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Mirrors > Home > MPE Home > Th. List > hlcms | Structured version Visualization version GIF version |
Description: Every subcomplex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.) |
Ref | Expression |
---|---|
hlcms | ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlbn 25418 | . 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban) | |
2 | bncms 25399 | . 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 CMetSpccms 25387 Bancbn 25388 ℂHilchl 25389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-iota 6527 df-fv 6583 df-bn 25391 df-hl 25392 |
This theorem is referenced by: pjthlem2 25493 |
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