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Mirrors > Home > MPE Home > Th. List > lmclimf | Structured version Visualization version GIF version |
Description: Relate a limit on the metric space of complex numbers to our complex number limit notation. (Contributed by NM, 24-Jul-2007.) (Revised by Mario Carneiro, 1-May-2014.) |
Ref | Expression |
---|---|
lmclim.2 | ⊢ 𝐽 = (TopOpen‘ℂfld) |
lmclim.3 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
lmclimf | ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → 𝐹:𝑍⟶ℂ) | |
2 | lmclim.3 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | uzssz 12822 | . . . . 5 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
4 | zsscn 12545 | . . . . 5 ⊢ ℤ ⊆ ℂ | |
5 | 3, 4 | sstri 3984 | . . . 4 ⊢ (ℤ≥‘𝑀) ⊆ ℂ |
6 | 2, 5 | eqsstri 4009 | . . 3 ⊢ 𝑍 ⊆ ℂ |
7 | cnex 11170 | . . . 4 ⊢ ℂ ∈ V | |
8 | elpm2r 8819 | . . . 4 ⊢ (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝑍⟶ℂ ∧ 𝑍 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ)) | |
9 | 7, 7, 8 | mpanl12 700 | . . 3 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑍 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ)) |
10 | 1, 6, 9 | sylancl 586 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ)) |
11 | fdm 6710 | . . . 4 ⊢ (𝐹:𝑍⟶ℂ → dom 𝐹 = 𝑍) | |
12 | eqimss2 4034 | . . . 4 ⊢ (dom 𝐹 = 𝑍 → 𝑍 ⊆ dom 𝐹) | |
13 | 1, 11, 12 | 3syl 18 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → 𝑍 ⊆ dom 𝐹) |
14 | lmclim.2 | . . . 4 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
15 | 14, 2 | lmclim 24744 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑍 ⊆ dom 𝐹) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐹 ⇝ 𝑃))) |
16 | 13, 15 | syldan 591 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐹 ⇝ 𝑃))) |
17 | 10, 16 | mpbirand 705 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3470 ⊆ wss 3941 class class class wbr 5138 dom cdm 5666 ⟶wf 6525 ‘cfv 6529 (class class class)co 7390 ↑pm cpm 8801 ℂcc 11087 ℤcz 12537 ℤ≥cuz 12801 ⇝ cli 15407 TopOpenctopn 17346 ℂfldccnfld 20873 ⇝𝑡clm 22654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 ax-pre-mulgt0 11166 ax-pre-sup 11167 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-tp 4624 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7836 df-1st 7954 df-2nd 7955 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-1o 8445 df-er 8683 df-map 8802 df-pm 8803 df-en 8920 df-dom 8921 df-sdom 8922 df-fin 8923 df-sup 9416 df-inf 9417 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11425 df-neg 11426 df-div 11851 df-nn 12192 df-2 12254 df-3 12255 df-4 12256 df-5 12257 df-6 12258 df-7 12259 df-8 12260 df-9 12261 df-n0 12452 df-z 12538 df-dec 12657 df-uz 12802 df-q 12912 df-rp 12954 df-xneg 13071 df-xadd 13072 df-xmul 13073 df-fz 13464 df-seq 13946 df-exp 14007 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-clim 15411 df-struct 17059 df-slot 17094 df-ndx 17106 df-base 17124 df-plusg 17189 df-mulr 17190 df-starv 17191 df-tset 17195 df-ple 17196 df-ds 17198 df-unif 17199 df-rest 17347 df-topn 17348 df-topgen 17368 df-psmet 20865 df-xmet 20866 df-met 20867 df-bl 20868 df-mopn 20869 df-cnfld 20874 df-top 22320 df-topon 22337 df-bases 22373 df-lm 22657 |
This theorem is referenced by: lmlim 32742 climreeq 44088 |
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