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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminflelimsupuz | Structured version Visualization version GIF version |
Description: The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminflelimsupuz.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
liminflelimsupuz.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
liminflelimsupuz.3 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
Ref | Expression |
---|---|
liminflelimsupuz | ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | liminflelimsupuz.3 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
2 | liminflelimsupuz.2 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | 2 | fvexi 6659 | . . . 4 ⊢ 𝑍 ∈ V |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑍 ∈ V) |
5 | 1, 4 | fexd 6967 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
6 | liminflelimsupuz.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | 6, 2 | uzubico2 42207 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)𝑗 ∈ 𝑍) |
8 | 1 | ffnd 6488 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝑍) |
9 | 8 | adantr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 Fn 𝑍) |
10 | simpr 488 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
11 | id 22 | . . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ 𝑍) | |
12 | 2, 11 | uzxrd 42101 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ*) |
13 | pnfxr 10684 | . . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ* | |
14 | 13 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → +∞ ∈ ℝ*) |
15 | 12 | xrleidd 12533 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ≤ 𝑗) |
16 | 2, 11 | uzred 42080 | . . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ) |
17 | ltpnf 12503 | . . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℝ → 𝑗 < +∞) | |
18 | 16, 17 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → 𝑗 < +∞) |
19 | 12, 14, 12, 15, 18 | elicod 12775 | . . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (𝑗[,)+∞)) |
20 | 19 | adantl 485 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (𝑗[,)+∞)) |
21 | 9, 10, 20 | fnfvimad 6974 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (𝐹 “ (𝑗[,)+∞))) |
22 | 1 | ffvelrnda 6828 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ*) |
23 | 21, 22 | elind 4121 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*)) |
24 | 23 | ne0d 4251 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) |
25 | 24 | ex 416 | . . . . . 6 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
26 | 25 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ (𝑘[,)+∞)) → (𝑗 ∈ 𝑍 → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
27 | 26 | reximdva 3233 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (∃𝑗 ∈ (𝑘[,)+∞)𝑗 ∈ 𝑍 → ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
28 | 27 | ralimdva 3144 | . . 3 ⊢ (𝜑 → (∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)𝑗 ∈ 𝑍 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
29 | 7, 28 | mpd 15 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) |
30 | 5, 29 | liminflelimsup 42418 | 1 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ∩ cin 3880 ∅c0 4243 class class class wbr 5030 “ cima 5522 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 +∞cpnf 10661 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 ℤcz 11969 ℤ≥cuz 12231 [,)cico 12728 lim supclsp 14819 lim infclsi 42393 |
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-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-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 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-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-ioo 12730 df-ico 12732 df-fl 13157 df-ceil 13158 df-limsup 14820 df-liminf 42394 |
This theorem is referenced by: liminfgelimsupuz 42430 liminflimsupclim 42449 xlimliminflimsup 42504 |
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