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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 28590 | . . 3 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))) | |
2 | 1 | simplbi 492 | . 2 ⊢ (𝐻 ∈ Sℋ → (𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻)) |
3 | 2 | simpld 489 | 1 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ⊆ wss 3769 × cxp 5310 “ cima 5315 ℂcc 10222 ℋchba 28301 +ℎ cva 28302 ·ℎ csm 28303 0ℎc0v 28306 Sℋ csh 28310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-hilex 28381 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-xp 5318 df-cnv 5320 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-sh 28589 |
This theorem is referenced by: shel 28593 shex 28594 shssii 28595 shsubcl 28602 chss 28611 shsspwh 28628 hhsssh 28651 shocel 28666 shocsh 28668 ocss 28669 shocss 28670 shocorth 28676 shococss 28678 shorth 28679 shoccl 28689 shsel 28698 shintcli 28713 spanid 28731 shjval 28735 shjcl 28740 shlej1 28744 shlub 28798 chscllem2 29022 chscllem4 29024 |
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