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Mirrors > Home > MPE Home > Th. List > limsupcl | Structured version Visualization version GIF version |
Description: Closure of the superior limit. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 12-Sep-2020.) |
Ref | Expression |
---|---|
limsupcl | ⊢ (𝐹 ∈ 𝑉 → (lim sup‘𝐹) ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3510 | . 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | |
2 | df-limsup 14816 | . . . 4 ⊢ lim sup = (𝑓 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) | |
3 | eqid 2818 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
4 | inss2 4203 | . . . . . . . 8 ⊢ ((𝑓 “ (𝑘[,)+∞)) ∩ ℝ*) ⊆ ℝ* | |
5 | supxrcl 12696 | . . . . . . . 8 ⊢ (((𝑓 “ (𝑘[,)+∞)) ∩ ℝ*) ⊆ ℝ* → sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*) | |
6 | 4, 5 | mp1i 13 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ → sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*) |
7 | 3, 6 | fmpti 6868 | . . . . . 6 ⊢ (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ* |
8 | frn 6513 | . . . . . 6 ⊢ ((𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ* → ran (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) ⊆ ℝ*) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ ran (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) ⊆ ℝ* |
10 | infxrcl 12714 | . . . . 5 ⊢ (ran (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) ⊆ ℝ* → inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) ∈ ℝ*) | |
11 | 9, 10 | mp1i 13 | . . . 4 ⊢ (𝑓 ∈ V → inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑓 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) ∈ ℝ*) |
12 | 2, 11 | fmpti 6868 | . . 3 ⊢ lim sup:V⟶ℝ* |
13 | 12 | ffvelrni 6842 | . 2 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*) |
14 | 1, 13 | syl 17 | 1 ⊢ (𝐹 ∈ 𝑉 → (lim sup‘𝐹) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 ↦ cmpt 5137 ran crn 5549 “ cima 5551 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 supcsup 8892 infcinf 8893 ℝcr 10524 +∞cpnf 10660 ℝ*cxr 10662 < clt 10663 [,)cico 12728 lim supclsp 14815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-limsup 14816 |
This theorem is referenced by: limsuplt 14824 limsupbnd1 14827 caucvgrlem 15017 limsupre 41798 limsupcld 41847 limsupcli 41914 limsupval4 41951 liminfreuzlem 41959 |
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