Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfgelimsupuz | Structured version Visualization version GIF version |
Description: The inferior limit is greater than or equal to the superior limit if and only if they are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfgelimsupuz.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
liminfgelimsupuz.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
liminfgelimsupuz.3 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
Ref | Expression |
---|---|
liminfgelimsupuz | ⊢ (𝜑 → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | liminfgelimsupuz.3 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
2 | liminfgelimsupuz.2 | . . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | 2 | fvexi 6782 | . . . . . . 7 ⊢ 𝑍 ∈ V |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ V) |
5 | 1, 4 | fexd 7097 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ V) |
6 | 5 | liminfcld 43265 | . . . 4 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ*) |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → (lim inf‘𝐹) ∈ ℝ*) |
8 | 5 | limsupcld 43185 | . . . 4 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ*) |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → (lim sup‘𝐹) ∈ ℝ*) |
10 | liminfgelimsupuz.1 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
11 | 10, 2, 1 | liminflelimsupuz 43280 | . . . 4 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) |
12 | 11 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) |
13 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → (lim sup‘𝐹) ≤ (lim inf‘𝐹)) | |
14 | 7, 9, 12, 13 | xrletrid 12871 | . 2 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → (lim inf‘𝐹) = (lim sup‘𝐹)) |
15 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) → (lim sup‘𝐹) ∈ ℝ*) |
16 | id 22 | . . . . 5 ⊢ ((lim inf‘𝐹) = (lim sup‘𝐹) → (lim inf‘𝐹) = (lim sup‘𝐹)) | |
17 | 16 | eqcomd 2745 | . . . 4 ⊢ ((lim inf‘𝐹) = (lim sup‘𝐹) → (lim sup‘𝐹) = (lim inf‘𝐹)) |
18 | 17 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) → (lim sup‘𝐹) = (lim inf‘𝐹)) |
19 | 15, 18 | xreqled 42823 | . 2 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) → (lim sup‘𝐹) ≤ (lim inf‘𝐹)) |
20 | 14, 19 | impbida 797 | 1 ⊢ (𝜑 → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1541 ∈ wcel 2109 Vcvv 3430 class class class wbr 5078 ⟶wf 6426 ‘cfv 6430 ℝ*cxr 10992 ≤ cle 10994 ℤcz 12302 ℤ≥cuz 12564 lim supclsp 15160 lim infclsi 43246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-sup 9162 df-inf 9163 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-n0 12217 df-z 12303 df-uz 12565 df-ioo 13065 df-ico 13067 df-fl 13493 df-ceil 13494 df-limsup 15161 df-liminf 43247 |
This theorem is referenced by: climliminflimsup2 43304 climliminflimsup3 43305 climliminflimsup4 43306 xlimlimsupleliminf 43358 |
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