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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 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 | simpld 498 | 1 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ 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: shel 28994 shex 28995 shssii 28996 shsubcl 29003 chss 29012 shsspwh 29029 hhsssh 29052 shocel 29065 shocsh 29067 ocss 29068 shocss 29069 shocorth 29075 shococss 29077 shorth 29078 shoccl 29088 shsel 29097 shintcli 29112 spanid 29130 shjval 29134 shjcl 29139 shlej1 29143 shlub 29197 chscllem2 29421 chscllem4 29423 |
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