![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
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 31239 | . 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ) | |
2 | 1 | sselda 3995 | 1 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ℋchba 30948 Sℋ csh 30957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-hilex 31028 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-sh 31236 |
This theorem is referenced by: shuni 31329 shsel3 31344 shscom 31348 shsel1 31350 elspancl 31366 pjpjpre 31448 spansnss 31600 sh1dle 32380 |
Copyright terms: Public domain | W3C validator |