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Mirrors > Home > MPE Home > Th. List > hlcph | Structured version Visualization version GIF version |
Description: Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
hlcph | โข (๐ โ โHil โ ๐ โ โPreHil) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishl 25289 | . 2 โข (๐ โ โHil โ (๐ โ Ban โง ๐ โ โPreHil)) | |
2 | 1 | simprbi 496 | 1 โข (๐ โ โHil โ ๐ โ โPreHil) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wcel 2099 โPreHilccph 25093 Bancbn 25260 โHilchl 25261 |
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 3473 df-in 3954 df-hl 25264 |
This theorem is referenced by: hlphl 25292 hlprlem 25294 cmslsschl 25304 chlcsschl 25305 pjthlem1 25364 pjthlem2 25365 cldcss 25368 |
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