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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 15529 | . . . . 5 ⊢ Rel ⇝ | |
| 2 | 1 | brrelex1i 5740 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐹 ∈ V) | 
| 3 | eqidd 2737 | . . . 4 ⊢ ((𝐹 ⇝ 𝐴 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
| 4 | 2, 3 | clim 15531 | . . 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 2107 ∀wral 3060 ∃wrex 3069 Vcvv 3479 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 < clt 11296 − cmin 11493 ℤcz 12615 ℤ≥cuz 12879 ℝ+crp 13035 abscabs 15274 ⇝ cli 15521 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-cnex 11212 ax-resscn 11213 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-neg 11496 df-z 12616 df-uz 12880 df-clim 15525 | 
| This theorem is referenced by: rlimclim 15583 climrlim2 15584 climuni 15589 fclim 15590 climeu 15592 climreu 15593 2clim 15609 climcn1lem 15640 climadd 15669 climmul 15670 climsub 15671 climaddc2 15673 climcau 15708 clim2div 15926 ntrivcvgtail 15937 ntrivcvgmullem 15938 mbflim 25704 ulmcau 26439 emcllem6 27045 dchrmusum2 27539 dchrvmasumiflem1 27546 dchrvmasumiflem2 27547 dchrisum0lem1b 27560 dchrmusumlem 27567 iprodefisum 35742 climrec 45623 climexp 45625 climsuse 45628 climneg 45630 climdivf 45632 climleltrp 45696 climuzlem 45763 climxlim2lem 45865 climxlim2 45866 sge0isum 46447 | 
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