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 42963 | . 2 ⊢ (𝜑 → ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋)) |
7 | fveq2 6695 | . . . . 5 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗)) | |
8 | 7 | raleqdv 3315 | . . . 4 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋))) |
9 | nfcv 2897 | . . . . . . . 8 ⊢ Ⅎ𝑘lim inf | |
10 | liminflt.k | . . . . . . . 8 ⊢ Ⅎ𝑘𝐹 | |
11 | 9, 10 | nffv 6705 | . . . . . . 7 ⊢ Ⅎ𝑘(lim inf‘𝐹) |
12 | nfcv 2897 | . . . . . . 7 ⊢ Ⅎ𝑘 < | |
13 | nfcv 2897 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝑙 | |
14 | 10, 13 | nffv 6705 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑙) |
15 | nfcv 2897 | . . . . . . . 8 ⊢ Ⅎ𝑘 + | |
16 | nfcv 2897 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑋 | |
17 | 14, 15, 16 | nfov 7221 | . . . . . . 7 ⊢ Ⅎ𝑘((𝐹‘𝑙) + 𝑋) |
18 | 11, 12, 17 | nfbr 5086 | . . . . . 6 ⊢ Ⅎ𝑘(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) |
19 | nfv 1922 | . . . . . 6 ⊢ Ⅎ𝑙(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋) | |
20 | fveq2 6695 | . . . . . . . 8 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘)) | |
21 | 20 | oveq1d 7206 | . . . . . . 7 ⊢ (𝑙 = 𝑘 → ((𝐹‘𝑙) + 𝑋) = ((𝐹‘𝑘) + 𝑋)) |
22 | 21 | breq2d 5051 | . . . . . 6 ⊢ (𝑙 = 𝑘 → ((lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) ↔ (lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋))) |
23 | 18, 19, 22 | cbvralw 3339 | . . . . 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 282 | . . 3 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋))) |
26 | 25 | cbvrexvw 3349 | . 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 221 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 Ⅎwnfc 2877 ∀wral 3051 ∃wrex 3052 class class class wbr 5039 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℝcr 10693 + caddc 10697 < clt 10832 ℤcz 12141 ℤ≥cuz 12403 ℝ+crp 12551 lim infclsi 42910 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-n0 12056 df-z 12142 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-ico 12906 df-fz 13061 df-fzo 13204 df-fl 13332 df-ceil 13333 df-limsup 14997 df-liminf 42911 |
This theorem is referenced by: liminflimsupclim 42966 |
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