Axiom ax-hcompl 31810

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 31534	. . 3 class Cauchy
3	1, 2	wcel 2144	. 2 wff 𝐹 ∈ Cauchy
4		vx	. . . . 5 setvar 𝑥
5	4	cv 1568	. . . 4 class 𝑥
6		chli 31535	. . . 4 class ⇝𝑣
7	1, 5, 6	wbr 5102	. . 3 wff 𝐹 ⇝𝑣 𝑥
8		chba 31527	. . 3 class ℋ
9	7, 4, 8	wrex 3086	. 2 wff ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥
10	3, 9	wi 4	1 wff (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥)

This axiom is used by:  hhcms  31811  isch3  31849  hhsscms  31886  occllem  31911  occl  31912  chscllem2  32246