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 )