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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 14631 | . . . . 5 ⊢ Rel ⇝ | |
2 | 1 | brrelex1i 5406 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐹 ∈ V) |
3 | eqidd 2778 | . . . 4 ⊢ ((𝐹 ⇝ 𝐴 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
4 | 2, 3 | clim 14633 | . . 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 259 | . 2 ⊢ (𝐹 ⇝ 𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))) |
6 | 5 | simpld 490 | 1 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2106 ∀wral 3089 ∃wrex 3090 Vcvv 3397 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 < clt 10411 − cmin 10606 ℤcz 11728 ℤ≥cuz 11992 ℝ+crp 12137 abscabs 14381 ⇝ cli 14623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-cnex 10328 ax-resscn 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-ov 6925 df-neg 10609 df-z 11729 df-uz 11993 df-clim 14627 |
This theorem is referenced by: rlimclim 14685 climrlim2 14686 climuni 14691 fclim 14692 climeu 14694 climreu 14695 2clim 14711 climcn1lem 14741 climadd 14770 climmul 14771 climsub 14772 climaddc2 14774 climcau 14809 clim2div 15024 ntrivcvgtail 15035 ntrivcvgmullem 15036 mbflim 23872 ulmcau 24586 emcllem6 25179 dchrmusum2 25635 dchrvmasumiflem1 25642 dchrvmasumiflem2 25643 dchrisum0lem1b 25656 dchrmusumlem 25663 iprodefisum 32221 climrec 40725 climexp 40727 climsuse 40730 climneg 40732 climdivf 40734 climleltrp 40798 climuzlem 40865 climxlim2lem 40967 climxlim2 40968 sge0isum 41550 |
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