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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 31236 | . . 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 2105 ⊆ wss 3962 × cxp 5686 “ cima 5691 ℂcc 11150 ℋchba 30947 +ℎ cva 30948 ·ℎ csm 30949 0ℎc0v 30952 Sℋ csh 30956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-hilex 31027 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-sh 31235 |
This theorem is referenced by: ch0 31256 hhssabloilem 31289 hhssnv 31292 oc0 31318 ocin 31324 shscli 31345 shsel1 31349 shintcli 31357 shunssi 31396 omlsii 31431 sh0le 31468 imaelshi 32086 shatomistici 32389 |
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