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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimresdm | Structured version Visualization version GIF version |
Description: A function converges in the extended reals iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
Ref | Expression |
---|---|
xlimresdm.1 | ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
xlimresdm.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
Ref | Expression |
---|---|
xlimresdm | ⊢ (𝜑 → (𝐹 ∈ dom ~~>* ↔ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ~~>*)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimrel 44151 | . . 3 ⊢ Rel ~~>* | |
2 | xlimdm 44188 | . . . . . 6 ⊢ (𝐹 ∈ dom ~~>* ↔ 𝐹~~>*(~~>*‘𝐹)) | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ dom ~~>* ↔ 𝐹~~>*(~~>*‘𝐹))) |
4 | 3 | biimpa 478 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ~~>*) → 𝐹~~>*(~~>*‘𝐹)) |
5 | xlimresdm.1 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) | |
6 | 5 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ~~>*) → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
7 | xlimresdm.2 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
8 | 7 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ~~>*) → 𝑀 ∈ ℤ) |
9 | 6, 8 | xlimres 44152 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ~~>*) → (𝐹~~>*(~~>*‘𝐹) ↔ (𝐹 ↾ (ℤ≥‘𝑀))~~>*(~~>*‘𝐹))) |
10 | 4, 9 | mpbid 231 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ~~>*) → (𝐹 ↾ (ℤ≥‘𝑀))~~>*(~~>*‘𝐹)) |
11 | releldm 5903 | . . 3 ⊢ ((Rel ~~>* ∧ (𝐹 ↾ (ℤ≥‘𝑀))~~>*(~~>*‘𝐹)) → (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ~~>*) | |
12 | 1, 10, 11 | sylancr 588 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ~~>*) → (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ~~>*) |
13 | xlimdm 44188 | . . . . . 6 ⊢ ((𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ~~>* ↔ (𝐹 ↾ (ℤ≥‘𝑀))~~>*(~~>*‘(𝐹 ↾ (ℤ≥‘𝑀)))) | |
14 | 13 | biimpi 215 | . . . . 5 ⊢ ((𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ~~>* → (𝐹 ↾ (ℤ≥‘𝑀))~~>*(~~>*‘(𝐹 ↾ (ℤ≥‘𝑀)))) |
15 | 14 | adantl 483 | . . . 4 ⊢ ((𝜑 ∧ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ~~>*) → (𝐹 ↾ (ℤ≥‘𝑀))~~>*(~~>*‘(𝐹 ↾ (ℤ≥‘𝑀)))) |
16 | 5, 7 | xlimres 44152 | . . . . 5 ⊢ (𝜑 → (𝐹~~>*(~~>*‘(𝐹 ↾ (ℤ≥‘𝑀))) ↔ (𝐹 ↾ (ℤ≥‘𝑀))~~>*(~~>*‘(𝐹 ↾ (ℤ≥‘𝑀))))) |
17 | 16 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ~~>*) → (𝐹~~>*(~~>*‘(𝐹 ↾ (ℤ≥‘𝑀))) ↔ (𝐹 ↾ (ℤ≥‘𝑀))~~>*(~~>*‘(𝐹 ↾ (ℤ≥‘𝑀))))) |
18 | 15, 17 | mpbird 257 | . . 3 ⊢ ((𝜑 ∧ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ~~>*) → 𝐹~~>*(~~>*‘(𝐹 ↾ (ℤ≥‘𝑀)))) |
19 | releldm 5903 | . . 3 ⊢ ((Rel ~~>* ∧ 𝐹~~>*(~~>*‘(𝐹 ↾ (ℤ≥‘𝑀)))) → 𝐹 ∈ dom ~~>*) | |
20 | 1, 18, 19 | sylancr 588 | . 2 ⊢ ((𝜑 ∧ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ~~>*) → 𝐹 ∈ dom ~~>*) |
21 | 12, 20 | impbida 800 | 1 ⊢ (𝜑 → (𝐹 ∈ dom ~~>* ↔ (𝐹 ↾ (ℤ≥‘𝑀)) ∈ dom ~~>*)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 class class class wbr 5109 dom cdm 5637 ↾ cres 5639 Rel wrel 5642 ‘cfv 6500 (class class class)co 7361 ↑pm cpm 8772 ℂcc 11057 ℝ*cxr 11196 ℤcz 12507 ℤ≥cuz 12771 ~~>*clsxlim 44149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-pm 8774 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fi 9355 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-z 12508 df-uz 12772 df-topgen 17333 df-ordt 17391 df-ps 18463 df-tsr 18464 df-top 22266 df-topon 22283 df-bases 22319 df-lm 22603 df-haus 22689 df-xlim 44150 |
This theorem is referenced by: xlimliminflimsup 44193 |
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