| Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > HSE Home > Th. List > sh0 | Structured version Visualization version GIF version | ||
| Description: The zero vector belongs to any subspace of a Hilbert space. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| sh0 | ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | issh 31227 | . . 3 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))) | |
| 2 | 1 | simplbi 497 | . 2 ⊢ (𝐻 ∈ Sℋ → (𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻)) |
| 3 | 2 | simprd 495 | 1 ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 × cxp 5683 “ cima 5688 ℂcc 11153 ℋchba 30938 +ℎ cva 30939 ·ℎ csm 30940 0ℎc0v 30943 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: ch0 31247 hhssabloilem 31280 hhssnv 31283 oc0 31309 ocin 31315 shscli 31336 shsel1 31340 shintcli 31348 shunssi 31387 omlsii 31422 sh0le 31459 imaelshi 32077 shatomistici 32380 |
| Copyright terms: Public domain | W3C validator |