Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 1805 | . . . 4 ⊢ Ⅎ𝑘⊤ | |
2 | nnex 11644 | . . . . 5 ⊢ ℕ ∈ V | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ ∈ V) |
4 | liminf10ex.1 | . . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) | |
5 | 0xr 10688 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ*) |
7 | 1xr 10700 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
8 | 7 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ*) |
9 | 6, 8 | ifcld 4512 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → if(2 ∥ 𝑛, 0, 1) ∈ ℝ*) |
10 | 4, 9 | fmpti 6876 | . . . . 5 ⊢ 𝐹:ℕ⟶ℝ* |
11 | 10 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐹:ℕ⟶ℝ*) |
12 | eqid 2821 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) | |
13 | 1, 3, 11, 12 | liminfval5 42066 | . . 3 ⊢ (⊤ → (lim inf‘𝐹) = sup(ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < )) |
14 | 13 | mptru 1544 | . 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 42073 | . . . . . . . 8 ⊢ (𝑘 ∈ ℝ → (𝐹 “ (𝑘[,)+∞)) = {0, 1}) |
17 | 16 | infeq1d 8941 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = inf({0, 1}, ℝ*, < )) |
18 | xrltso 12535 | . . . . . . . . 9 ⊢ < Or ℝ* | |
19 | infpr 8967 | . . . . . . . . 9 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ* ∧ 1 ∈ ℝ*) → inf({0, 1}, ℝ*, < ) = if(0 < 1, 0, 1)) | |
20 | 18, 5, 7, 19 | mp3an 1457 | . . . . . . . 8 ⊢ inf({0, 1}, ℝ*, < ) = if(0 < 1, 0, 1) |
21 | 0lt1 11162 | . . . . . . . . 9 ⊢ 0 < 1 | |
22 | 21 | iftruei 4474 | . . . . . . . 8 ⊢ if(0 < 1, 0, 1) = 0 |
23 | 20, 22 | eqtri 2844 | . . . . . . 7 ⊢ inf({0, 1}, ℝ*, < ) = 0 |
24 | 17, 23 | syl6eq 2872 | . . . . . 6 ⊢ (𝑘 ∈ ℝ → inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = 0) |
25 | 24 | mpteq2ia 5157 | . . . . 5 ⊢ (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ 0) |
26 | 25 | rneqi 5807 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = ran (𝑘 ∈ ℝ ↦ 0) |
27 | eqid 2821 | . . . . . 6 ⊢ (𝑘 ∈ ℝ ↦ 0) = (𝑘 ∈ ℝ ↦ 0) | |
28 | ren0 41695 | . . . . . . 7 ⊢ ℝ ≠ ∅ | |
29 | 28 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ≠ ∅) |
30 | 27, 29 | rnmptc 6969 | . . . . 5 ⊢ (⊤ → ran (𝑘 ∈ ℝ ↦ 0) = {0}) |
31 | 30 | mptru 1544 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ 0) = {0} |
32 | 26, 31 | eqtri 2844 | . . 3 ⊢ ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = {0} |
33 | 32 | supeq1i 8911 | . 2 ⊢ sup(ran (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = sup({0}, ℝ*, < ) |
34 | supsn 8936 | . . 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 2848 | 1 ⊢ (lim inf‘𝐹) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∅c0 4291 ifcif 4467 {csn 4567 {cpr 4569 class class class wbr 5066 ↦ cmpt 5146 Or wor 5473 ran crn 5556 “ cima 5558 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 supcsup 8904 infcinf 8905 ℝcr 10536 0cc0 10537 1c1 10538 +∞cpnf 10672 ℝ*cxr 10674 < clt 10675 ℕcn 11638 2c2 11693 [,)cico 12741 ∥ cdvds 15607 lim infclsi 42052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-ico 12745 df-fl 13163 df-ceil 13164 df-dvds 15608 df-liminf 42053 |
This theorem is referenced by: liminfltlimsupex 42082 |
Copyright terms: Public domain | W3C validator |