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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 1806 | . . . 4 ⊢ Ⅎ𝑘⊤ | |
2 | nnex 12155 | . . . . 5 ⊢ ℕ ∈ V | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ ∈ V) |
4 | liminf10ex.1 | . . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) | |
5 | 0xr 11198 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ*) |
7 | 1xr 11210 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
8 | 7 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ*) |
9 | 6, 8 | ifcld 4530 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → if(2 ∥ 𝑛, 0, 1) ∈ ℝ*) |
10 | 4, 9 | fmpti 7056 | . . . . 5 ⊢ 𝐹:ℕ⟶ℝ* |
11 | 10 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐹:ℕ⟶ℝ*) |
12 | eqid 2736 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) | |
13 | 1, 3, 11, 12 | liminfval5 43938 | . . 3 ⊢ (⊤ → (lim inf‘𝐹) = sup(ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < )) |
14 | 13 | mptru 1548 | . 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 43945 | . . . . . . . 8 ⊢ (𝑘 ∈ ℝ → (𝐹 “ (𝑘[,)+∞)) = {0, 1}) |
17 | 16 | infeq1d 9409 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = inf({0, 1}, ℝ*, < )) |
18 | xrltso 13052 | . . . . . . . . 9 ⊢ < Or ℝ* | |
19 | infpr 9435 | . . . . . . . . 9 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ* ∧ 1 ∈ ℝ*) → inf({0, 1}, ℝ*, < ) = if(0 < 1, 0, 1)) | |
20 | 18, 5, 7, 19 | mp3an 1461 | . . . . . . . 8 ⊢ inf({0, 1}, ℝ*, < ) = if(0 < 1, 0, 1) |
21 | 0lt1 11673 | . . . . . . . . 9 ⊢ 0 < 1 | |
22 | 21 | iftruei 4491 | . . . . . . . 8 ⊢ if(0 < 1, 0, 1) = 0 |
23 | 20, 22 | eqtri 2764 | . . . . . . 7 ⊢ inf({0, 1}, ℝ*, < ) = 0 |
24 | 17, 23 | eqtrdi 2792 | . . . . . 6 ⊢ (𝑘 ∈ ℝ → inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = 0) |
25 | 24 | mpteq2ia 5206 | . . . . 5 ⊢ (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ 0) |
26 | 25 | rneqi 5890 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = ran (𝑘 ∈ ℝ ↦ 0) |
27 | eqid 2736 | . . . . . 6 ⊢ (𝑘 ∈ ℝ ↦ 0) = (𝑘 ∈ ℝ ↦ 0) | |
28 | ren0 43573 | . . . . . . 7 ⊢ ℝ ≠ ∅ | |
29 | 28 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ≠ ∅) |
30 | 27, 29 | rnmptc 7152 | . . . . 5 ⊢ (⊤ → ran (𝑘 ∈ ℝ ↦ 0) = {0}) |
31 | 30 | mptru 1548 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ 0) = {0} |
32 | 26, 31 | eqtri 2764 | . . 3 ⊢ ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = {0} |
33 | 32 | supeq1i 9379 | . 2 ⊢ sup(ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = sup({0}, ℝ*, < ) |
34 | supsn 9404 | . . 3 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
35 | 18, 5, 34 | mp2an 690 | . 2 ⊢ sup({0}, ℝ*, < ) = 0 |
36 | 14, 33, 35 | 3eqtri 2768 | 1 ⊢ (lim inf‘𝐹) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ≠ wne 2941 Vcvv 3443 ∅c0 4280 ifcif 4484 {csn 4584 {cpr 4586 class class class wbr 5103 ↦ cmpt 5186 Or wor 5542 ran crn 5632 “ cima 5634 ⟶wf 6489 ‘cfv 6493 (class class class)co 7353 supcsup 9372 infcinf 9373 ℝcr 11046 0cc0 11047 1c1 11048 +∞cpnf 11182 ℝ*cxr 11184 < clt 11185 ℕcn 12149 2c2 12204 [,)cico 13258 ∥ cdvds 16128 lim infclsi 43924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9374 df-inf 9375 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-n0 12410 df-z 12496 df-uz 12760 df-rp 12908 df-ico 13262 df-fl 13689 df-ceil 13690 df-dvds 16129 df-liminf 43925 |
This theorem is referenced by: liminfltlimsupex 43954 |
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