![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupgt | Structured version Visualization version GIF version |
Description: Given a sequence of real numbers, there exists an upper part of the sequence that's appxoximated from below by the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
limsupgt.k | ⊢ Ⅎ𝑘𝐹 |
limsupgt.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
limsupgt.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
limsupgt.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
limsupgt.r | ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) |
limsupgt.x | ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
Ref | Expression |
---|---|
limsupgt | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupgt.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | limsupgt.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | limsupgt.f | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
4 | limsupgt.r | . . 3 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) | |
5 | limsupgt.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ+) | |
6 | 1, 2, 3, 4, 5 | limsupgtlem 42419 | . 2 ⊢ (𝜑 → ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)((𝐹‘𝑙) − 𝑋) < (lim sup‘𝐹)) |
7 | limsupgt.k | . . . . . . . . 9 ⊢ Ⅎ𝑘𝐹 | |
8 | nfcv 2955 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝑙 | |
9 | 7, 8 | nffv 6655 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑙) |
10 | nfcv 2955 | . . . . . . . 8 ⊢ Ⅎ𝑘 − | |
11 | nfcv 2955 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑋 | |
12 | 9, 10, 11 | nfov 7165 | . . . . . . 7 ⊢ Ⅎ𝑘((𝐹‘𝑙) − 𝑋) |
13 | nfcv 2955 | . . . . . . 7 ⊢ Ⅎ𝑘 < | |
14 | nfcv 2955 | . . . . . . . 8 ⊢ Ⅎ𝑘lim sup | |
15 | 14, 7 | nffv 6655 | . . . . . . 7 ⊢ Ⅎ𝑘(lim sup‘𝐹) |
16 | 12, 13, 15 | nfbr 5077 | . . . . . 6 ⊢ Ⅎ𝑘((𝐹‘𝑙) − 𝑋) < (lim sup‘𝐹) |
17 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑙((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹) | |
18 | fveq2 6645 | . . . . . . . 8 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘)) | |
19 | 18 | oveq1d 7150 | . . . . . . 7 ⊢ (𝑙 = 𝑘 → ((𝐹‘𝑙) − 𝑋) = ((𝐹‘𝑘) − 𝑋)) |
20 | 19 | breq1d 5040 | . . . . . 6 ⊢ (𝑙 = 𝑘 → (((𝐹‘𝑙) − 𝑋) < (lim sup‘𝐹) ↔ ((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹))) |
21 | 16, 17, 20 | cbvralw 3387 | . . . . 5 ⊢ (∀𝑙 ∈ (ℤ≥‘𝑖)((𝐹‘𝑙) − 𝑋) < (lim sup‘𝐹) ↔ ∀𝑘 ∈ (ℤ≥‘𝑖)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)) |
22 | 21 | a1i 11 | . . . 4 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)((𝐹‘𝑙) − 𝑋) < (lim sup‘𝐹) ↔ ∀𝑘 ∈ (ℤ≥‘𝑖)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹))) |
23 | fveq2 6645 | . . . . 5 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗)) | |
24 | 23 | raleqdv 3364 | . . . 4 ⊢ (𝑖 = 𝑗 → (∀𝑘 ∈ (ℤ≥‘𝑖)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹))) |
25 | 22, 24 | bitrd 282 | . . 3 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)((𝐹‘𝑙) − 𝑋) < (lim sup‘𝐹) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹))) |
26 | 25 | cbvrexvw 3397 | . 2 ⊢ (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)((𝐹‘𝑙) − 𝑋) < (lim sup‘𝐹) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)) |
27 | 6, 26 | sylib 221 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 Ⅎwnfc 2936 ∀wral 3106 ∃wrex 3107 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 < clt 10664 − cmin 10859 ℤcz 11969 ℤ≥cuz 12231 ℝ+crp 12377 lim supclsp 14819 |
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-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
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-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 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-int 4839 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-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-xadd 12496 df-ico 12732 df-fz 12886 df-fzo 13029 df-fl 13157 df-ceil 13158 df-limsup 14820 |
This theorem is referenced by: liminfltlem 42446 liminflimsupclim 42449 |
Copyright terms: Public domain | W3C validator |