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Mirrors > Home > HSE Home > Th. List > shss | Structured version Visualization version GIF version |
Description: A subspace is a subset of Hilbert space. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shss | ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issh 28913 | . . 3 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))) | |
2 | 1 | simplbi 498 | . 2 ⊢ (𝐻 ∈ Sℋ → (𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻)) |
3 | 2 | simpld 495 | 1 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ⊆ wss 3935 × cxp 5547 “ cima 5552 ℂcc 10524 ℋchba 28624 +ℎ cva 28625 ·ℎ csm 28626 0ℎc0v 28629 Sℋ csh 28633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-hilex 28704 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-br 5059 df-opab 5121 df-xp 5555 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-sh 28912 |
This theorem is referenced by: shel 28916 shex 28917 shssii 28918 shsubcl 28925 chss 28934 shsspwh 28951 hhsssh 28974 shocel 28987 shocsh 28989 ocss 28990 shocss 28991 shocorth 28997 shococss 28999 shorth 29000 shoccl 29010 shsel 29019 shintcli 29034 spanid 29052 shjval 29056 shjcl 29061 shlej1 29065 shlub 29119 chscllem2 29343 chscllem4 29345 |
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