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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 4003 | . 2 ⊢ (𝜑 → (0[,)+∞) ⊆ (0[,)+∞)) | |
2 | 0e0icopnf 13422 | . . 3 ⊢ 0 ∈ (0[,)+∞) | |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ (0[,)+∞)) |
4 | rpssre 12968 | . . 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 486 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+) | |
9 | rpreccl 12987 | . . . . . 6 ⊢ (𝑦 ∈ ℝ+ → (1 / 𝑦) ∈ ℝ+) | |
10 | 9 | adantl 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (1 / 𝑦) ∈ ℝ+) |
11 | 10 | rpred 13003 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (1 / 𝑦) ∈ ℝ) |
12 | 10 | rpge0d 13007 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 0 ≤ (1 / 𝑦)) |
13 | elrege0 13418 | . . . 4 ⊢ ((1 / 𝑦) ∈ (0[,)+∞) ↔ ((1 / 𝑦) ∈ ℝ ∧ 0 ≤ (1 / 𝑦))) | |
14 | 11, 12, 13 | sylanbrc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → (1 / 𝑦) ∈ (0[,)+∞)) |
15 | 8, 14 | 2thd 265 | . 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 26438 | 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3946 ifcif 4524 class class class wbr 5144 ↦ cmpt 5227 ‘cfv 6535 (class class class)co 7396 ℂcc 11095 ℝcr 11096 0cc0 11097 1c1 11098 +∞cpnf 11232 ≤ cle 11236 / cdiv 11858 ℝ+crp 12961 [,)cico 13313 ⇝𝑟 crli 15416 ↾t crest 17353 TopOpenctopn 17354 ℂfldccnfld 20918 CnP ccnp 22698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-pm 8811 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-sup 9424 df-inf 9425 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-q 12920 df-rp 12962 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13315 df-ico 13317 df-fz 13472 df-seq 13954 df-exp 14015 df-cj 15033 df-re 15034 df-im 15035 df-sqrt 15169 df-abs 15170 df-rlim 15420 df-struct 17067 df-slot 17102 df-ndx 17114 df-base 17132 df-plusg 17197 df-mulr 17198 df-starv 17199 df-tset 17203 df-ple 17204 df-ds 17206 df-unif 17207 df-rest 17355 df-topn 17356 df-topgen 17376 df-psmet 20910 df-xmet 20911 df-met 20912 df-bl 20913 df-mopn 20914 df-cnfld 20919 df-top 22365 df-topon 22382 df-bases 22418 df-cnp 22701 |
This theorem is referenced by: efrlim 26441 |
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