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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 1787 | . . . 4 ⊢ Ⅎ𝑘⊤ | |
2 | nnex 11494 | . . . . 5 ⊢ ℕ ∈ V | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ ∈ V) |
4 | liminf10ex.1 | . . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) | |
5 | 0xr 10537 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ*) |
7 | 1xr 10549 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
8 | 7 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ*) |
9 | 6, 8 | ifcld 4428 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → if(2 ∥ 𝑛, 0, 1) ∈ ℝ*) |
10 | 4, 9 | fmpti 6742 | . . . . 5 ⊢ 𝐹:ℕ⟶ℝ* |
11 | 10 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐹:ℕ⟶ℝ*) |
12 | eqid 2794 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) | |
13 | 1, 3, 11, 12 | liminfval5 41601 | . . 3 ⊢ (⊤ → (lim inf‘𝐹) = sup(ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < )) |
14 | 13 | mptru 1529 | . 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 41608 | . . . . . . . 8 ⊢ (𝑘 ∈ ℝ → (𝐹 “ (𝑘[,)+∞)) = {0, 1}) |
17 | 16 | infeq1d 8790 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = inf({0, 1}, ℝ*, < )) |
18 | xrltso 12384 | . . . . . . . . . 10 ⊢ < Or ℝ* | |
19 | infpr 8816 | . . . . . . . . . 10 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ* ∧ 1 ∈ ℝ*) → inf({0, 1}, ℝ*, < ) = if(0 < 1, 0, 1)) | |
20 | 18, 5, 7, 19 | mp3an 1453 | . . . . . . . . 9 ⊢ inf({0, 1}, ℝ*, < ) = if(0 < 1, 0, 1) |
21 | 0lt1 11012 | . . . . . . . . . 10 ⊢ 0 < 1 | |
22 | 21 | iftruei 4390 | . . . . . . . . 9 ⊢ if(0 < 1, 0, 1) = 0 |
23 | 20, 22 | eqtri 2818 | . . . . . . . 8 ⊢ inf({0, 1}, ℝ*, < ) = 0 |
24 | 23 | a1i 11 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → inf({0, 1}, ℝ*, < ) = 0) |
25 | 17, 24 | eqtrd 2830 | . . . . . 6 ⊢ (𝑘 ∈ ℝ → inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = 0) |
26 | 25 | mpteq2ia 5054 | . . . . 5 ⊢ (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ 0) |
27 | 26 | rneqi 5692 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = ran (𝑘 ∈ ℝ ↦ 0) |
28 | eqid 2794 | . . . . . 6 ⊢ (𝑘 ∈ ℝ ↦ 0) = (𝑘 ∈ ℝ ↦ 0) | |
29 | 0re 10492 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
30 | 29 | a1i 11 | . . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℝ) → 0 ∈ ℝ) |
31 | ren0 41229 | . . . . . . 7 ⊢ ℝ ≠ ∅ | |
32 | 31 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ≠ ∅) |
33 | 28, 30, 32 | rnmptc 6839 | . . . . 5 ⊢ (⊤ → ran (𝑘 ∈ ℝ ↦ 0) = {0}) |
34 | 33 | mptru 1529 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ 0) = {0} |
35 | 27, 34 | eqtri 2818 | . . 3 ⊢ ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = {0} |
36 | 35 | supeq1i 8760 | . 2 ⊢ sup(ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = sup({0}, ℝ*, < ) |
37 | supsn 8785 | . . 3 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
38 | 18, 5, 37 | mp2an 688 | . 2 ⊢ sup({0}, ℝ*, < ) = 0 |
39 | 14, 36, 38 | 3eqtri 2822 | 1 ⊢ (lim inf‘𝐹) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1522 ⊤wtru 1523 ∈ wcel 2080 ≠ wne 2983 Vcvv 3436 ∅c0 4213 ifcif 4383 {csn 4474 {cpr 4476 class class class wbr 4964 ↦ cmpt 5043 Or wor 5364 ran crn 5447 “ cima 5449 ⟶wf 6224 ‘cfv 6228 (class class class)co 7019 supcsup 8753 infcinf 8754 ℝcr 10385 0cc0 10386 1c1 10387 +∞cpnf 10521 ℝ*cxr 10523 < clt 10524 ℕcn 11488 2c2 11542 [,)cico 12590 ∥ cdvds 15440 lim infclsi 41587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 ax-pre-sup 10464 |
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 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-er 8142 df-en 8361 df-dom 8362 df-sdom 8363 df-sup 8755 df-inf 8756 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-div 11148 df-nn 11489 df-2 11550 df-n0 11748 df-z 11832 df-uz 12094 df-rp 12240 df-ico 12594 df-fl 13012 df-ceil 13013 df-dvds 15441 df-liminf 41588 |
This theorem is referenced by: liminfltlimsupex 41617 |
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