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 6202 | . . . 4 ⊢ 𝑍 ∈ V |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑍 ∈ V) |
5 | 1, 4 | fexd 39296 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
6 | liminflelimsupuz.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | 6, 2 | uzubico2 39797 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)𝑗 ∈ 𝑍) |
8 | 1 | ffnd 6046 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝑍) |
9 | 8 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 Fn 𝑍) |
10 | simpr 477 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
11 | id 22 | . . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ 𝑍) | |
12 | 2, 11 | uzxrd 39692 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ*) |
13 | pnfxr 10092 | . . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ* | |
14 | 13 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → +∞ ∈ ℝ*) |
15 | 12 | xrleidd 39610 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ≤ 𝑗) |
16 | 2, 11 | uzred 39670 | . . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ) |
17 | ltpnf 11954 | . . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℝ → 𝑗 < +∞) | |
18 | 16, 17 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → 𝑗 < +∞) |
19 | 12, 14, 12, 15, 18 | elicod 12224 | . . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (𝑗[,)+∞)) |
20 | 19 | adantl 482 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (𝑗[,)+∞)) |
21 | 9, 10, 20 | fnfvima2 39478 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (𝐹 “ (𝑗[,)+∞))) |
22 | 1 | ffvelrnda 6359 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ*) |
23 | 21, 22 | elind 3798 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*)) |
24 | ne0i 3921 | . . . . . . . 8 ⊢ ((𝐹‘𝑗) ∈ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) | |
25 | 23, 24 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) |
26 | 25 | ex 450 | . . . . . 6 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
27 | 26 | ad2antrr 762 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ (𝑘[,)+∞)) → (𝑗 ∈ 𝑍 → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
28 | 27 | reximdva 3017 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (∃𝑗 ∈ (𝑘[,)+∞)𝑗 ∈ 𝑍 → ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
29 | 28 | ralimdva 2962 | . . 3 ⊢ (𝜑 → (∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)𝑗 ∈ 𝑍 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
30 | 7, 29 | mpd 15 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) |
31 | 5, 30 | liminflelimsup 40008 | 1 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ∩ cin 3573 ∅c0 3915 class class class wbr 4653 “ cima 5117 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 +∞cpnf 10071 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 ℤcz 11377 ℤ≥cuz 11687 [,)cico 12177 lim supclsp 14201 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-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-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-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 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-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-ioo 12179 df-ico 12181 df-fl 12593 df-ceil 12594 df-limsup 14202 df-liminf 39984 |
This theorem is referenced by: liminfgelimsupuz 40020 liminflimsupclim 40039 |
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