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| 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 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 | simpld 494 | 1 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ 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: shel 31230 shex 31231 shssii 31232 shsubcl 31239 chss 31248 shsspwh 31265 hhsssh 31288 shocel 31301 shocsh 31303 ocss 31304 shocss 31305 shocorth 31311 shococss 31313 shorth 31314 shoccl 31324 shsel 31333 shintcli 31348 spanid 31366 shjval 31370 shjcl 31375 shlej1 31379 shlub 31433 chscllem2 31657 chscllem4 31659 | 
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