![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 6905 | . . . . . . 7 ⊢ 𝑍 ∈ V |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ V) |
5 | 1, 4 | fexd 7231 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ V) |
6 | 5 | liminfcld 44945 | . . . 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 44865 | . . . 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 44960 | . . . 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 13141 | . 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 2737 | . . . 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 44499 | . 2 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) → (lim sup‘𝐹) ≤ (lim inf‘𝐹)) |
20 | 14, 19 | impbida 798 | 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 1540 ∈ wcel 2105 Vcvv 3473 class class class wbr 5148 ⟶wf 6539 ‘cfv 6543 ℝ*cxr 11254 ≤ cle 11256 ℤcz 12565 ℤ≥cuz 12829 lim supclsp 15421 lim infclsi 44926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-ioo 13335 df-ico 13337 df-fl 13764 df-ceil 13765 df-limsup 15422 df-liminf 44927 |
This theorem is referenced by: climliminflimsup2 44984 climliminflimsup3 44985 climliminflimsup4 44986 xlimlimsupleliminf 45038 |
Copyright terms: Public domain | W3C validator |