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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 25236 | . 2 β’ (π β βHil β π β βPreHil) | |
2 | cphphl 25043 | . 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 2098 PreHilcphl 21506 βPreHilccph 25038 βHilchl 25206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-nul 5297 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-rab 3425 df-v 3468 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-xp 5673 df-cnv 5675 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fv 6542 df-ov 7405 df-cph 25040 df-hl 25209 |
This theorem is referenced by: chlcsschl 25250 pjthlem1 25309 pjth 25311 pjth2 25312 cldcss 25313 hlhil 25315 |
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