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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 27449 | . . 3 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))) | |
2 | 1 | simplbi 475 | . 2 ⊢ (𝐻 ∈ Sℋ → (𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻)) |
3 | 2 | simprd 478 | 1 ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ⊆ wss 3540 × cxp 5036 “ cima 5041 ℂcc 9813 ℋchil 27160 +ℎ cva 27161 ·ℎ csm 27162 0ℎc0v 27165 Sℋ csh 27169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-hilex 27240 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-sh 27448 |
This theorem is referenced by: ch0 27469 hhssabloilem 27502 hhssnv 27505 oc0 27533 ocin 27539 shscli 27560 shsel1 27564 shintcli 27572 shunssi 27611 omlsii 27646 sh0le 27683 imaelshi 28301 shatomistici 28604 |
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