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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 25398 | . 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil) | |
| 2 | cphphl 25205 | . 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 2108 PreHilcphl 21642 ℂPreHilccph 25200 ℂHilchl 25368 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fv 6569 df-ov 7434 df-cph 25202 df-hl 25371 |
| This theorem is referenced by: chlcsschl 25412 pjthlem1 25471 pjth 25473 pjth2 25474 cldcss 25475 hlhil 25477 |
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