Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsup10ex | Structured version Visualization version GIF version |
Description: The superior limit of a function that alternates between two values. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
limsup10ex.1 | ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) |
Ref | Expression |
---|---|
limsup10ex | ⊢ (lim sup‘𝐹) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1796 | . . . 4 ⊢ Ⅎ𝑘⊤ | |
2 | nnex 11632 | . . . . 5 ⊢ ℕ ∈ V | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ ∈ V) |
4 | limsup10ex.1 | . . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) | |
5 | 0xr 10676 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ*) |
7 | 1xr 10688 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
8 | 7 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ*) |
9 | 6, 8 | ifcld 4508 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → if(2 ∥ 𝑛, 0, 1) ∈ ℝ*) |
10 | 4, 9 | fmpti 6868 | . . . . 5 ⊢ 𝐹:ℕ⟶ℝ* |
11 | 10 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐹:ℕ⟶ℝ*) |
12 | eqid 2818 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) | |
13 | 1, 3, 11, 12 | limsupval3 41849 | . . 3 ⊢ (⊤ → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < )) |
14 | 13 | mptru 1535 | . 2 ⊢ (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) |
15 | id 22 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℝ) | |
16 | 4, 15 | limsup10exlem 41929 | . . . . . . . 8 ⊢ (𝑘 ∈ ℝ → (𝐹 “ (𝑘[,)+∞)) = {0, 1}) |
17 | 16 | supeq1d 8898 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = sup({0, 1}, ℝ*, < )) |
18 | xrltso 12522 | . . . . . . . . . 10 ⊢ < Or ℝ* | |
19 | suppr 8923 | . . . . . . . . . 10 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ* ∧ 1 ∈ ℝ*) → sup({0, 1}, ℝ*, < ) = if(1 < 0, 0, 1)) | |
20 | 18, 5, 7, 19 | mp3an 1452 | . . . . . . . . 9 ⊢ sup({0, 1}, ℝ*, < ) = if(1 < 0, 0, 1) |
21 | 0le1 11151 | . . . . . . . . . . 11 ⊢ 0 ≤ 1 | |
22 | 0re 10631 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℝ | |
23 | 1re 10629 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℝ | |
24 | lenlt 10707 | . . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → (0 ≤ 1 ↔ ¬ 1 < 0)) | |
25 | 22, 23, 24 | mp2an 688 | . . . . . . . . . . 11 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0) |
26 | 21, 25 | mpbi 231 | . . . . . . . . . 10 ⊢ ¬ 1 < 0 |
27 | 26 | iffalsei 4473 | . . . . . . . . 9 ⊢ if(1 < 0, 0, 1) = 1 |
28 | 20, 27 | eqtri 2841 | . . . . . . . 8 ⊢ sup({0, 1}, ℝ*, < ) = 1 |
29 | 28 | a1i 11 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → sup({0, 1}, ℝ*, < ) = 1) |
30 | 17, 29 | eqtrd 2853 | . . . . . 6 ⊢ (𝑘 ∈ ℝ → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = 1) |
31 | 30 | mpteq2ia 5148 | . . . . 5 ⊢ (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ 1) |
32 | 31 | rneqi 5800 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = ran (𝑘 ∈ ℝ ↦ 1) |
33 | eqid 2818 | . . . . . 6 ⊢ (𝑘 ∈ ℝ ↦ 1) = (𝑘 ∈ ℝ ↦ 1) | |
34 | 23 | elexi 3511 | . . . . . . 7 ⊢ 1 ∈ V |
35 | 34 | a1i 11 | . . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℝ) → 1 ∈ V) |
36 | ren0 41551 | . . . . . . 7 ⊢ ℝ ≠ ∅ | |
37 | 36 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ≠ ∅) |
38 | 33, 35, 37 | rnmptc 6961 | . . . . 5 ⊢ (⊤ → ran (𝑘 ∈ ℝ ↦ 1) = {1}) |
39 | 38 | mptru 1535 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ 1) = {1} |
40 | 32, 39 | eqtri 2841 | . . 3 ⊢ ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = {1} |
41 | 40 | infeq1i 8930 | . 2 ⊢ inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = inf({1}, ℝ*, < ) |
42 | infsn 8957 | . . 3 ⊢ (( < Or ℝ* ∧ 1 ∈ ℝ*) → inf({1}, ℝ*, < ) = 1) | |
43 | 18, 7, 42 | mp2an 688 | . 2 ⊢ inf({1}, ℝ*, < ) = 1 |
44 | 14, 41, 43 | 3eqtri 2845 | 1 ⊢ (lim sup‘𝐹) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∧ wa 396 = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ∅c0 4288 ifcif 4463 {csn 4557 {cpr 4559 class class class wbr 5057 ↦ cmpt 5137 Or wor 5466 ran crn 5549 “ cima 5551 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 supcsup 8892 infcinf 8893 ℝcr 10524 0cc0 10525 1c1 10526 +∞cpnf 10660 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 ℕcn 11626 2c2 11680 [,)cico 12728 lim supclsp 14815 ∥ cdvds 15595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-fl 13150 df-ceil 13151 df-limsup 14816 df-dvds 15596 |
This theorem is referenced by: liminfltlimsupex 41938 |
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