Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climliminf | Structured version Visualization version GIF version |
Description: A sequence of real numbers converges if and only if it converges to its inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
climliminf.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climliminf.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climliminf.3 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
Ref | Expression |
---|---|
climliminf | ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim inf‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climliminf.1 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | climliminf.2 | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | climliminf.3 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
4 | 1, 2, 3 | climlimsup 43693 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim sup‘𝐹))) |
5 | 4 | biimpd 228 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ → 𝐹 ⇝ (lim sup‘𝐹))) |
6 | 5 | imp 408 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ (lim sup‘𝐹)) |
7 | 1 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝑀 ∈ ℤ) |
8 | 3 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹:𝑍⟶ℝ) |
9 | simpr 486 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ ) | |
10 | 7, 2, 8, 9 | climliminflimsupd 43734 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (lim inf‘𝐹) = (lim sup‘𝐹)) |
11 | 6, 10 | breqtrrd 5131 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ (lim inf‘𝐹)) |
12 | climrel 15308 | . . . 4 ⊢ Rel ⇝ | |
13 | 12 | releldmi 5899 | . . 3 ⊢ (𝐹 ⇝ (lim inf‘𝐹) → 𝐹 ∈ dom ⇝ ) |
14 | 13 | adantl 483 | . 2 ⊢ ((𝜑 ∧ 𝐹 ⇝ (lim inf‘𝐹)) → 𝐹 ∈ dom ⇝ ) |
15 | 11, 14 | impbida 799 | 1 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim inf‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 dom cdm 5630 ⟶wf 6487 ‘cfv 6491 ℝcr 10983 ℤcz 12432 ℤ≥cuz 12695 lim supclsp 15286 ⇝ cli 15300 lim infclsi 43684 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 ax-pre-sup 11062 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-1st 7911 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-er 8581 df-pm 8701 df-en 8817 df-dom 8818 df-sdom 8819 df-sup 9311 df-inf 9312 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-div 11746 df-nn 12087 df-2 12149 df-3 12150 df-n0 12347 df-z 12433 df-uz 12696 df-q 12802 df-rp 12844 df-xneg 12961 df-ico 13198 df-fl 13625 df-seq 13835 df-exp 13896 df-cj 14917 df-re 14918 df-im 14919 df-sqrt 15053 df-abs 15054 df-limsup 15287 df-clim 15304 df-rlim 15305 df-liminf 43685 |
This theorem is referenced by: climliminflimsup 43741 dmclimxlim 43784 |
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