Theorem chcompl

Description: Completeness of a closed subspace of Hilbert space. (Contributed by NM, 4-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	chcompl	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐹 ∈ Cauchy ∧ 𝐹 : ℕ ⟶ 𝐻 ) → ∃ 𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		isch3	⊢ ( 𝐻 ∈ Cℋ ↔ ( 𝐻 ∈ Sℋ ∧ ∀ 𝑓 ∈ Cauchy ( 𝑓 : ℕ ⟶ 𝐻 → ∃ 𝑥 ∈ 𝐻 𝑓 ⇝𝑣 𝑥 ) ) )
2	1	simprbi	⊢ ( 𝐻 ∈ Cℋ → ∀ 𝑓 ∈ Cauchy ( 𝑓 : ℕ ⟶ 𝐻 → ∃ 𝑥 ∈ 𝐻 𝑓 ⇝𝑣 𝑥 ) )
3		feq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 : ℕ ⟶ 𝐻 ↔ 𝐹 : ℕ ⟶ 𝐻 ) )
4		breq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ⇝𝑣 𝑥 ↔ 𝐹 ⇝𝑣 𝑥 ) )
5	4	rexbidv	⊢ ( 𝑓 = 𝐹 → ( ∃ 𝑥 ∈ 𝐻 𝑓 ⇝𝑣 𝑥 ↔ ∃ 𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ) )
6	3 5	imbi12d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 : ℕ ⟶ 𝐻 → ∃ 𝑥 ∈ 𝐻 𝑓 ⇝𝑣 𝑥 ) ↔ ( 𝐹 : ℕ ⟶ 𝐻 → ∃ 𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ) ) )
7	6	rspccv	⊢ ( ∀ 𝑓 ∈ Cauchy ( 𝑓 : ℕ ⟶ 𝐻 → ∃ 𝑥 ∈ 𝐻 𝑓 ⇝𝑣 𝑥 ) → ( 𝐹 ∈ Cauchy → ( 𝐹 : ℕ ⟶ 𝐻 → ∃ 𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ) ) )
8	2 7	syl	⊢ ( 𝐻 ∈ Cℋ → ( 𝐹 ∈ Cauchy → ( 𝐹 : ℕ ⟶ 𝐻 → ∃ 𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ) ) )
9	8	3imp	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐹 ∈ Cauchy ∧ 𝐹 : ℕ ⟶ 𝐻 ) → ∃ 𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 )