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Metamath Proof Explorer

Theorem hoeq2

Description: A condition implying that two Hilbert space operators are equal. Lemma 3.2(S11) of Beran p. 95. (Contributed by NM, 15-Feb-2006) (New usage is discouraged.)

Ref Expression
Assertion hoeq2 ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑆 ‘ 𝑦 ) ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) ↔ 𝑆 = 𝑇 ) )

Proof

Step Hyp Ref Expression
1 ralcom ⊢ ( ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑆 ‘ 𝑦 ) ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ ℋ ∀ 𝑥 ∈ ℋ ( 𝑥 ·ih ( 𝑆 ‘ 𝑦 ) ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) )
2 1 a1i ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑆 ‘ 𝑦 ) ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ ℋ ∀ 𝑥 ∈ ℋ ( 𝑥 ·ih ( 𝑆 ‘ 𝑦 ) ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) ) )
3 ffvelcdm ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑦 ∈ ℋ ) → ( 𝑆 ‘ 𝑦 ) ∈ ℋ )
4 ffvelcdm ⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑦 ∈ ℋ ) → ( 𝑇 ‘ 𝑦 ) ∈ ℋ )
5 hial2eq2 ⊢ ( ( ( 𝑆 ‘ 𝑦 ) ∈ ℋ ∧ ( 𝑇 ‘ 𝑦 ) ∈ ℋ ) → ( ∀ 𝑥 ∈ ℋ ( 𝑥 ·ih ( 𝑆 ‘ 𝑦 ) ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) ↔ ( 𝑆 ‘ 𝑦 ) = ( 𝑇 ‘ 𝑦 ) ) )
6 hial2eq ⊢ ( ( ( 𝑆 ‘ 𝑦 ) ∈ ℋ ∧ ( 𝑇 ‘ 𝑦 ) ∈ ℋ ) → ( ∀ 𝑥 ∈ ℋ ( ( 𝑆 ‘ 𝑦 ) ·ih 𝑥 ) = ( ( 𝑇 ‘ 𝑦 ) ·ih 𝑥 ) ↔ ( 𝑆 ‘ 𝑦 ) = ( 𝑇 ‘ 𝑦 ) ) )
7 5 6 bitr4d ⊢ ( ( ( 𝑆 ‘ 𝑦 ) ∈ ℋ ∧ ( 𝑇 ‘ 𝑦 ) ∈ ℋ ) → ( ∀ 𝑥 ∈ ℋ ( 𝑥 ·ih ( 𝑆 ‘ 𝑦 ) ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ ℋ ( ( 𝑆 ‘ 𝑦 ) ·ih 𝑥 ) = ( ( 𝑇 ‘ 𝑦 ) ·ih 𝑥 ) ) )
8 3 4 7 syl2an ⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑦 ∈ ℋ ) ∧ ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑦 ∈ ℋ ) ) → ( ∀ 𝑥 ∈ ℋ ( 𝑥 ·ih ( 𝑆 ‘ 𝑦 ) ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ ℋ ( ( 𝑆 ‘ 𝑦 ) ·ih 𝑥 ) = ( ( 𝑇 ‘ 𝑦 ) ·ih 𝑥 ) ) )
9 8 anandirs ⊢ ( ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) ∧ 𝑦 ∈ ℋ ) → ( ∀ 𝑥 ∈ ℋ ( 𝑥 ·ih ( 𝑆 ‘ 𝑦 ) ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ ℋ ( ( 𝑆 ‘ 𝑦 ) ·ih 𝑥 ) = ( ( 𝑇 ‘ 𝑦 ) ·ih 𝑥 ) ) )
10 9 ralbidva ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ∀ 𝑦 ∈ ℋ ∀ 𝑥 ∈ ℋ ( 𝑥 ·ih ( 𝑆 ‘ 𝑦 ) ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ ℋ ∀ 𝑥 ∈ ℋ ( ( 𝑆 ‘ 𝑦 ) ·ih 𝑥 ) = ( ( 𝑇 ‘ 𝑦 ) ·ih 𝑥 ) ) )
11 hoeq1 ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ∀ 𝑦 ∈ ℋ ∀ 𝑥 ∈ ℋ ( ( 𝑆 ‘ 𝑦 ) ·ih 𝑥 ) = ( ( 𝑇 ‘ 𝑦 ) ·ih 𝑥 ) ↔ 𝑆 = 𝑇 ) )
12 2 10 11 3bitrd ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑆 ‘ 𝑦 ) ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) ↔ 𝑆 = 𝑇 ) )