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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 27451 | . 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐻 ⊆ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ⊆ wss 3540 ℋchil 27160 Sℋ csh 27169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-hilex 27240 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-sh 27448 |
This theorem is referenced by: sheli 27455 shelii 27456 chssii 27472 hhssabloilem 27502 hhssabloi 27503 hhssnv 27505 hhssba 27512 shunssji 27612 shsval3i 27631 shjshsi 27735 span0 27785 spanuni 27787 imaelshi 28301 nlelchi 28304 hmopidmchi 28394 pjimai 28419 shatomistici 28604 |
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