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| Mirrors > Home > HSE Home > Th. List > shssii | Structured version Visualization version GIF version | ||
| Description: A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| shssi.1 | ⊢ 𝐻 ∈ Sℋ |
| Ref | Expression |
|---|---|
| shssii | ⊢ 𝐻 ⊆ ℋ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | shssi.1 | . 2 ⊢ 𝐻 ∈ Sℋ | |
| 2 | shss 31229 | . 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐻 ⊆ ℋ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ⊆ wss 3951 ℋchba 30938 Sℋ csh 30947 |
| 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-sep 5296 ax-hilex 31018 |
| 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-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-sh 31226 |
| This theorem is referenced by: sheli 31233 shelii 31234 chssii 31250 hhssabloilem 31280 hhssabloi 31281 hhssnv 31283 hhssba 31290 shunssji 31388 shsval3i 31407 shjshsi 31511 span0 31561 spanuni 31563 imaelshi 32077 nlelchi 32080 hmopidmchi 32170 pjimai 32195 shatomistici 32380 |
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