This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem hoeq1

Description: A condition implying that two Hilbert space operators are equal. Lemma 3.2(S9) of Beran p. 95. (Contributed by NM, 15-Feb-2006) (New usage is discouraged.)

Ref Expression
Assertion hoeq1 ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( ( 𝑆 ‘ 𝑥 ) ·ih 𝑦 ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) ↔ 𝑆 = 𝑇 ) )

Proof

Step Hyp Ref Expression
1 ffvelcdm ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑥 ∈ ℋ ) → ( 𝑆 ‘ 𝑥 ) ∈ ℋ )
2 ffvelcdm ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑥 ∈ ℋ ) → ( 𝑇 ‘ 𝑥 ) ∈ ℋ )
3 hial2eq ⊢ ( ( ( 𝑆 ‘ 𝑥 ) ∈ ℋ ∧ ( 𝑇 ‘ 𝑥 ) ∈ ℋ ) → ( ∀ 𝑦 ∈ ℋ ( ( 𝑆 ‘ 𝑥 ) ·ih 𝑦 ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) ↔ ( 𝑆 ‘ 𝑥 ) = ( 𝑇 ‘ 𝑥 ) ) )
4 1 2 3 syl2an ⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑥 ∈ ℋ ) ∧ ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑥 ∈ ℋ ) ) → ( ∀ 𝑦 ∈ ℋ ( ( 𝑆 ‘ 𝑥 ) ·ih 𝑦 ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) ↔ ( 𝑆 ‘ 𝑥 ) = ( 𝑇 ‘ 𝑥 ) ) )
5 4 anandirs ⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) ∧ 𝑥 ∈ ℋ ) → ( ∀ 𝑦 ∈ ℋ ( ( 𝑆 ‘ 𝑥 ) ·ih 𝑦 ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) ↔ ( 𝑆 ‘ 𝑥 ) = ( 𝑇 ‘ 𝑥 ) ) )
6 5 ralbidva ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( ( 𝑆 ‘ 𝑥 ) ·ih 𝑦 ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) ↔ ∀ 𝑥 ∈ ℋ ( 𝑆 ‘ 𝑥 ) = ( 𝑇 ‘ 𝑥 ) ) )
7 ffn ⊢ ( 𝑆 : ℋ ⟶ ℋ → 𝑆 Fn ℋ )
8 ffn ⊢ ( 𝑇 : ℋ ⟶ ℋ → 𝑇 Fn ℋ )
9 eqfnfv ⊢ ( ( 𝑆 Fn ℋ ∧ 𝑇 Fn ℋ ) → ( 𝑆 = 𝑇 ↔ ∀ 𝑥 ∈ ℋ ( 𝑆 ‘ 𝑥 ) = ( 𝑇 ‘ 𝑥 ) ) )
10 7 8 9 syl2an ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝑆 = 𝑇 ↔ ∀ 𝑥 ∈ ℋ ( 𝑆 ‘ 𝑥 ) = ( 𝑇 ‘ 𝑥 ) ) )
11 6 10 bitr4d ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( ( 𝑆 ‘ 𝑥 ) ·ih 𝑦 ) = ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑦 ) ↔ 𝑆 = 𝑇 ) )