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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 30439 | . . 3 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))) | |
2 | 1 | simplbi 499 | . 2 ⊢ (𝐻 ∈ Sℋ → (𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻)) |
3 | 2 | simpld 496 | 1 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3947 × cxp 5673 “ cima 5678 ℂcc 11104 ℋchba 30150 +ℎ cva 30151 ·ℎ csm 30152 0ℎc0v 30155 Sℋ csh 30159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5298 ax-hilex 30230 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-cnv 5683 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-sh 30438 |
This theorem is referenced by: shel 30442 shex 30443 shssii 30444 shsubcl 30451 chss 30460 shsspwh 30477 hhsssh 30500 shocel 30513 shocsh 30515 ocss 30516 shocss 30517 shocorth 30523 shococss 30525 shorth 30526 shoccl 30536 shsel 30545 shintcli 30560 spanid 30578 shjval 30582 shjcl 30587 shlej1 30591 shlub 30645 chscllem2 30869 chscllem4 30871 |
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