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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 1799 | . . . 4 ⊢ Ⅎ𝑘⊤ | |
2 | nnex 12272 | . . . . 5 ⊢ ℕ ∈ V | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ ∈ V) |
4 | limsup10ex.1 | . . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) | |
5 | 0xr 11313 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ*) |
7 | 1xr 11325 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
8 | 7 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ*) |
9 | 6, 8 | ifcld 4579 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → if(2 ∥ 𝑛, 0, 1) ∈ ℝ*) |
10 | 4, 9 | fmpti 7128 | . . . . 5 ⊢ 𝐹:ℕ⟶ℝ* |
11 | 10 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐹:ℕ⟶ℝ*) |
12 | eqid 2726 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) | |
13 | 1, 3, 11, 12 | limsupval3 45331 | . . 3 ⊢ (⊤ → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < )) |
14 | 13 | mptru 1541 | . 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 45411 | . . . . . . . 8 ⊢ (𝑘 ∈ ℝ → (𝐹 “ (𝑘[,)+∞)) = {0, 1}) |
17 | 16 | supeq1d 9491 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = sup({0, 1}, ℝ*, < )) |
18 | xrltso 13176 | . . . . . . . . 9 ⊢ < Or ℝ* | |
19 | suppr 9516 | . . . . . . . . 9 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ* ∧ 1 ∈ ℝ*) → sup({0, 1}, ℝ*, < ) = if(1 < 0, 0, 1)) | |
20 | 18, 5, 7, 19 | mp3an 1458 | . . . . . . . 8 ⊢ sup({0, 1}, ℝ*, < ) = if(1 < 0, 0, 1) |
21 | 0le1 11789 | . . . . . . . . . 10 ⊢ 0 ≤ 1 | |
22 | 0re 11268 | . . . . . . . . . . 11 ⊢ 0 ∈ ℝ | |
23 | 1re 11266 | . . . . . . . . . . 11 ⊢ 1 ∈ ℝ | |
24 | 22, 23 | lenlti 11386 | . . . . . . . . . 10 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0) |
25 | 21, 24 | mpbi 229 | . . . . . . . . 9 ⊢ ¬ 1 < 0 |
26 | 25 | iffalsei 4543 | . . . . . . . 8 ⊢ if(1 < 0, 0, 1) = 1 |
27 | 20, 26 | eqtri 2754 | . . . . . . 7 ⊢ sup({0, 1}, ℝ*, < ) = 1 |
28 | 17, 27 | eqtrdi 2782 | . . . . . 6 ⊢ (𝑘 ∈ ℝ → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = 1) |
29 | 28 | mpteq2ia 5258 | . . . . 5 ⊢ (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ 1) |
30 | 29 | rneqi 5945 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = ran (𝑘 ∈ ℝ ↦ 1) |
31 | eqid 2726 | . . . . . 6 ⊢ (𝑘 ∈ ℝ ↦ 1) = (𝑘 ∈ ℝ ↦ 1) | |
32 | ren0 45035 | . . . . . . 7 ⊢ ℝ ≠ ∅ | |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ≠ ∅) |
34 | 31, 33 | rnmptc 7226 | . . . . 5 ⊢ (⊤ → ran (𝑘 ∈ ℝ ↦ 1) = {1}) |
35 | 34 | mptru 1541 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ 1) = {1} |
36 | 30, 35 | eqtri 2754 | . . 3 ⊢ ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = {1} |
37 | 36 | infeq1i 9523 | . 2 ⊢ inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = inf({1}, ℝ*, < ) |
38 | infsn 9550 | . . 3 ⊢ (( < Or ℝ* ∧ 1 ∈ ℝ*) → inf({1}, ℝ*, < ) = 1) | |
39 | 18, 7, 38 | mp2an 690 | . 2 ⊢ inf({1}, ℝ*, < ) = 1 |
40 | 14, 37, 39 | 3eqtri 2758 | 1 ⊢ (lim sup‘𝐹) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 ∅c0 4325 ifcif 4533 {csn 4633 {cpr 4635 class class class wbr 5155 ↦ cmpt 5238 Or wor 5595 ran crn 5685 “ cima 5687 ⟶wf 6552 ‘cfv 6556 (class class class)co 7426 supcsup 9485 infcinf 9486 ℝcr 11159 0cc0 11160 1c1 11161 +∞cpnf 11297 ℝ*cxr 11299 < clt 11300 ≤ cle 11301 ℕcn 12266 2c2 12321 [,)cico 13382 lim supclsp 15474 ∥ cdvds 16258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 ax-pre-sup 11238 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-sup 9487 df-inf 9488 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-div 11924 df-nn 12267 df-2 12329 df-n0 12527 df-z 12613 df-uz 12877 df-rp 13031 df-ico 13386 df-fl 13814 df-ceil 13815 df-limsup 15475 df-dvds 16259 |
This theorem is referenced by: liminfltlimsupex 45420 |
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