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Mirrors > Home > HSE Home > Th. List > hlim2 | Structured version Visualization version GIF version |
Description: The limit of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlim2 | ⊢ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝐹 ⇝𝑣 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5034 | . . . . 5 ⊢ (𝑤 = 𝐴 → (𝐹 ⇝𝑣 𝑤 ↔ 𝐹 ⇝𝑣 𝐴)) | |
2 | oveq2 7143 | . . . . . . . . 9 ⊢ (𝑤 = 𝐴 → ((𝐹‘𝑧) −ℎ 𝑤) = ((𝐹‘𝑧) −ℎ 𝐴)) | |
3 | 2 | fveq2d 6649 | . . . . . . . 8 ⊢ (𝑤 = 𝐴 → (normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) = (normℎ‘((𝐹‘𝑧) −ℎ 𝐴))) |
4 | 3 | breq1d 5040 | . . . . . . 7 ⊢ (𝑤 = 𝐴 → ((normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥 ↔ (normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
5 | 4 | rexralbidv 3260 | . . . . . 6 ⊢ (𝑤 = 𝐴 → (∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥 ↔ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
6 | 5 | ralbidv 3162 | . . . . 5 ⊢ (𝑤 = 𝐴 → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
7 | 1, 6 | bibi12d 349 | . . . 4 ⊢ (𝑤 = 𝐴 → ((𝐹 ⇝𝑣 𝑤 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥) ↔ (𝐹 ⇝𝑣 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥))) |
8 | 7 | imbi2d 344 | . . 3 ⊢ (𝑤 = 𝐴 → ((𝐹:ℕ⟶ ℋ → (𝐹 ⇝𝑣 𝑤 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥)) ↔ (𝐹:ℕ⟶ ℋ → (𝐹 ⇝𝑣 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)))) |
9 | vex 3444 | . . . . . 6 ⊢ 𝑤 ∈ V | |
10 | 9 | hlimi 28971 | . . . . 5 ⊢ (𝐹 ⇝𝑣 𝑤 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥)) |
11 | 10 | baib 539 | . . . 4 ⊢ ((𝐹:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) → (𝐹 ⇝𝑣 𝑤 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥)) |
12 | 11 | expcom 417 | . . 3 ⊢ (𝑤 ∈ ℋ → (𝐹:ℕ⟶ ℋ → (𝐹 ⇝𝑣 𝑤 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝑤)) < 𝑥))) |
13 | 8, 12 | vtoclga 3522 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐹:ℕ⟶ ℋ → (𝐹 ⇝𝑣 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥))) |
14 | 13 | impcom 411 | 1 ⊢ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝐹 ⇝𝑣 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 < clt 10664 ℕcn 11625 ℤ≥cuz 12231 ℝ+crp 12377 ℋchba 28702 normℎcno 28706 −ℎ cmv 28708 ⇝𝑣 chli 28710 |
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-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 df-hlim 28755 |
This theorem is referenced by: (None) |
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