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 31190 | . . . 4 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
3 | 1, 2 | eqtri 2847 | . . 3 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) |
4 | letopon 21816 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) | |
5 | iccssxr 12822 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
6 | resttopon 21772 | . . . 4 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) ∧ (0[,]+∞) ⊆ ℝ*) → ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞))) | |
7 | 4, 5, 6 | mp2an 690 | . . 3 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞)) |
8 | 3, 7 | eqeltri 2912 | . 2 ⊢ 𝐽 ∈ (TopOn‘(0[,]+∞)) |
9 | lmlimxrge0.f | . 2 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) | |
10 | lmlimxrge0.p | . 2 ⊢ (𝜑 → 𝑃 ∈ 𝑋) | |
11 | fvex 6686 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ V | |
12 | lmlimxrge0.x | . . . . 5 ⊢ 𝑋 ⊆ (0[,)+∞) | |
13 | icossicc 12827 | . . . . 5 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
14 | 12, 13 | sstri 3979 | . . . 4 ⊢ 𝑋 ⊆ (0[,]+∞) |
15 | ovex 7192 | . . . 4 ⊢ (0[,]+∞) ∈ V | |
16 | restabs 21776 | . . . 4 ⊢ (((ordTop‘ ≤ ) ∈ V ∧ 𝑋 ⊆ (0[,]+∞) ∧ (0[,]+∞) ∈ V) → (((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t 𝑋) = ((ordTop‘ ≤ ) ↾t 𝑋)) | |
17 | 11, 14, 15, 16 | mp3an 1457 | . . 3 ⊢ (((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t 𝑋) = ((ordTop‘ ≤ ) ↾t 𝑋) |
18 | 3 | oveq1i 7169 | . . 3 ⊢ (𝐽 ↾t 𝑋) = (((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t 𝑋) |
19 | rge0ssre 12847 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
20 | 12, 19 | sstri 3979 | . . . 4 ⊢ 𝑋 ⊆ ℝ |
21 | eqid 2824 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
22 | eqid 2824 | . . . . 5 ⊢ (ordTop‘ ≤ ) = (ordTop‘ ≤ ) | |
23 | 21, 22 | xrrest2 23419 | . . . 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 2857 | . 2 ⊢ (𝐽 ↾t 𝑋) = ((TopOpen‘ℂfld) ↾t 𝑋) |
26 | ax-resscn 10597 | . . 3 ⊢ ℝ ⊆ ℂ | |
27 | 20, 26 | sstri 3979 | . 2 ⊢ 𝑋 ⊆ ℂ |
28 | 8, 9, 10, 25, 27 | lmlim 31194 | 1 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ⊆ wss 3939 class class class wbr 5069 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 0cc0 10540 +∞cpnf 10675 ℝ*cxr 10677 ≤ cle 10679 ℕcn 11641 [,)cico 12743 [,]cicc 12744 ⇝ cli 14844 ↾s cress 16487 ↾t crest 16697 TopOpenctopn 16698 ordTopcordt 16775 ℝ*𝑠cxrs 16776 ℂfldccnfld 20548 TopOnctopon 21521 ⇝𝑡clm 21837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fi 8878 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-rest 16699 df-topn 16700 df-topgen 16720 df-ordt 16777 df-xrs 16778 df-ps 17813 df-tsr 17814 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-lm 21840 df-xms 22933 df-ms 22934 |
This theorem is referenced by: esumcvg 31349 dstfrvclim1 31739 |
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