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 29989 | . . . 4 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
3 | 1, 2 | eqtri 2644 | . . 3 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) |
4 | letopon 21009 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) | |
5 | iccssxr 12256 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
6 | resttopon 20965 | . . . 4 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) ∧ (0[,]+∞) ⊆ ℝ*) → ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞))) | |
7 | 4, 5, 6 | mp2an 708 | . . 3 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞)) |
8 | 3, 7 | eqeltri 2697 | . 2 ⊢ 𝐽 ∈ (TopOn‘(0[,]+∞)) |
9 | lmlimxrge0.f | . 2 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) | |
10 | lmlimxrge0.p | . 2 ⊢ (𝜑 → 𝑃 ∈ 𝑋) | |
11 | fvex 6201 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ V | |
12 | lmlimxrge0.x | . . . . 5 ⊢ 𝑋 ⊆ (0[,)+∞) | |
13 | icossicc 12260 | . . . . 5 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
14 | 12, 13 | sstri 3612 | . . . 4 ⊢ 𝑋 ⊆ (0[,]+∞) |
15 | ovex 6678 | . . . 4 ⊢ (0[,]+∞) ∈ V | |
16 | restabs 20969 | . . . 4 ⊢ (((ordTop‘ ≤ ) ∈ V ∧ 𝑋 ⊆ (0[,]+∞) ∧ (0[,]+∞) ∈ V) → (((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t 𝑋) = ((ordTop‘ ≤ ) ↾t 𝑋)) | |
17 | 11, 14, 15, 16 | mp3an 1424 | . . 3 ⊢ (((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t 𝑋) = ((ordTop‘ ≤ ) ↾t 𝑋) |
18 | 3 | oveq1i 6660 | . . 3 ⊢ (𝐽 ↾t 𝑋) = (((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t 𝑋) |
19 | rge0ssre 12280 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
20 | 12, 19 | sstri 3612 | . . . 4 ⊢ 𝑋 ⊆ ℝ |
21 | eqid 2622 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
22 | eqid 2622 | . . . . 5 ⊢ (ordTop‘ ≤ ) = (ordTop‘ ≤ ) | |
23 | 21, 22 | xrrest2 22611 | . . . 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 2654 | . 2 ⊢ (𝐽 ↾t 𝑋) = ((TopOpen‘ℂfld) ↾t 𝑋) |
26 | ax-resscn 9993 | . . 3 ⊢ ℝ ⊆ ℂ | |
27 | 20, 26 | sstri 3612 | . 2 ⊢ 𝑋 ⊆ ℂ |
28 | 8, 9, 10, 25, 27 | lmlim 29993 | 1 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 +∞cpnf 10071 ℝ*cxr 10073 ≤ cle 10075 ℕcn 11020 [,)cico 12177 [,]cicc 12178 ⇝ cli 14215 ↾s cress 15858 ↾t crest 16081 TopOpenctopn 16082 ordTopcordt 16159 ℝ*𝑠cxrs 16160 ℂfldccnfld 19746 TopOnctopon 20715 ⇝𝑡clm 21030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-topn 16084 df-topgen 16104 df-ordt 16161 df-xrs 16162 df-ps 17200 df-tsr 17201 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-lm 21033 df-xms 22125 df-ms 22126 |
This theorem is referenced by: esumcvg 30148 dstfrvclim1 30539 |
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