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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 28625 | . 2 ⊢ 𝐻 ⊆ ℋ |
3 | sheli.1 | . 2 ⊢ 𝐴 ∈ 𝐻 | |
4 | 2, 3 | sselii 3824 | 1 ⊢ 𝐴 ∈ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2166 ℋchba 28331 Sℋ csh 28340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-hilex 28411 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-xp 5348 df-cnv 5350 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-sh 28619 |
This theorem is referenced by: omlsilem 28816 |
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