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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminflelimsup | Structured version Visualization version GIF version |
Description: The superior limit is greater than or equal to the inferior limit. The second hypothesis is needed (see liminflelimsupcex 42439 for a counterexample). The inequality can be strict, see liminfltlimsupex 42423. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminflelimsup.1 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
liminflelimsup.2 | ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) |
Ref | Expression |
---|---|
liminflelimsup | ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | liminflelimsup.1 | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
2 | liminflelimsup.2 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) | |
3 | oveq1 7142 | . . . . . 6 ⊢ (𝑘 = 𝑖 → (𝑘[,)+∞) = (𝑖[,)+∞)) | |
4 | 3 | rexeqdv 3365 | . . . . 5 ⊢ (𝑘 = 𝑖 → (∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅ ↔ ∃𝑗 ∈ (𝑖[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)) |
5 | oveq1 7142 | . . . . . . . . . 10 ⊢ (𝑗 = 𝑙 → (𝑗[,)+∞) = (𝑙[,)+∞)) | |
6 | 5 | imaeq2d 5896 | . . . . . . . . 9 ⊢ (𝑗 = 𝑙 → (𝐹 “ (𝑗[,)+∞)) = (𝐹 “ (𝑙[,)+∞))) |
7 | 6 | ineq1d 4138 | . . . . . . . 8 ⊢ (𝑗 = 𝑙 → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ*)) |
8 | 7 | neeq1d 3046 | . . . . . . 7 ⊢ (𝑗 = 𝑙 → (((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅ ↔ ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ*) ≠ ∅)) |
9 | 8 | cbvrexvw 3397 | . . . . . 6 ⊢ (∃𝑗 ∈ (𝑖[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅ ↔ ∃𝑙 ∈ (𝑖[,)+∞)((𝐹 “ (𝑙[,)+∞)) ∩ ℝ*) ≠ ∅) |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝑘 = 𝑖 → (∃𝑗 ∈ (𝑖[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅ ↔ ∃𝑙 ∈ (𝑖[,)+∞)((𝐹 “ (𝑙[,)+∞)) ∩ ℝ*) ≠ ∅)) |
11 | 4, 10 | bitrd 282 | . . . 4 ⊢ (𝑘 = 𝑖 → (∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅ ↔ ∃𝑙 ∈ (𝑖[,)+∞)((𝐹 “ (𝑙[,)+∞)) ∩ ℝ*) ≠ ∅)) |
12 | 11 | cbvralvw 3396 | . . 3 ⊢ (∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅ ↔ ∀𝑖 ∈ ℝ ∃𝑙 ∈ (𝑖[,)+∞)((𝐹 “ (𝑙[,)+∞)) ∩ ℝ*) ≠ ∅) |
13 | 2, 12 | sylib 221 | . 2 ⊢ (𝜑 → ∀𝑖 ∈ ℝ ∃𝑙 ∈ (𝑖[,)+∞)((𝐹 “ (𝑙[,)+∞)) ∩ ℝ*) ≠ ∅) |
14 | 1, 13 | liminflelimsuplem 42417 | 1 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ∩ cin 3880 ∅c0 4243 class class class wbr 5030 “ cima 5522 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 +∞cpnf 10661 ℝ*cxr 10663 ≤ cle 10665 [,)cico 12728 lim supclsp 14819 lim infclsi 42393 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-ico 12732 df-limsup 14820 df-liminf 42394 |
This theorem is referenced by: liminfgelimsup 42424 liminflelimsupuz 42427 |
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