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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 24731 | . 2 β’ (π β βHil β π β βPreHil) | |
2 | cphphl 24538 | . 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 2107 PreHilcphl 21031 βPreHilccph 24533 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 ax-nul 5264 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-rab 3409 df-v 3448 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-xp 5640 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fv 6505 df-ov 7361 df-cph 24535 df-hl 24704 |
This theorem is referenced by: chlcsschl 24745 pjthlem1 24804 pjth 24806 pjth2 24807 cldcss 24808 hlhil 24810 |
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