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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 24729 | . 2 β’ (π β βHil β (π β Ban β§ π β βPreHil)) | |
2 | 1 | simplbi 499 | 1 β’ (π β βHil β π β Ban) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2107 βPreHilccph 24533 Bancbn 24700 βHilchl 24701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3448 df-in 3918 df-hl 24704 |
This theorem is referenced by: hlcms 24733 hlprlem 24734 cmslsschl 24744 chlcsschl 24745 |
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