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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 1877 | . . . 4 ⊢ Ⅎ𝑘⊤ | |
2 | nnex 11227 | . . . . 5 ⊢ ℕ ∈ V | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ ∈ V) |
4 | limsup10ex.1 | . . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) | |
5 | 0xr 10287 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ*) |
7 | 1re 10240 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
8 | 7 | rexri 10298 | . . . . . . . 8 ⊢ 1 ∈ ℝ* |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ*) |
10 | 6, 9 | ifcld 4268 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → if(2 ∥ 𝑛, 0, 1) ∈ ℝ*) |
11 | 4, 10 | fmpti 6525 | . . . . 5 ⊢ 𝐹:ℕ⟶ℝ* |
12 | 11 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐹:ℕ⟶ℝ*) |
13 | eqid 2770 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) | |
14 | 1, 3, 12, 13 | limsupval3 40436 | . . 3 ⊢ (⊤ → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < )) |
15 | 14 | trud 1640 | . 2 ⊢ (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) |
16 | id 22 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℝ) | |
17 | 4, 16 | limsup10exlem 40516 | . . . . . . . 8 ⊢ (𝑘 ∈ ℝ → (𝐹 “ (𝑘[,)+∞)) = {0, 1}) |
18 | 17 | supeq1d 8507 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = sup({0, 1}, ℝ*, < )) |
19 | xrltso 12178 | . . . . . . . . . 10 ⊢ < Or ℝ* | |
20 | suppr 8532 | . . . . . . . . . 10 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ* ∧ 1 ∈ ℝ*) → sup({0, 1}, ℝ*, < ) = if(1 < 0, 0, 1)) | |
21 | 19, 5, 8, 20 | mp3an 1571 | . . . . . . . . 9 ⊢ sup({0, 1}, ℝ*, < ) = if(1 < 0, 0, 1) |
22 | 0le1 10752 | . . . . . . . . . . 11 ⊢ 0 ≤ 1 | |
23 | 0re 10241 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℝ | |
24 | lenlt 10317 | . . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → (0 ≤ 1 ↔ ¬ 1 < 0)) | |
25 | 23, 7, 24 | mp2an 664 | . . . . . . . . . . 11 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0) |
26 | 22, 25 | mpbi 220 | . . . . . . . . . 10 ⊢ ¬ 1 < 0 |
27 | 26 | iffalsei 4233 | . . . . . . . . 9 ⊢ if(1 < 0, 0, 1) = 1 |
28 | 21, 27 | eqtri 2792 | . . . . . . . 8 ⊢ sup({0, 1}, ℝ*, < ) = 1 |
29 | 28 | a1i 11 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → sup({0, 1}, ℝ*, < ) = 1) |
30 | 18, 29 | eqtrd 2804 | . . . . . 6 ⊢ (𝑘 ∈ ℝ → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = 1) |
31 | 30 | mpteq2ia 4872 | . . . . 5 ⊢ (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ 1) |
32 | 31 | rneqi 5490 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = ran (𝑘 ∈ ℝ ↦ 1) |
33 | eqid 2770 | . . . . . 6 ⊢ (𝑘 ∈ ℝ ↦ 1) = (𝑘 ∈ ℝ ↦ 1) | |
34 | 7 | elexi 3362 | . . . . . . 7 ⊢ 1 ∈ V |
35 | 34 | a1i 11 | . . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℝ) → 1 ∈ V) |
36 | ren0 40136 | . . . . . . 7 ⊢ ℝ ≠ ∅ | |
37 | 36 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ≠ ∅) |
38 | 33, 35, 37 | rnmptc 39867 | . . . . 5 ⊢ (⊤ → ran (𝑘 ∈ ℝ ↦ 1) = {1}) |
39 | 38 | trud 1640 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ 1) = {1} |
40 | 32, 39 | eqtri 2792 | . . 3 ⊢ ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = {1} |
41 | 40 | infeq1i 8539 | . 2 ⊢ inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = inf({1}, ℝ*, < ) |
42 | infsn 8565 | . . 3 ⊢ (( < Or ℝ* ∧ 1 ∈ ℝ*) → inf({1}, ℝ*, < ) = 1) | |
43 | 19, 8, 42 | mp2an 664 | . 2 ⊢ inf({1}, ℝ*, < ) = 1 |
44 | 15, 41, 43 | 3eqtri 2796 | 1 ⊢ (lim sup‘𝐹) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∧ wa 382 = wceq 1630 ⊤wtru 1631 ∈ wcel 2144 ≠ wne 2942 Vcvv 3349 ∅c0 4061 ifcif 4223 {csn 4314 {cpr 4316 class class class wbr 4784 ↦ cmpt 4861 Or wor 5169 ran crn 5250 “ cima 5252 ⟶wf 6027 ‘cfv 6031 (class class class)co 6792 supcsup 8501 infcinf 8502 ℝcr 10136 0cc0 10137 1c1 10138 +∞cpnf 10272 ℝ*cxr 10274 < clt 10275 ≤ cle 10276 ℕcn 11221 2c2 11271 [,)cico 12381 lim supclsp 14408 ∥ cdvds 15188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-pre-sup 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-sup 8503 df-inf 8504 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-div 10886 df-nn 11222 df-2 11280 df-n0 11494 df-z 11579 df-uz 11888 df-rp 12035 df-ico 12385 df-fl 12800 df-ceil 12801 df-limsup 14409 df-dvds 15189 |
This theorem is referenced by: liminfltlimsupex 40525 |
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