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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 43977 | . 2 ⊢ (𝜑 → ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋)) |
7 | fveq2 6839 | . . . . 5 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗)) | |
8 | 7 | raleqdv 3311 | . . . 4 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋))) |
9 | nfcv 2905 | . . . . . . . 8 ⊢ Ⅎ𝑘lim inf | |
10 | liminflt.k | . . . . . . . 8 ⊢ Ⅎ𝑘𝐹 | |
11 | 9, 10 | nffv 6849 | . . . . . . 7 ⊢ Ⅎ𝑘(lim inf‘𝐹) |
12 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑘 < | |
13 | nfcv 2905 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝑙 | |
14 | 10, 13 | nffv 6849 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑙) |
15 | nfcv 2905 | . . . . . . . 8 ⊢ Ⅎ𝑘 + | |
16 | nfcv 2905 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑋 | |
17 | 14, 15, 16 | nfov 7383 | . . . . . . 7 ⊢ Ⅎ𝑘((𝐹‘𝑙) + 𝑋) |
18 | 11, 12, 17 | nfbr 5150 | . . . . . 6 ⊢ Ⅎ𝑘(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) |
19 | nfv 1917 | . . . . . 6 ⊢ Ⅎ𝑙(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋) | |
20 | fveq2 6839 | . . . . . . . 8 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘)) | |
21 | 20 | oveq1d 7368 | . . . . . . 7 ⊢ (𝑙 = 𝑘 → ((𝐹‘𝑙) + 𝑋) = ((𝐹‘𝑘) + 𝑋)) |
22 | 21 | breq2d 5115 | . . . . . 6 ⊢ (𝑙 = 𝑘 → ((lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) ↔ (lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋))) |
23 | 18, 19, 22 | cbvralw 3287 | . . . . 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 278 | . . 3 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋))) |
26 | 25 | cbvrexvw 3224 | . 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 217 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 Ⅎwnfc 2885 ∀wral 3062 ∃wrex 3071 class class class wbr 5103 ⟶wf 6489 ‘cfv 6493 (class class class)co 7353 ℝcr 11046 + caddc 11050 < clt 11185 ℤcz 12495 ℤ≥cuz 12759 ℝ+crp 12907 lim infclsi 43924 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9374 df-inf 9375 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-n0 12410 df-z 12496 df-uz 12760 df-q 12866 df-rp 12908 df-xneg 13025 df-xadd 13026 df-ico 13262 df-fz 13417 df-fzo 13560 df-fl 13689 df-ceil 13690 df-limsup 15345 df-liminf 43925 |
This theorem is referenced by: liminflimsupclim 43980 |
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