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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 21387 | . . . 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 21482 | . 2 ⊢ (𝜑 → (𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑀))(⇝𝑡‘(ordTop‘ ≤ ))𝐴)) |
6 | df-xlim 40834 | . . 3 ⊢ ~~>* = (⇝𝑡‘(ordTop‘ ≤ )) | |
7 | 6 | breqi 4881 | . 2 ⊢ (𝐹~~>*𝐴 ↔ 𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
8 | 6 | breqi 4881 | . 2 ⊢ ((𝐹 ↾ (ℤ≥‘𝑀))~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑀))(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
9 | 5, 7, 8 | 3bitr4g 306 | 1 ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑀))~~>*𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2164 class class class wbr 4875 ↾ cres 5348 ‘cfv 6127 (class class class)co 6910 ↑pm cpm 8128 ℂcc 10257 ℝ*cxr 10397 ≤ cle 10399 ℤcz 11711 ℤ≥cuz 11975 ordTopcordt 16519 TopOnctopon 21092 ⇝𝑡clm 21408 ~~>*clsxlim 40833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-pre-lttri 10333 ax-pre-lttrn 10334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fi 8592 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-neg 10595 df-z 11712 df-uz 11976 df-topgen 16464 df-ordt 16521 df-ps 17560 df-tsr 17561 df-top 21076 df-topon 21093 df-bases 21128 df-lm 21411 df-xlim 40834 |
This theorem is referenced by: xlimconst2 40850 xlimclim2lem 40854 climxlim2 40861 |
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