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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfreuz | Structured version Visualization version GIF version |
Description: Given a function on the reals, its inferior limit is real if and only if two condition holds: 1. there is a real number that is greater than or equal to the function, infinitely often; 2. there is a real number that is smaller than or equal to the function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfreuz.1 | ⊢ Ⅎ𝑗𝐹 |
liminfreuz.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
liminfreuz.3 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
liminfreuz.4 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
Ref | Expression |
---|---|
liminfreuz | ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2908 | . . 3 ⊢ Ⅎ𝑙𝐹 | |
2 | liminfreuz.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | liminfreuz.3 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | liminfreuz.4 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
5 | 1, 2, 3, 4 | liminfreuzlem 45725 | . 2 ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑙 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑙)))) |
6 | breq2 5170 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥)) | |
7 | 6 | rexbidv 3185 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥)) |
8 | 7 | ralbidv 3184 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥)) |
9 | fveq2 6922 | . . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (ℤ≥‘𝑖) = (ℤ≥‘𝑘)) | |
10 | 9 | rexeqdv 3335 | . . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥)) |
11 | liminfreuz.1 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝐹 | |
12 | nfcv 2908 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑙 | |
13 | 11, 12 | nffv 6932 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝐹‘𝑙) |
14 | nfcv 2908 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤ | |
15 | nfcv 2908 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑥 | |
16 | 13, 14, 15 | nfbr 5213 | . . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥 |
17 | nfv 1913 | . . . . . . . . . . 11 ⊢ Ⅎ𝑙(𝐹‘𝑗) ≤ 𝑥 | |
18 | fveq2 6922 | . . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗)) | |
19 | 18 | breq1d 5176 | . . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)) |
20 | 16, 17, 19 | cbvrexw 3313 | . . . . . . . . . 10 ⊢ (∃𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) |
21 | 20 | a1i 11 | . . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
22 | 10, 21 | bitrd 279 | . . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
23 | 22 | cbvralvw 3243 | . . . . . . 7 ⊢ (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
25 | 8, 24 | bitrd 279 | . . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
26 | 25 | cbvrexvw 3244 | . . . 4 ⊢ (∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) |
27 | breq1 5169 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑙))) | |
28 | 27 | ralbidv 3184 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑙 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑙))) |
29 | 15, 14, 13 | nfbr 5213 | . . . . . . . 8 ⊢ Ⅎ𝑗 𝑥 ≤ (𝐹‘𝑙) |
30 | nfv 1913 | . . . . . . . 8 ⊢ Ⅎ𝑙 𝑥 ≤ (𝐹‘𝑗) | |
31 | 18 | breq2d 5178 | . . . . . . . 8 ⊢ (𝑙 = 𝑗 → (𝑥 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑗))) |
32 | 29, 30, 31 | cbvralw 3312 | . . . . . . 7 ⊢ (∀𝑙 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))) |
34 | 28, 33 | bitrd 279 | . . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))) |
35 | 34 | cbvrexvw 3244 | . . . 4 ⊢ (∃𝑦 ∈ ℝ ∀𝑙 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) |
36 | 26, 35 | anbi12i 627 | . . 3 ⊢ ((∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑙 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑙)) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))) |
37 | 36 | a1i 11 | . 2 ⊢ (𝜑 → ((∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑙 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑙)) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))) |
38 | 5, 37 | bitrd 279 | 1 ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 Ⅎwnfc 2893 ∀wral 3067 ∃wrex 3076 class class class wbr 5166 ⟶wf 6571 ‘cfv 6575 ℝcr 11185 ≤ cle 11327 ℤcz 12641 ℤ≥cuz 12905 lim infclsi 45674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-pre-sup 11264 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-isom 6584 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-sup 9513 df-inf 9514 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-div 11950 df-nn 12296 df-n0 12556 df-z 12642 df-uz 12906 df-q 13016 df-xneg 13177 df-ico 13415 df-fz 13570 df-fzo 13714 df-fl 13845 df-ceil 13846 df-limsup 15519 df-liminf 45675 |
This theorem is referenced by: (None) |
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