| 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 1804 | . . . 4 ⊢ Ⅎ𝑘⊤ | |
| 2 | nnex 12272 | . . . . 5 ⊢ ℕ ∈ V | |
| 3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ ∈ V) |
| 4 | limsup10ex.1 | . . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) | |
| 5 | 0xr 11308 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
| 6 | 5 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ*) |
| 7 | 1xr 11320 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
| 8 | 7 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ*) |
| 9 | 6, 8 | ifcld 4572 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → if(2 ∥ 𝑛, 0, 1) ∈ ℝ*) |
| 10 | 4, 9 | fmpti 7132 | . . . . 5 ⊢ 𝐹:ℕ⟶ℝ* |
| 11 | 10 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐹:ℕ⟶ℝ*) |
| 12 | eqid 2737 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) | |
| 13 | 1, 3, 11, 12 | limsupval3 45707 | . . 3 ⊢ (⊤ → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < )) |
| 14 | 13 | mptru 1547 | . 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 45787 | . . . . . . . 8 ⊢ (𝑘 ∈ ℝ → (𝐹 “ (𝑘[,)+∞)) = {0, 1}) |
| 17 | 16 | supeq1d 9486 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = sup({0, 1}, ℝ*, < )) |
| 18 | xrltso 13183 | . . . . . . . . 9 ⊢ < Or ℝ* | |
| 19 | suppr 9511 | . . . . . . . . 9 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ* ∧ 1 ∈ ℝ*) → sup({0, 1}, ℝ*, < ) = if(1 < 0, 0, 1)) | |
| 20 | 18, 5, 7, 19 | mp3an 1463 | . . . . . . . 8 ⊢ sup({0, 1}, ℝ*, < ) = if(1 < 0, 0, 1) |
| 21 | 0le1 11786 | . . . . . . . . . 10 ⊢ 0 ≤ 1 | |
| 22 | 0re 11263 | . . . . . . . . . . 11 ⊢ 0 ∈ ℝ | |
| 23 | 1re 11261 | . . . . . . . . . . 11 ⊢ 1 ∈ ℝ | |
| 24 | 22, 23 | lenlti 11381 | . . . . . . . . . 10 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0) |
| 25 | 21, 24 | mpbi 230 | . . . . . . . . 9 ⊢ ¬ 1 < 0 |
| 26 | 25 | iffalsei 4535 | . . . . . . . 8 ⊢ if(1 < 0, 0, 1) = 1 |
| 27 | 20, 26 | eqtri 2765 | . . . . . . 7 ⊢ sup({0, 1}, ℝ*, < ) = 1 |
| 28 | 17, 27 | eqtrdi 2793 | . . . . . 6 ⊢ (𝑘 ∈ ℝ → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = 1) |
| 29 | 28 | mpteq2ia 5245 | . . . . 5 ⊢ (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ 1) |
| 30 | 29 | rneqi 5948 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = ran (𝑘 ∈ ℝ ↦ 1) |
| 31 | eqid 2737 | . . . . . 6 ⊢ (𝑘 ∈ ℝ ↦ 1) = (𝑘 ∈ ℝ ↦ 1) | |
| 32 | ren0 45413 | . . . . . . 7 ⊢ ℝ ≠ ∅ | |
| 33 | 32 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ≠ ∅) |
| 34 | 31, 33 | rnmptc 7227 | . . . . 5 ⊢ (⊤ → ran (𝑘 ∈ ℝ ↦ 1) = {1}) |
| 35 | 34 | mptru 1547 | . . . 4 ⊢ ran (𝑘 ∈ ℝ ↦ 1) = {1} |
| 36 | 30, 35 | eqtri 2765 | . . 3 ⊢ ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = {1} |
| 37 | 36 | infeq1i 9518 | . 2 ⊢ inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = inf({1}, ℝ*, < ) |
| 38 | infsn 9545 | . . 3 ⊢ (( < Or ℝ* ∧ 1 ∈ ℝ*) → inf({1}, ℝ*, < ) = 1) | |
| 39 | 18, 7, 38 | mp2an 692 | . 2 ⊢ inf({1}, ℝ*, < ) = 1 |
| 40 | 14, 37, 39 | 3eqtri 2769 | 1 ⊢ (lim sup‘𝐹) = 1 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 ifcif 4525 {csn 4626 {cpr 4628 class class class wbr 5143 ↦ cmpt 5225 Or wor 5591 ran crn 5686 “ cima 5688 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supcsup 9480 infcinf 9481 ℝcr 11154 0cc0 11155 1c1 11156 +∞cpnf 11292 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 ℕcn 12266 2c2 12321 [,)cico 13389 lim supclsp 15506 ∥ cdvds 16290 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ico 13393 df-fl 13832 df-ceil 13833 df-limsup 15507 df-dvds 16291 |
| This theorem is referenced by: liminfltlimsupex 45796 |
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