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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 25409 | . 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban) | |
2 | bncms 25390 | . 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 2103 CMetSpccms 25378 Bancbn 25379 ℂHilchl 25380 |
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 2105 ax-9 2113 ax-ext 2705 |
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 2712 df-cleq 2726 df-clel 2813 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-iota 6524 df-fv 6580 df-bn 25382 df-hl 25383 |
This theorem is referenced by: pjthlem2 25484 |
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