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Mirrors > Home > MPE Home > Th. List > hlbn | Structured version Visualization version GIF version |
Description: Every subcomplex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) |
Ref | Expression |
---|---|
hlbn | β’ (π β βHil β π β Ban) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishl 25283 | . 2 β’ (π β βHil β (π β Ban β§ π β βPreHil)) | |
2 | 1 | simplbi 497 | 1 β’ (π β βHil β π β Ban) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2099 βPreHilccph 25087 Bancbn 25254 βHilchl 25255 |
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 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3472 df-in 3952 df-hl 25258 |
This theorem is referenced by: hlcms 25287 hlprlem 25288 cmslsschl 25298 chlcsschl 25299 |
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