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Mirrors > Home > HSE Home > Th. List > ax-hcompl | Structured version Visualization version GIF version |
Description: Completeness of a Hilbert space. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax-hcompl | ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cF | . . 3 class 𝐹 | |
2 | ccauold 29276 | . . 3 class Cauchy | |
3 | 1, 2 | wcel 2110 | . 2 wff 𝐹 ∈ Cauchy |
4 | vx | . . . . 5 setvar 𝑥 | |
5 | 4 | cv 1541 | . . . 4 class 𝑥 |
6 | chli 29277 | . . . 4 class ⇝𝑣 | |
7 | 1, 5, 6 | wbr 5079 | . . 3 wff 𝐹 ⇝𝑣 𝑥 |
8 | chba 29269 | . . 3 class ℋ | |
9 | 7, 4, 8 | wrex 3067 | . 2 wff ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥 |
10 | 3, 9 | wi 4 | 1 wff (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) |
Colors of variables: wff setvar class |
This axiom is referenced by: hhcms 29553 isch3 29591 hhsscms 29628 occllem 29653 occl 29654 chscllem2 29988 |
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