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| Mirrors > Home > HSE Home > Th. List > shelii | Structured version Visualization version GIF version | ||
| Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| shssi.1 | ⊢ 𝐻 ∈ Sℋ | 
| sheli.1 | ⊢ 𝐴 ∈ 𝐻 | 
| Ref | Expression | 
|---|---|
| shelii | ⊢ 𝐴 ∈ ℋ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | shssi.1 | . . 3 ⊢ 𝐻 ∈ Sℋ | |
| 2 | 1 | shssii 31232 | . 2 ⊢ 𝐻 ⊆ ℋ | 
| 3 | sheli.1 | . 2 ⊢ 𝐴 ∈ 𝐻 | |
| 4 | 2, 3 | sselii 3980 | 1 ⊢ 𝐴 ∈ ℋ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2108 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: omlsilem 31421 | 
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