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| Mirrors > Home > MPE Home > Th. List > Mathboxes > liminflelimsupuz | Structured version Visualization version GIF version | ||
| Description: The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| Ref | Expression |
|---|---|
| liminflelimsupuz.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
| liminflelimsupuz.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
| liminflelimsupuz.3 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
| Ref | Expression |
|---|---|
| liminflelimsupuz | ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | liminflelimsupuz.3 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
| 2 | liminflelimsupuz.2 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 3 | 2 | fvexi 6920 | . . . 4 ⊢ 𝑍 ∈ V |
| 4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑍 ∈ V) |
| 5 | 1, 4 | fexd 7247 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
| 6 | liminflelimsupuz.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 7 | 6, 2 | uzubico2 45583 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)𝑗 ∈ 𝑍) |
| 8 | 1 | ffnd 6737 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝑍) |
| 9 | 8 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 Fn 𝑍) |
| 10 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
| 11 | id 22 | . . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ 𝑍) | |
| 12 | 2, 11 | uzxrd 45473 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ*) |
| 13 | pnfxr 11315 | . . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ* | |
| 14 | 13 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → +∞ ∈ ℝ*) |
| 15 | 12 | xrleidd 13194 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ≤ 𝑗) |
| 16 | 2, 11 | uzred 45454 | . . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ) |
| 17 | ltpnf 13162 | . . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℝ → 𝑗 < +∞) | |
| 18 | 16, 17 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → 𝑗 < +∞) |
| 19 | 12, 14, 12, 15, 18 | elicod 13437 | . . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (𝑗[,)+∞)) |
| 20 | 19 | adantl 481 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (𝑗[,)+∞)) |
| 21 | 9, 10, 20 | fnfvimad 7254 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (𝐹 “ (𝑗[,)+∞))) |
| 22 | 1 | ffvelcdmda 7104 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ*) |
| 23 | 21, 22 | elind 4200 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*)) |
| 24 | 23 | ne0d 4342 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) |
| 25 | 24 | ex 412 | . . . . . 6 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
| 26 | 25 | ad2antrr 726 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ (𝑘[,)+∞)) → (𝑗 ∈ 𝑍 → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
| 27 | 26 | reximdva 3168 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (∃𝑗 ∈ (𝑘[,)+∞)𝑗 ∈ 𝑍 → ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
| 28 | 27 | ralimdva 3167 | . . 3 ⊢ (𝜑 → (∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)𝑗 ∈ 𝑍 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
| 29 | 7, 28 | mpd 15 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) |
| 30 | 5, 29 | liminflelimsup 45791 | 1 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ∩ cin 3950 ∅c0 4333 class class class wbr 5143 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 +∞cpnf 11292 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 ℤcz 12613 ℤ≥cuz 12878 [,)cico 13389 lim supclsp 15506 lim infclsi 45766 |
| 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-1st 8014 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-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-ioo 13391 df-ico 13393 df-fl 13832 df-ceil 13833 df-limsup 15507 df-liminf 45767 |
| This theorem is referenced by: liminfgelimsupuz 45803 liminflimsupclim 45822 xlimliminflimsup 45877 |
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