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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 1786 | . . . 4 ⊢ Ⅎ𝑘⊤ | |
2 | nnex 11492 | . . . . 5 ⊢ ℕ ∈ V | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ ∈ V) |
4 | limsup10ex.1 | . . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) | |
5 | 0xr 10534 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ*) |
7 | 1xr 10547 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
8 | 7 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ*) |
9 | 6, 8 | ifcld 4426 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → if(2 ∥ 𝑛, 0, 1) ∈ ℝ*) |
10 | 4, 9 | fmpti 6739 | . . . . 5 ⊢ 𝐹:ℕ⟶ℝ* |
11 | 10 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐹:ℕ⟶ℝ*) |
12 | eqid 2795 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) | |
13 | 1, 3, 11, 12 | limsupval3 41515 | . . 3 ⊢ (⊤ → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < )) |
14 | 13 | mptru 1529 | . 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 41595 | . . . . . . . 8 ⊢ (𝑘 ∈ ℝ → (𝐹 “ (𝑘[,)+∞)) = {0, 1}) |
17 | 16 | supeq1d 8756 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = sup({0, 1}, ℝ*, < )) |
18 | xrltso 12384 | . . . . . . . . . 10 ⊢ < Or ℝ* | |
19 | suppr 8781 | . . . . . . . . . 10 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ* ∧ 1 ∈ ℝ*) → sup({0, 1}, ℝ*, < ) = if(1 < 0, 0, 1)) | |
20 | 18, 5, 7, 19 | mp3an 1453 | . . . . . . . . 9 ⊢ sup({0, 1}, ℝ*, < ) = if(1 < 0, 0, 1) |
21 | 0le1 11011 | . . . . . . . . . . 11 ⊢ 0 ≤ 1 | |
22 | 0re 10489 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℝ | |
23 | 1re 10487 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℝ | |
24 | lenlt 10566 | . . . . . . . . . . . 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 4391 | . . . . . . . . 9 ⊢ if(1 < 0, 0, 1) = 1 |
28 | 20, 27 | eqtri 2819 | . . . . . . . 8 ⊢ sup({0, 1}, ℝ*, < ) = 1 |
29 | 28 | a1i 11 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → sup({0, 1}, ℝ*, < ) = 1) |
30 | 17, 29 | eqtrd 2831 | . . . . . 6 ⊢ (𝑘 ∈ ℝ → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = 1) |
31 | 30 | mpteq2ia 5051 | . . . . 5 ⊢ (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ 1) |
32 | 31 | rneqi 5689 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = ran (𝑘 ∈ ℝ ↦ 1) |
33 | eqid 2795 | . . . . . 6 ⊢ (𝑘 ∈ ℝ ↦ 1) = (𝑘 ∈ ℝ ↦ 1) | |
34 | 23 | elexi 3456 | . . . . . . 7 ⊢ 1 ∈ V |
35 | 34 | a1i 11 | . . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℝ) → 1 ∈ V) |
36 | ren0 41216 | . . . . . . 7 ⊢ ℝ ≠ ∅ | |
37 | 36 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ≠ ∅) |
38 | 33, 35, 37 | rnmptc 6836 | . . . . 5 ⊢ (⊤ → ran (𝑘 ∈ ℝ ↦ 1) = {1}) |
39 | 38 | mptru 1529 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ 1) = {1} |
40 | 32, 39 | eqtri 2819 | . . 3 ⊢ ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = {1} |
41 | 40 | infeq1i 8788 | . 2 ⊢ inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = inf({1}, ℝ*, < ) |
42 | infsn 8815 | . . 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 2823 | 1 ⊢ (lim sup‘𝐹) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∧ wa 396 = wceq 1522 ⊤wtru 1523 ∈ wcel 2081 ≠ wne 2984 Vcvv 3437 ∅c0 4211 ifcif 4381 {csn 4472 {cpr 4474 class class class wbr 4962 ↦ cmpt 5041 Or wor 5361 ran crn 5444 “ cima 5446 ⟶wf 6221 ‘cfv 6225 (class class class)co 7016 supcsup 8750 infcinf 8751 ℝcr 10382 0cc0 10383 1c1 10384 +∞cpnf 10518 ℝ*cxr 10520 < clt 10521 ≤ cle 10522 ℕcn 11486 2c2 11540 [,)cico 12590 lim supclsp 14661 ∥ cdvds 15440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-sup 8752 df-inf 8753 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-n0 11746 df-z 11830 df-uz 12094 df-rp 12240 df-ico 12594 df-fl 13012 df-ceil 13013 df-limsup 14662 df-dvds 15441 |
This theorem is referenced by: liminfltlimsupex 41604 |
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