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Mirrors > Home > MPE Home > Th. List > climcl | Structured version Visualization version GIF version |
Description: Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
climcl | ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climrel 15525 | . . . . 5 ⊢ Rel ⇝ | |
2 | 1 | brrelex1i 5745 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐹 ∈ V) |
3 | eqidd 2736 | . . . 4 ⊢ ((𝐹 ⇝ 𝐴 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
4 | 2, 3 | clim 15527 | . . 3 ⊢ (𝐹 ⇝ 𝐴 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))) |
5 | 4 | ibi 267 | . 2 ⊢ (𝐹 ⇝ 𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))) |
6 | 5 | simpld 494 | 1 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 Vcvv 3478 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 < clt 11293 − cmin 11490 ℤcz 12611 ℤ≥cuz 12876 ℝ+crp 13032 abscabs 15270 ⇝ cli 15517 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-neg 11493 df-z 12612 df-uz 12877 df-clim 15521 |
This theorem is referenced by: rlimclim 15579 climrlim2 15580 climuni 15585 fclim 15586 climeu 15588 climreu 15589 2clim 15605 climcn1lem 15636 climadd 15665 climmul 15666 climsub 15667 climaddc2 15669 climcau 15704 clim2div 15922 ntrivcvgtail 15933 ntrivcvgmullem 15934 mbflim 25717 ulmcau 26453 emcllem6 27059 dchrmusum2 27553 dchrvmasumiflem1 27560 dchrvmasumiflem2 27561 dchrisum0lem1b 27574 dchrmusumlem 27581 iprodefisum 35721 climrec 45559 climexp 45561 climsuse 45564 climneg 45566 climdivf 45568 climleltrp 45632 climuzlem 45699 climxlim2lem 45801 climxlim2 45802 sge0isum 46383 |
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