (New usage is discouraged.)

Ref Expression
Hypotheses hoeq.1 ⊢ 𝑆 : ℋ ⟶ ℋ
hoeq.2 ⊢ 𝑇 : ℋ ⟶ ℋ
Assertion hoeqi ( ∀ 𝑥 ∈ ℋ ( 𝑆 ‘ 𝑥 ) = ( 𝑇 ‘ 𝑥 ) ↔ 𝑆 = 𝑇 )

Proof

Step Hyp Ref Expression
1 hoeq.1 ⊢ 𝑆 : ℋ ⟶ ℋ
2 hoeq.2 ⊢ 𝑇 : ℋ ⟶ ℋ
3 hoeq ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ∀ 𝑥 ∈ ℋ ( 𝑆 ‘ 𝑥 ) = ( 𝑇 ‘ 𝑥 ) ↔ 𝑆 = 𝑇 ) )
4 1 2 3 mp2an ⊢ ( ∀ 𝑥 ∈ ℋ ( 𝑆 ‘ 𝑥 ) = ( 𝑇 ‘ 𝑥 ) ↔ 𝑆 = 𝑇 )