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Mirrors > Home > HSE Home > Th. List > hlimi | Structured version Visualization version GIF version |
Description: Express the predicate: The limit of vector sequence 𝐹 in a Hilbert space is 𝐴, i.e. 𝐹 converges to 𝐴. This means that for any real 𝑥, no matter how small, there always exists an integer 𝑦 such that the norm of any later vector in the sequence minus the limit is less than 𝑥. Definition of converge in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlim.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
hlimi | ⊢ (𝐹 ⇝𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-hlim 30802 | . . . 4 ⊢ ⇝𝑣 = {〈𝑓, 𝑤〉 ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥)} | |
2 | 1 | relopabiv 5826 | . . 3 ⊢ Rel ⇝𝑣 |
3 | 2 | brrelex1i 5738 | . 2 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ V) |
4 | nnex 12256 | . . . 4 ⊢ ℕ ∈ V | |
5 | fex 7244 | . . . 4 ⊢ ((𝐹:ℕ⟶ ℋ ∧ ℕ ∈ V) → 𝐹 ∈ V) | |
6 | 4, 5 | mpan2 689 | . . 3 ⊢ (𝐹:ℕ⟶ ℋ → 𝐹 ∈ V) |
7 | 6 | ad2antrr 724 | . 2 ⊢ (((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥) → 𝐹 ∈ V) |
8 | hlim.1 | . . 3 ⊢ 𝐴 ∈ V | |
9 | feq1 6708 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓:ℕ⟶ ℋ ↔ 𝐹:ℕ⟶ ℋ)) | |
10 | eleq1 2817 | . . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ ℋ ↔ 𝐴 ∈ ℋ)) | |
11 | 9, 10 | bi2anan9 636 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ↔ (𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ))) |
12 | fveq1 6901 | . . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑧) = (𝐹‘𝑧)) | |
13 | oveq12 7435 | . . . . . . . . . 10 ⊢ (((𝑓‘𝑧) = (𝐹‘𝑧) ∧ 𝑤 = 𝐴) → ((𝑓‘𝑧) −ℎ 𝑤) = ((𝐹‘𝑧) −ℎ 𝐴)) | |
14 | 12, 13 | sylan 578 | . . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → ((𝑓‘𝑧) −ℎ 𝑤) = ((𝐹‘𝑧) −ℎ 𝐴)) |
15 | 14 | fveq2d 6906 | . . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → (normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) = (normℎ‘((𝐹‘𝑧) −ℎ 𝐴))) |
16 | 15 | breq1d 5162 | . . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → ((normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥 ↔ (normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
17 | 16 | rexralbidv 3218 | . . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → (∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥 ↔ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
18 | 17 | ralbidv 3175 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 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)) |
19 | 11, 18 | anbi12d 630 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → (((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥) ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥))) |
20 | 19, 1 | brabga 5540 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝐴 ∈ V) → (𝐹 ⇝𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥))) |
21 | 8, 20 | mpan2 689 | . 2 ⊢ (𝐹 ∈ V → (𝐹 ⇝𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥))) |
22 | 3, 7, 21 | pm5.21nii 377 | 1 ⊢ (𝐹 ⇝𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ∃wrex 3067 Vcvv 3473 class class class wbr 5152 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 < clt 11286 ℕcn 12250 ℤ≥cuz 12860 ℝ+crp 13014 ℋchba 30749 normℎcno 30753 −ℎ cmv 30755 ⇝𝑣 chli 30757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-1cn 11204 ax-addcl 11206 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-nn 12251 df-hlim 30802 |
This theorem is referenced by: hlimseqi 31019 hlimveci 31020 hlimconvi 31021 hlim2 31022 |
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