Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimres | Structured version Visualization version GIF version |
Description: A function converges iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimres.1 | ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
xlimres.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
Ref | Expression |
---|---|
xlimres | ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑀))~~>*𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | letopon 21741 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*)) |
3 | xlimres.1 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) | |
4 | xlimres.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
5 | 2, 3, 4 | lmres 21836 | . 2 ⊢ (𝜑 → (𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑀))(⇝𝑡‘(ordTop‘ ≤ ))𝐴)) |
6 | df-xlim 41976 | . . 3 ⊢ ~~>* = (⇝𝑡‘(ordTop‘ ≤ )) | |
7 | 6 | breqi 5063 | . 2 ⊢ (𝐹~~>*𝐴 ↔ 𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
8 | 6 | breqi 5063 | . 2 ⊢ ((𝐹 ↾ (ℤ≥‘𝑀))~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑀))(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
9 | 5, 7, 8 | 3bitr4g 315 | 1 ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑀))~~>*𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 class class class wbr 5057 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 ↑pm cpm 8396 ℂcc 10523 ℝ*cxr 10662 ≤ cle 10664 ℤcz 11969 ℤ≥cuz 12231 ordTopcordt 16760 TopOnctopon 21446 ⇝𝑡clm 21762 ~~>*clsxlim 41975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fi 8863 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-neg 10861 df-z 11970 df-uz 12232 df-topgen 16705 df-ordt 16762 df-ps 17798 df-tsr 17799 df-top 21430 df-topon 21447 df-bases 21482 df-lm 21765 df-xlim 41976 |
This theorem is referenced by: xlimconst2 41992 xlimclim2lem 41996 climxlim2 42003 xlimresdm 42016 xlimliminflimsup 42019 |
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