Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmlimxrge0 | Structured version Visualization version GIF version |
Description: Relate a limit in the nonnegative extended reals to a complex limit, provided the considered function is a real function. (Contributed by Thierry Arnoux, 11-Jul-2017.) |
Ref | Expression |
---|---|
lmlimxrge0.j | ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
lmlimxrge0.f | ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) |
lmlimxrge0.p | ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
lmlimxrge0.x | ⊢ 𝑋 ⊆ (0[,)+∞) |
Ref | Expression |
---|---|
lmlimxrge0 | ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmlimxrge0.j | . . . 4 ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
2 | xrge0topn 31579 | . . . 4 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
3 | 1, 2 | eqtri 2762 | . . 3 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) |
4 | letopon 22074 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) | |
5 | iccssxr 13001 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
6 | resttopon 22030 | . . . 4 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) ∧ (0[,]+∞) ⊆ ℝ*) → ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞))) | |
7 | 4, 5, 6 | mp2an 692 | . . 3 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞)) |
8 | 3, 7 | eqeltri 2830 | . 2 ⊢ 𝐽 ∈ (TopOn‘(0[,]+∞)) |
9 | lmlimxrge0.f | . 2 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) | |
10 | lmlimxrge0.p | . 2 ⊢ (𝜑 → 𝑃 ∈ 𝑋) | |
11 | fvex 6719 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ V | |
12 | lmlimxrge0.x | . . . . 5 ⊢ 𝑋 ⊆ (0[,)+∞) | |
13 | icossicc 13007 | . . . . 5 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
14 | 12, 13 | sstri 3900 | . . . 4 ⊢ 𝑋 ⊆ (0[,]+∞) |
15 | ovex 7235 | . . . 4 ⊢ (0[,]+∞) ∈ V | |
16 | restabs 22034 | . . . 4 ⊢ (((ordTop‘ ≤ ) ∈ V ∧ 𝑋 ⊆ (0[,]+∞) ∧ (0[,]+∞) ∈ V) → (((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t 𝑋) = ((ordTop‘ ≤ ) ↾t 𝑋)) | |
17 | 11, 14, 15, 16 | mp3an 1463 | . . 3 ⊢ (((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t 𝑋) = ((ordTop‘ ≤ ) ↾t 𝑋) |
18 | 3 | oveq1i 7212 | . . 3 ⊢ (𝐽 ↾t 𝑋) = (((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t 𝑋) |
19 | rge0ssre 13027 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
20 | 12, 19 | sstri 3900 | . . . 4 ⊢ 𝑋 ⊆ ℝ |
21 | eqid 2734 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
22 | eqid 2734 | . . . . 5 ⊢ (ordTop‘ ≤ ) = (ordTop‘ ≤ ) | |
23 | 21, 22 | xrrest2 23677 | . . . 4 ⊢ (𝑋 ⊆ ℝ → ((TopOpen‘ℂfld) ↾t 𝑋) = ((ordTop‘ ≤ ) ↾t 𝑋)) |
24 | 20, 23 | ax-mp 5 | . . 3 ⊢ ((TopOpen‘ℂfld) ↾t 𝑋) = ((ordTop‘ ≤ ) ↾t 𝑋) |
25 | 17, 18, 24 | 3eqtr4i 2772 | . 2 ⊢ (𝐽 ↾t 𝑋) = ((TopOpen‘ℂfld) ↾t 𝑋) |
26 | ax-resscn 10769 | . . 3 ⊢ ℝ ⊆ ℂ | |
27 | 20, 26 | sstri 3900 | . 2 ⊢ 𝑋 ⊆ ℂ |
28 | 8, 9, 10, 25, 27 | lmlim 31583 | 1 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 Vcvv 3401 ⊆ wss 3857 class class class wbr 5043 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ℂcc 10710 ℝcr 10711 0cc0 10712 +∞cpnf 10847 ℝ*cxr 10849 ≤ cle 10851 ℕcn 11813 [,)cico 12920 [,]cicc 12921 ⇝ cli 15028 ↾s cress 16685 ↾t crest 16897 TopOpenctopn 16898 ordTopcordt 16976 ℝ*𝑠cxrs 16977 ℂfldccnfld 20335 TopOnctopon 21779 ⇝𝑡clm 22095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-pm 8500 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fi 9016 df-sup 9047 df-inf 9048 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ioc 12923 df-ico 12924 df-icc 12925 df-fz 13079 df-seq 13558 df-exp 13619 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-clim 15032 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-rest 16899 df-topn 16900 df-topgen 16920 df-ordt 16978 df-xrs 16979 df-ps 18044 df-tsr 18045 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-cnfld 20336 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-lm 22098 df-xms 23190 df-ms 23191 |
This theorem is referenced by: esumcvg 31738 dstfrvclim1 32128 |
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