Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminf10ex | Structured version Visualization version GIF version |
Description: The inferior limit of a function that alternates between two values. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminf10ex.1 | ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) |
Ref | Expression |
---|---|
liminf10ex | ⊢ (lim inf‘𝐹) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1812 | . . . 4 ⊢ Ⅎ𝑘⊤ | |
2 | nnex 11819 | . . . . 5 ⊢ ℕ ∈ V | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ ∈ V) |
4 | liminf10ex.1 | . . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) | |
5 | 0xr 10863 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ*) |
7 | 1xr 10875 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
8 | 7 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ*) |
9 | 6, 8 | ifcld 4475 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → if(2 ∥ 𝑛, 0, 1) ∈ ℝ*) |
10 | 4, 9 | fmpti 6918 | . . . . 5 ⊢ 𝐹:ℕ⟶ℝ* |
11 | 10 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐹:ℕ⟶ℝ*) |
12 | eqid 2734 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) | |
13 | 1, 3, 11, 12 | liminfval5 42935 | . . 3 ⊢ (⊤ → (lim inf‘𝐹) = sup(ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < )) |
14 | 13 | mptru 1550 | . 2 ⊢ (lim inf‘𝐹) = sup(ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) |
15 | id 22 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℝ) | |
16 | 4, 15 | limsup10exlem 42942 | . . . . . . . 8 ⊢ (𝑘 ∈ ℝ → (𝐹 “ (𝑘[,)+∞)) = {0, 1}) |
17 | 16 | infeq1d 9082 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = inf({0, 1}, ℝ*, < )) |
18 | xrltso 12714 | . . . . . . . . 9 ⊢ < Or ℝ* | |
19 | infpr 9108 | . . . . . . . . 9 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ* ∧ 1 ∈ ℝ*) → inf({0, 1}, ℝ*, < ) = if(0 < 1, 0, 1)) | |
20 | 18, 5, 7, 19 | mp3an 1463 | . . . . . . . 8 ⊢ inf({0, 1}, ℝ*, < ) = if(0 < 1, 0, 1) |
21 | 0lt1 11337 | . . . . . . . . 9 ⊢ 0 < 1 | |
22 | 21 | iftruei 4436 | . . . . . . . 8 ⊢ if(0 < 1, 0, 1) = 0 |
23 | 20, 22 | eqtri 2762 | . . . . . . 7 ⊢ inf({0, 1}, ℝ*, < ) = 0 |
24 | 17, 23 | eqtrdi 2790 | . . . . . 6 ⊢ (𝑘 ∈ ℝ → inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = 0) |
25 | 24 | mpteq2ia 5135 | . . . . 5 ⊢ (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ 0) |
26 | 25 | rneqi 5795 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = ran (𝑘 ∈ ℝ ↦ 0) |
27 | eqid 2734 | . . . . . 6 ⊢ (𝑘 ∈ ℝ ↦ 0) = (𝑘 ∈ ℝ ↦ 0) | |
28 | ren0 42567 | . . . . . . 7 ⊢ ℝ ≠ ∅ | |
29 | 28 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ≠ ∅) |
30 | 27, 29 | rnmptc 7011 | . . . . 5 ⊢ (⊤ → ran (𝑘 ∈ ℝ ↦ 0) = {0}) |
31 | 30 | mptru 1550 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ 0) = {0} |
32 | 26, 31 | eqtri 2762 | . . 3 ⊢ ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = {0} |
33 | 32 | supeq1i 9052 | . 2 ⊢ sup(ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = sup({0}, ℝ*, < ) |
34 | supsn 9077 | . . 3 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
35 | 18, 5, 34 | mp2an 692 | . 2 ⊢ sup({0}, ℝ*, < ) = 0 |
36 | 14, 33, 35 | 3eqtri 2766 | 1 ⊢ (lim inf‘𝐹) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ⊤wtru 1544 ∈ wcel 2110 ≠ wne 2935 Vcvv 3401 ∅c0 4227 ifcif 4429 {csn 4531 {cpr 4533 class class class wbr 5043 ↦ cmpt 5124 Or wor 5456 ran crn 5541 “ cima 5543 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 supcsup 9045 infcinf 9046 ℝcr 10711 0cc0 10712 1c1 10713 +∞cpnf 10847 ℝ*cxr 10849 < clt 10850 ℕcn 11813 2c2 11868 [,)cico 12920 ∥ cdvds 15796 lim infclsi 42921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-sup 9047 df-inf 9048 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-ico 12924 df-fl 13350 df-ceil 13351 df-dvds 15797 df-liminf 42922 |
This theorem is referenced by: liminfltlimsupex 42951 |
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