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 40036 | . 2 ⊢ (𝜑 → ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋)) |
7 | fveq2 6191 | . . . . 5 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗)) | |
8 | 7 | raleqdv 3144 | . . . 4 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋))) |
9 | nfcv 2764 | . . . . . . . 8 ⊢ Ⅎ𝑘lim inf | |
10 | liminflt.k | . . . . . . . 8 ⊢ Ⅎ𝑘𝐹 | |
11 | 9, 10 | nffv 6198 | . . . . . . 7 ⊢ Ⅎ𝑘(lim inf‘𝐹) |
12 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑘 < | |
13 | nfcv 2764 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝑙 | |
14 | 10, 13 | nffv 6198 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑙) |
15 | nfcv 2764 | . . . . . . . 8 ⊢ Ⅎ𝑘 + | |
16 | nfcv 2764 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑋 | |
17 | 14, 15, 16 | nfov 6676 | . . . . . . 7 ⊢ Ⅎ𝑘((𝐹‘𝑙) + 𝑋) |
18 | 11, 12, 17 | nfbr 4699 | . . . . . 6 ⊢ Ⅎ𝑘(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) |
19 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑙(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋) | |
20 | fveq2 6191 | . . . . . . . 8 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘)) | |
21 | 20 | oveq1d 6665 | . . . . . . 7 ⊢ (𝑙 = 𝑘 → ((𝐹‘𝑙) + 𝑋) = ((𝐹‘𝑘) + 𝑋)) |
22 | 21 | breq2d 4665 | . . . . . 6 ⊢ (𝑙 = 𝑘 → ((lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) ↔ (lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋))) |
23 | 18, 19, 22 | cbvral 3167 | . . . . 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 268 | . . 3 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(lim inf‘𝐹) < ((𝐹‘𝑙) + 𝑋) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋))) |
26 | 25 | cbvrexv 3172 | . 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 208 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 Ⅎwnfc 2751 ∀wral 2912 ∃wrex 2913 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 + caddc 9939 < clt 10074 ℤcz 11377 ℤ≥cuz 11687 ℝ+crp 11832 lim infclsi 39983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-ico 12181 df-fz 12327 df-fzo 12466 df-fl 12593 df-ceil 12594 df-limsup 14202 df-liminf 39984 |
This theorem is referenced by: liminflimsupclim 40039 |
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