Axiom ax-hcompl 30443

Description: Completeness of a Hilbert space. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-hcompl	⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥)

Distinct variable group:   𝑥,𝐹

Detailed syntax breakdown of Axiom ax-hcompl

Step	Hyp	Ref	Expression
1		cF	. . 3 class 𝐹
2		ccauold 30167	. . 3 class Cauchy
3	1, 2	wcel 2107	. 2 wff 𝐹 ∈ Cauchy
4		vx	. . . . 5 setvar 𝑥
5	4	cv 1541	. . . 4 class 𝑥
6		chli 30168	. . . 4 class ⇝𝑣
7	1, 5, 6	wbr 5148	. . 3 wff 𝐹 ⇝𝑣 𝑥
8		chba 30160	. . 3 class ℋ
9	7, 4, 8	wrex 3071	. 2 wff ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥
10	3, 9	wi 4	1 wff (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥)

This axiom is referenced by:  hhcms  30444  isch3  30482  hhsscms  30519  occllem  30544  occl  30545  chscllem2  30879