![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > climliminflimsup2 | Structured version Visualization version GIF version |
Description: A sequence of real numbers converges if and only if its superior limit is real and it is less than or equal to its inferior limit (in such a case, they are actually equal, see liminfgelimsupuz 45176). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
climliminflimsup2.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climliminflimsup2.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climliminflimsup2.3 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
Ref | Expression |
---|---|
climliminflimsup2 | ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climliminflimsup2.1 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | climliminflimsup2.2 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | climliminflimsup2.3 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
4 | 1, 2, 3 | climliminflimsup 45196 | . 2 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))) |
5 | 1 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝑀 ∈ ℤ) |
6 | 3 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝐹:𝑍⟶ℝ) |
7 | simprl 770 | . . . . . . 7 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) ∈ ℝ) | |
8 | simprr 772 | . . . . . . 7 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ≤ (lim inf‘𝐹)) | |
9 | 5, 2, 6, 7, 8 | liminflimsupclim 45195 | . . . . . 6 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝐹 ∈ dom ⇝ ) |
10 | 1 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝑀 ∈ ℤ) |
11 | 3 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹:𝑍⟶ℝ) |
12 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ ) | |
13 | 10, 2, 11, 12 | climliminflimsupd 45189 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (lim inf‘𝐹) = (lim sup‘𝐹)) |
14 | 13 | eqcomd 2734 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (lim sup‘𝐹) = (lim inf‘𝐹)) |
15 | 9, 14 | syldan 590 | . . . . 5 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) = (lim inf‘𝐹)) |
16 | 15, 7 | eqeltrd 2829 | . . . 4 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ∈ ℝ) |
17 | 16, 8 | jca 511 | . . 3 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) |
18 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → (lim sup‘𝐹) ≤ (lim inf‘𝐹)) | |
19 | 1 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → 𝑀 ∈ ℤ) |
20 | 3 | frexr 44767 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
21 | 20 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → 𝐹:𝑍⟶ℝ*) |
22 | 19, 2, 21 | liminfgelimsupuz 45176 | . . . . . . 7 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹))) |
23 | 18, 22 | mpbid 231 | . . . . . 6 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → (lim inf‘𝐹) = (lim sup‘𝐹)) |
24 | 23 | adantrl 715 | . . . . 5 ⊢ ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) = (lim sup‘𝐹)) |
25 | simprl 770 | . . . . 5 ⊢ ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ∈ ℝ) | |
26 | 24, 25 | eqeltrd 2829 | . . . 4 ⊢ ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) ∈ ℝ) |
27 | simprr 772 | . . . 4 ⊢ ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ≤ (lim inf‘𝐹)) | |
28 | 26, 27 | jca 511 | . . 3 ⊢ ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) |
29 | 17, 28 | impbida 800 | . 2 ⊢ (𝜑 → (((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))) |
30 | 4, 29 | bitrd 279 | 1 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 dom cdm 5678 ⟶wf 6544 ‘cfv 6548 ℝcr 11138 ℝ*cxr 11278 ≤ cle 11280 ℤcz 12589 ℤ≥cuz 12853 lim supclsp 15447 ⇝ cli 15461 lim infclsi 45139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-ioo 13361 df-ico 13363 df-fz 13518 df-fzo 13661 df-fl 13790 df-ceil 13791 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-limsup 15448 df-clim 15465 df-rlim 15466 df-liminf 45140 |
This theorem is referenced by: climliminflimsup4 45199 |
Copyright terms: Public domain | W3C validator |