Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminflt | Structured version Visualization version GIF version |
Description: Given a sequence of real numbers, there exists an upper part of the sequence that's approximated from above by the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminflt.k | ⊢ Ⅎ𝑘𝐹 |
liminflt.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
liminflt.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
liminflt.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
liminflt.r | ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ) |
liminflt.x | ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
Ref | Expression |
---|---|
liminflt | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | liminflt.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | liminflt.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | liminflt.f | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
4 | liminflt.r | . . 3 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ) | |
5 | liminflt.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ+) | |
6 | 1, 2, 3, 4, 5 | liminfltlem 42083 | . 2 ⊢ (𝜑 → ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋)) |
7 | fveq2 6669 | . . . . 5 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗)) | |
8 | 7 | raleqdv 3415 | . . . 4 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋))) |
9 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑘lim inf | |
10 | liminflt.k | . . . . . . . 8 ⊢ Ⅎ𝑘𝐹 | |
11 | 9, 10 | nffv 6679 | . . . . . . 7 ⊢ Ⅎ𝑘(lim inf‘𝐹) |
12 | nfcv 2977 | . . . . . . 7 ⊢ Ⅎ𝑘 < | |
13 | nfcv 2977 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝑙 | |
14 | 10, 13 | nffv 6679 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑙) |
15 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑘 + | |
16 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑋 | |
17 | 14, 15, 16 | nfov 7185 | . . . . . . 7 ⊢ Ⅎ𝑘((𝐹‘𝑙) + 𝑋) |
18 | 11, 12, 17 | nfbr 5112 | . . . . . 6 ⊢ Ⅎ𝑘(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) |
19 | nfv 1911 | . . . . . 6 ⊢ Ⅎ𝑙(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋) | |
20 | fveq2 6669 | . . . . . . . 8 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘)) | |
21 | 20 | oveq1d 7170 | . . . . . . 7 ⊢ (𝑙 = 𝑘 → ((𝐹‘𝑙) + 𝑋) = ((𝐹‘𝑘) + 𝑋)) |
22 | 21 | breq2d 5077 | . . . . . 6 ⊢ (𝑙 = 𝑘 → ((lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) ↔ (lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋))) |
23 | 18, 19, 22 | cbvralw 3441 | . . . . 5 ⊢ (∀𝑙 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋)) |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋))) |
25 | 8, 24 | bitrd 281 | . . 3 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋))) |
26 | 25 | cbvrexvw 3450 | . 2 ⊢ (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋)) |
27 | 6, 26 | sylib 220 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 Ⅎwnfc 2961 ∀wral 3138 ∃wrex 3139 class class class wbr 5065 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ℝcr 10535 + caddc 10539 < clt 10674 ℤcz 11980 ℤ≥cuz 12242 ℝ+crp 12388 lim infclsi 42030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-ico 12743 df-fz 12892 df-fzo 13033 df-fl 13161 df-ceil 13162 df-limsup 14827 df-liminf 42031 |
This theorem is referenced by: liminflimsupclim 42086 |
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