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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 28991 | . . 3 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))) | |
2 | 1 | simplbi 501 | . 2 ⊢ (𝐻 ∈ Sℋ → (𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻)) |
3 | 2 | simprd 499 | 1 ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ⊆ wss 3881 × cxp 5517 “ cima 5522 ℂcc 10524 ℋchba 28702 +ℎ cva 28703 ·ℎ csm 28704 0ℎc0v 28707 Sℋ csh 28711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-hilex 28782 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-sh 28990 |
This theorem is referenced by: ch0 29011 hhssabloilem 29044 hhssnv 29047 oc0 29073 ocin 29079 shscli 29100 shsel1 29104 shintcli 29112 shunssi 29151 omlsii 29186 sh0le 29223 imaelshi 29841 shatomistici 30144 |
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