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 6679 | . . . 4 ⊢ 𝑍 ∈ V |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑍 ∈ V) |
5 | 1, 4 | fexd 41372 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
6 | liminflelimsupuz.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | 6, 2 | uzubico2 41838 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)𝑗 ∈ 𝑍) |
8 | 1 | ffnd 6510 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝑍) |
9 | 8 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 Fn 𝑍) |
10 | simpr 487 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
11 | id 22 | . . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ 𝑍) | |
12 | 2, 11 | uzxrd 41730 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ*) |
13 | pnfxr 10689 | . . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ* | |
14 | 13 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → +∞ ∈ ℝ*) |
15 | 12 | xrleidd 12539 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ≤ 𝑗) |
16 | 2, 11 | uzred 41709 | . . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ) |
17 | ltpnf 12509 | . . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℝ → 𝑗 < +∞) | |
18 | 16, 17 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → 𝑗 < +∞) |
19 | 12, 14, 12, 15, 18 | elicod 12781 | . . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (𝑗[,)+∞)) |
20 | 19 | adantl 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (𝑗[,)+∞)) |
21 | 9, 10, 20 | fnfvimad 6990 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (𝐹 “ (𝑗[,)+∞))) |
22 | 1 | ffvelrnda 6846 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ*) |
23 | 21, 22 | elind 4171 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*)) |
24 | 23 | ne0d 4301 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) |
25 | 24 | ex 415 | . . . . . 6 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
26 | 25 | ad2antrr 724 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ (𝑘[,)+∞)) → (𝑗 ∈ 𝑍 → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
27 | 26 | reximdva 3274 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (∃𝑗 ∈ (𝑘[,)+∞)𝑗 ∈ 𝑍 → ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
28 | 27 | ralimdva 3177 | . . 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 42049 | 1 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3495 ∩ cin 3935 ∅c0 4291 class class class wbr 5059 “ cima 5553 Fn wfn 6345 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 +∞cpnf 10666 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 ℤcz 11975 ℤ≥cuz 12237 [,)cico 12734 lim supclsp 14821 lim infclsi 42024 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-ioo 12736 df-ico 12738 df-fl 13156 df-ceil 13157 df-limsup 14822 df-liminf 42025 |
This theorem is referenced by: liminfgelimsupuz 42061 liminflimsupclim 42080 xlimliminflimsup 42135 |
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