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Metamath Proof Explorer

Theorem eleigveccl

Description: Closure of an eigenvector of a Hilbert space operator. (Contributed by NM, 23-Mar-2006) (New usage is discouraged.)

Ref Expression
Assertion eleigveccl ⊢ T : ℋ ⟶ ℋ ∧ A ∈ eigvec ⁡ T → A ∈ ℋ

Proof

Step Hyp Ref Expression
1 eleigvec2 ⊢ T : ℋ ⟶ ℋ → A ∈ eigvec ⁡ T ↔ A ∈ ℋ ∧ A ≠ 0 ℎ ∧ T ⁡ A ∈ span ⁡ A
2 1 biimpa ⊢ T : ℋ ⟶ ℋ ∧ A ∈ eigvec ⁡ T → A ∈ ℋ ∧ A ≠ 0 ℎ ∧ T ⁡ A ∈ span ⁡ A
3 2 simp1d ⊢ T : ℋ ⟶ ℋ ∧ A ∈ eigvec ⁡ T → A ∈ ℋ