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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 40786 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim sup‘𝐹))) |
5 | 4 | biimpd 221 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ → 𝐹 ⇝ (lim sup‘𝐹))) |
6 | 5 | imp 397 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ (lim sup‘𝐹)) |
7 | 1 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝑀 ∈ ℤ) |
8 | 3 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹:𝑍⟶ℝ) |
9 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ ) | |
10 | 7, 2, 8, 9 | climliminflimsupd 40827 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (lim inf‘𝐹) = (lim sup‘𝐹)) |
11 | 6, 10 | breqtrrd 4900 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ (lim inf‘𝐹)) |
12 | climrel 14599 | . . . 4 ⊢ Rel ⇝ | |
13 | 12 | releldmi 5594 | . . 3 ⊢ (𝐹 ⇝ (lim inf‘𝐹) → 𝐹 ∈ dom ⇝ ) |
14 | 13 | adantl 475 | . 2 ⊢ ((𝜑 ∧ 𝐹 ⇝ (lim inf‘𝐹)) → 𝐹 ∈ dom ⇝ ) |
15 | 11, 14 | impbida 837 | 1 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim inf‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 class class class wbr 4872 dom cdm 5341 ⟶wf 6118 ‘cfv 6122 ℝcr 10250 ℤcz 11703 ℤ≥cuz 11967 lim supclsp 14577 ⇝ cli 14591 lim infclsi 40777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-er 8008 df-pm 8124 df-en 8222 df-dom 8223 df-sdom 8224 df-sup 8616 df-inf 8617 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-n0 11618 df-z 11704 df-uz 11968 df-q 12071 df-rp 12112 df-xneg 12231 df-ico 12468 df-fl 12887 df-seq 13095 df-exp 13154 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 df-abs 14352 df-limsup 14578 df-clim 14595 df-rlim 14596 df-liminf 40778 |
This theorem is referenced by: climliminflimsup 40834 |
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