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Mirrors > Home > HSE Home > Th. List > hlimveci | Structured version Visualization version GIF version |
Description: Closure of the limit of a sequence on Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlim.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
hlimveci | ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐴 ∈ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlim.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | 1 | hlimi 28617 | . . 3 ⊢ (𝐹 ⇝𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
3 | 2 | simplbi 493 | . 2 ⊢ (𝐹 ⇝𝑣 𝐴 → (𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ)) |
4 | 3 | simprd 491 | 1 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐴 ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2106 ∀wral 3089 ∃wrex 3090 Vcvv 3397 class class class wbr 4886 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 < clt 10411 ℕcn 11374 ℤ≥cuz 11992 ℝ+crp 12137 ℋchba 28348 normℎcno 28352 −ℎ cmv 28354 ⇝𝑣 chli 28356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-1cn 10330 ax-addcl 10332 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-nn 11375 df-hlim 28401 |
This theorem is referenced by: hlimf 28666 helch 28672 occllem 28734 nlelchi 29492 hmopidmchi 29582 |
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