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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 23649 | . 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban) | |
2 | bncms 23630 | . 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 2081 CMetSpccms 23618 Bancbn 23619 ℂHilchl 23620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-iota 6189 df-fv 6233 df-bn 23622 df-hl 23623 |
This theorem is referenced by: pjthlem2 23724 |
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