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Mirrors > Home > HSE Home > Th. List > shel | Structured version Visualization version GIF version |
Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shel | ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shss 28914 | . 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ) | |
2 | 1 | sselda 3964 | 1 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 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: shuni 29004 shsel3 29019 shscom 29023 shsel1 29025 elspancl 29041 pjpjpre 29123 spansnss 29275 sh1dle 30055 |
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