Theorem hlimveci

Description: Closure of the limit of a sequence on Hilbert space. (Contributed by NM, 16-Aug-1999) (New usage is discouraged.)

	Ref	Expression
Hypothesis	hlim.1	⊢ 𝐴 ∈ V
Assertion	hlimveci	⊢ ( 𝐹 ⇝𝑣 𝐴 → 𝐴 ∈ ℋ )

Proof

Step	Hyp	Ref	Expression
1		hlim.1	⊢ 𝐴 ∈ V
2	1	hlimi	⊢ ( 𝐹 ⇝𝑣 𝐴 ↔ ( ( 𝐹 : ℕ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑧 ) −ℎ 𝐴 ) ) < 𝑥 ) )
3	2	simplbi	⊢ ( 𝐹 ⇝𝑣 𝐴 → ( 𝐹 : ℕ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) )
4	3	simprd	⊢ ( 𝐹 ⇝𝑣 𝐴 → 𝐴 ∈ ℋ )