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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 43012). (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 43032 | . 2 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))) |
5 | 1 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝑀 ∈ ℤ) |
6 | 3 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝐹:𝑍⟶ℝ) |
7 | simprl 771 | . . . . . . 7 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) ∈ ℝ) | |
8 | simprr 773 | . . . . . . 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 43031 | . . . . . 6 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝐹 ∈ dom ⇝ ) |
10 | 1 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝑀 ∈ ℤ) |
11 | 3 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹:𝑍⟶ℝ) |
12 | simpr 488 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ ) | |
13 | 10, 2, 11, 12 | climliminflimsupd 43025 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (lim inf‘𝐹) = (lim sup‘𝐹)) |
14 | 13 | eqcomd 2743 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (lim sup‘𝐹) = (lim inf‘𝐹)) |
15 | 9, 14 | syldan 594 | . . . . 5 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) = (lim inf‘𝐹)) |
16 | 15, 7 | eqeltrd 2838 | . . . 4 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ∈ ℝ) |
17 | 16, 8 | jca 515 | . . 3 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) |
18 | simpr 488 | . . . . . . 7 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → (lim sup‘𝐹) ≤ (lim inf‘𝐹)) | |
19 | 1 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → 𝑀 ∈ ℤ) |
20 | 3 | frexr 42605 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
21 | 20 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → 𝐹:𝑍⟶ℝ*) |
22 | 19, 2, 21 | liminfgelimsupuz 43012 | . . . . . . 7 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹))) |
23 | 18, 22 | mpbid 235 | . . . . . 6 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → (lim inf‘𝐹) = (lim sup‘𝐹)) |
24 | 23 | adantrl 716 | . . . . 5 ⊢ ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) = (lim sup‘𝐹)) |
25 | simprl 771 | . . . . 5 ⊢ ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ∈ ℝ) | |
26 | 24, 25 | eqeltrd 2838 | . . . 4 ⊢ ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) ∈ ℝ) |
27 | simprr 773 | . . . 4 ⊢ ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ≤ (lim inf‘𝐹)) | |
28 | 26, 27 | jca 515 | . . 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 801 | . 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 282 | 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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 class class class wbr 5058 dom cdm 5556 ⟶wf 6381 ‘cfv 6385 ℝcr 10733 ℝ*cxr 10871 ≤ cle 10873 ℤcz 12181 ℤ≥cuz 12443 lim supclsp 15036 ⇝ cli 15050 lim infclsi 42975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5184 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 ax-un 7528 ax-cnex 10790 ax-resscn 10791 ax-1cn 10792 ax-icn 10793 ax-addcl 10794 ax-addrcl 10795 ax-mulcl 10796 ax-mulrcl 10797 ax-mulcom 10798 ax-addass 10799 ax-mulass 10800 ax-distr 10801 ax-i2m1 10802 ax-1ne0 10803 ax-1rid 10804 ax-rnegex 10805 ax-rrecex 10806 ax-cnre 10807 ax-pre-lttri 10808 ax-pre-lttrn 10809 ax-pre-ltadd 10810 ax-pre-mulgt0 10811 ax-pre-sup 10812 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-pss 3890 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-tp 4551 df-op 4553 df-uni 4825 df-iun 4911 df-br 5059 df-opab 5121 df-mpt 5141 df-tr 5167 df-id 5460 df-eprel 5465 df-po 5473 df-so 5474 df-fr 5514 df-we 5516 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-pred 6165 df-ord 6221 df-on 6222 df-lim 6223 df-suc 6224 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-isom 6394 df-riota 7175 df-ov 7221 df-oprab 7222 df-mpo 7223 df-om 7650 df-1st 7766 df-2nd 7767 df-wrecs 8052 df-recs 8113 df-rdg 8151 df-1o 8207 df-er 8396 df-pm 8516 df-en 8632 df-dom 8633 df-sdom 8634 df-fin 8635 df-sup 9063 df-inf 9064 df-pnf 10874 df-mnf 10875 df-xr 10876 df-ltxr 10877 df-le 10878 df-sub 11069 df-neg 11070 df-div 11495 df-nn 11836 df-2 11898 df-3 11899 df-n0 12096 df-z 12182 df-uz 12444 df-q 12550 df-rp 12592 df-xneg 12709 df-xadd 12710 df-ioo 12944 df-ico 12946 df-fz 13101 df-fzo 13244 df-fl 13372 df-ceil 13373 df-seq 13580 df-exp 13641 df-cj 14667 df-re 14668 df-im 14669 df-sqrt 14803 df-abs 14804 df-limsup 15037 df-clim 15054 df-rlim 15055 df-liminf 42976 |
This theorem is referenced by: climliminflimsup4 43035 |
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