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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 27213 | . . . 4 ⊢ ⇝𝑣 = {〈𝑓, 𝑤〉 ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥)} | |
2 | 1 | relopabi 5167 | . . 3 ⊢ Rel ⇝𝑣 |
3 | 2 | brrelexi 5082 | . 2 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ V) |
4 | nnex 10903 | . . . 4 ⊢ ℕ ∈ V | |
5 | fex 6394 | . . . 4 ⊢ ((𝐹:ℕ⟶ ℋ ∧ ℕ ∈ V) → 𝐹 ∈ V) | |
6 | 4, 5 | mpan2 703 | . . 3 ⊢ (𝐹:ℕ⟶ ℋ → 𝐹 ∈ V) |
7 | 6 | ad2antrr 758 | . 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 5939 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓:ℕ⟶ ℋ ↔ 𝐹:ℕ⟶ ℋ)) | |
10 | eleq1 2676 | . . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ ℋ ↔ 𝐴 ∈ ℋ)) | |
11 | 9, 10 | bi2anan9 913 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ↔ (𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ))) |
12 | fveq1 6102 | . . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑧) = (𝐹‘𝑧)) | |
13 | oveq12 6558 | . . . . . . . . . 10 ⊢ (((𝑓‘𝑧) = (𝐹‘𝑧) ∧ 𝑤 = 𝐴) → ((𝑓‘𝑧) −ℎ 𝑤) = ((𝐹‘𝑧) −ℎ 𝐴)) | |
14 | 12, 13 | sylan 487 | . . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → ((𝑓‘𝑧) −ℎ 𝑤) = ((𝐹‘𝑧) −ℎ 𝐴)) |
15 | 14 | fveq2d 6107 | . . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → (normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) = (normℎ‘((𝐹‘𝑧) −ℎ 𝐴))) |
16 | 15 | breq1d 4593 | . . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → ((normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥 ↔ (normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) |
17 | 16 | rexralbidv 3040 | . . . . . 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 2969 | . . . . 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 743 | . . . 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 4914 | . . 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 703 | . 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 367 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 < clt 9953 ℕcn 10897 ℤ≥cuz 11563 ℝ+crp 11708 ℋchil 27160 normℎcno 27164 −ℎ cmv 27166 ⇝𝑣 chli 27168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-hlim 27213 |
This theorem is referenced by: hlimseqi 27430 hlimveci 27431 hlimconvi 27432 hlim2 27433 |
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