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Mirrors > Home > MPE Home > Th. List > rlimcnp3 | Structured version Visualization version GIF version |
Description: Relate a limit of a real-valued sequence at infinity to the continuity of the function 𝑆(𝑦) = 𝑅(1 / 𝑦) at zero. (Contributed by Mario Carneiro, 1-Mar-2015.) |
Ref | Expression |
---|---|
rlimcnp3.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
rlimcnp3.r | ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝑆 ∈ ℂ) |
rlimcnp3.s | ⊢ (𝑦 = (1 / 𝑥) → 𝑆 = 𝑅) |
rlimcnp3.j | ⊢ 𝐽 = (TopOpen‘ℂfld) |
rlimcnp3.k | ⊢ 𝐾 = (𝐽 ↾t (0[,)+∞)) |
Ref | Expression |
---|---|
rlimcnp3 | ⊢ (𝜑 → ((𝑦 ∈ ℝ+ ↦ 𝑆) ⇝𝑟 𝐶 ↔ (𝑥 ∈ (0[,)+∞) ↦ if(𝑥 = 0, 𝐶, 𝑅)) ∈ ((𝐾 CnP 𝐽)‘0))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssidd 3989 | . 2 ⊢ (𝜑 → (0[,)+∞) ⊆ (0[,)+∞)) | |
2 | 0e0icopnf 12840 | . . 3 ⊢ 0 ∈ (0[,)+∞) | |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ (0[,)+∞)) |
4 | rpssre 12390 | . . 3 ⊢ ℝ+ ⊆ ℝ | |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → ℝ+ ⊆ ℝ) |
6 | rlimcnp3.c | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
7 | rlimcnp3.r | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝑆 ∈ ℂ) | |
8 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+) | |
9 | rpreccl 12409 | . . . . . 6 ⊢ (𝑦 ∈ ℝ+ → (1 / 𝑦) ∈ ℝ+) | |
10 | 9 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (1 / 𝑦) ∈ ℝ+) |
11 | 10 | rpred 12425 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (1 / 𝑦) ∈ ℝ) |
12 | 10 | rpge0d 12429 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 0 ≤ (1 / 𝑦)) |
13 | elrege0 12836 | . . . 4 ⊢ ((1 / 𝑦) ∈ (0[,)+∞) ↔ ((1 / 𝑦) ∈ ℝ ∧ 0 ≤ (1 / 𝑦))) | |
14 | 11, 12, 13 | sylanbrc 585 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (1 / 𝑦) ∈ (0[,)+∞)) |
15 | 8, 14 | 2thd 267 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (𝑦 ∈ ℝ+ ↔ (1 / 𝑦) ∈ (0[,)+∞))) |
16 | rlimcnp3.s | . 2 ⊢ (𝑦 = (1 / 𝑥) → 𝑆 = 𝑅) | |
17 | rlimcnp3.j | . 2 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
18 | rlimcnp3.k | . 2 ⊢ 𝐾 = (𝐽 ↾t (0[,)+∞)) | |
19 | 1, 3, 5, 6, 7, 15, 16, 17, 18 | rlimcnp2 25538 | 1 ⊢ (𝜑 → ((𝑦 ∈ ℝ+ ↦ 𝑆) ⇝𝑟 𝐶 ↔ (𝑥 ∈ (0[,)+∞) ↦ if(𝑥 = 0, 𝐶, 𝑅)) ∈ ((𝐾 CnP 𝐽)‘0))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ifcif 4466 class class class wbr 5058 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 +∞cpnf 10666 ≤ cle 10670 / cdiv 11291 ℝ+crp 12383 [,)cico 12734 ⇝𝑟 crli 14836 ↾t crest 16688 TopOpenctopn 16689 ℂfldccnfld 20539 CnP ccnp 21827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ico 12738 df-fz 12887 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-rlim 14840 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-rest 16690 df-topn 16691 df-topgen 16711 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-cnfld 20540 df-top 21496 df-topon 21513 df-bases 21548 df-cnp 21830 |
This theorem is referenced by: efrlim 25541 |
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