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| Mirrors > Home > HSE Home > Th. List > cheli | Structured version Visualization version GIF version | ||
| Description: A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| chssi.1 | ⊢ 𝐻 ∈ Cℋ | 
| Ref | Expression | 
|---|---|
| cheli | ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | chssi.1 | . . 3 ⊢ 𝐻 ∈ Cℋ | |
| 2 | 1 | chssii 31250 | . 2 ⊢ 𝐻 ⊆ ℋ | 
| 3 | 2 | sseli 3979 | 1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 ℋchba 30938 Cℋ cch 30948 | 
| 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-uni 4908 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-iota 6514 df-fv 6569 df-ov 7434 df-sh 31226 df-ch 31240 | 
| This theorem is referenced by: pjhthlem1 31410 pjhthlem2 31411 h1de2ci 31575 spanunsni 31598 spansncvi 31671 3oalem1 31681 pjcompi 31691 pjocini 31717 pjjsi 31719 pjrni 31721 pjdsi 31731 pjds3i 31732 mayete3i 31747 riesz3i 32081 pjnmopi 32167 pjnormssi 32187 pjimai 32195 pjclem4a 32217 pjclem4 32218 pj3lem1 32225 pj3si 32226 strlem1 32269 strlem3 32272 strlem5 32274 hstrlem3 32280 hstrlem5 32282 sumdmdii 32434 sumdmdlem 32437 sumdmdlem2 32438 | 
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