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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 25278 | . 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban) | |
2 | bncms 25259 | . 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 2099 CMetSpccms 25247 Bancbn 25248 ℂHilchl 25249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-iota 6494 df-fv 6550 df-bn 25251 df-hl 25252 |
This theorem is referenced by: pjthlem2 25353 |
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