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Mirrors > Home > MPE Home > Th. List > hlphl | Structured version Visualization version GIF version |
Description: Every subcomplex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
hlphl | β’ (π β βHil β π β PreHil) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlcph 25305 | . 2 β’ (π β βHil β π β βPreHil) | |
2 | cphphl 25112 | . 2 β’ (π β βPreHil β π β PreHil) | |
3 | 1, 2 | syl 17 | 1 β’ (π β βHil β π β PreHil) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2099 PreHilcphl 21556 βPreHilccph 25107 βHilchl 25275 |
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 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fv 6556 df-ov 7423 df-cph 25109 df-hl 25278 |
This theorem is referenced by: chlcsschl 25319 pjthlem1 25378 pjth 25380 pjth2 25381 cldcss 25382 hlhil 25384 |
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