Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
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 28914 | . 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐻 ⊆ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ⊆ wss 3933 ℋchba 28623 Sℋ csh 28632 |
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-sep 5194 ax-hilex 28703 |
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-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-sh 28911 |
This theorem is referenced by: sheli 28918 shelii 28919 chssii 28935 hhssabloilem 28965 hhssabloi 28966 hhssnv 28968 hhssba 28975 shunssji 29073 shsval3i 29092 shjshsi 29196 span0 29246 spanuni 29248 imaelshi 29762 nlelchi 29765 hmopidmchi 29855 pjimai 29880 shatomistici 30065 |
Copyright terms: Public domain | W3C validator |