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Mirrors > Home > HSE Home > Th. List > hcau | Structured version Visualization version GIF version |
Description: Member of the set of Cauchy sequences on a Hilbert space. Definition for Cauchy sequence in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hcau | ⊢ (𝐹 ∈ Cauchy ↔ (𝐹:ℕ⟶ ℋ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6716 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦)) | |
2 | fveq1 6716 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑧) = (𝐹‘𝑧)) | |
3 | 1, 2 | oveq12d 7231 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑦) −ℎ (𝑓‘𝑧)) = ((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) |
4 | 3 | fveq2d 6721 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (normℎ‘((𝑓‘𝑦) −ℎ (𝑓‘𝑧))) = (normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧)))) |
5 | 4 | breq1d 5063 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((normℎ‘((𝑓‘𝑦) −ℎ (𝑓‘𝑧))) < 𝑥 ↔ (normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥)) |
6 | 5 | rexralbidv 3220 | . . . 4 ⊢ (𝑓 = 𝐹 → (∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑦) −ℎ (𝑓‘𝑧))) < 𝑥 ↔ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥)) |
7 | 6 | ralbidv 3118 | . . 3 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑦) −ℎ (𝑓‘𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥)) |
8 | df-hcau 29054 | . . 3 ⊢ Cauchy = {𝑓 ∈ ( ℋ ↑m ℕ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑦) −ℎ (𝑓‘𝑧))) < 𝑥} | |
9 | 7, 8 | elrab2 3605 | . 2 ⊢ (𝐹 ∈ Cauchy ↔ (𝐹 ∈ ( ℋ ↑m ℕ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥)) |
10 | ax-hilex 29080 | . . . 4 ⊢ ℋ ∈ V | |
11 | nnex 11836 | . . . 4 ⊢ ℕ ∈ V | |
12 | 10, 11 | elmap 8552 | . . 3 ⊢ (𝐹 ∈ ( ℋ ↑m ℕ) ↔ 𝐹:ℕ⟶ ℋ) |
13 | 12 | anbi1i 627 | . 2 ⊢ ((𝐹 ∈ ( ℋ ↑m ℕ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥) ↔ (𝐹:ℕ⟶ ℋ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥)) |
14 | 9, 13 | bitri 278 | 1 ⊢ (𝐹 ∈ Cauchy ↔ (𝐹:ℕ⟶ ℋ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 class class class wbr 5053 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ↑m cmap 8508 < clt 10867 ℕcn 11830 ℤ≥cuz 12438 ℝ+crp 12586 ℋchba 29000 normℎcno 29004 −ℎ cmv 29006 Cauchyccauold 29007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-1cn 10787 ax-addcl 10789 ax-hilex 29080 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-map 8510 df-nn 11831 df-hcau 29054 |
This theorem is referenced by: hcauseq 29266 hcaucvg 29267 seq1hcau 29268 chscllem2 29719 |
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