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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 | ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlcph 23894 | . 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil) | |
2 | cphphl 23702 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 PreHilcphl 20696 ℂPreHilccph 23697 ℂHilchl 23864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fv 6356 df-ov 7148 df-cph 23699 df-hl 23867 |
This theorem is referenced by: chlcsschl 23908 pjthlem1 23967 pjth 23969 pjth2 23970 cldcss 23971 hlhil 23973 |
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