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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 14841 | . . . . 5 ⊢ Rel ⇝ | |
2 | 1 | brrelex1i 5572 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐹 ∈ V) |
3 | eqidd 2799 | . . . 4 ⊢ ((𝐹 ⇝ 𝐴 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
4 | 2, 3 | clim 14843 | . . 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 270 | . 2 ⊢ (𝐹 ⇝ 𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))) |
6 | 5 | simpld 498 | 1 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 Vcvv 3441 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 < clt 10664 − cmin 10859 ℤcz 11969 ℤ≥cuz 12231 ℝ+crp 12377 abscabs 14585 ⇝ cli 14833 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-neg 10862 df-z 11970 df-uz 12232 df-clim 14837 |
This theorem is referenced by: rlimclim 14895 climrlim2 14896 climuni 14901 fclim 14902 climeu 14904 climreu 14905 2clim 14921 climcn1lem 14951 climadd 14980 climmul 14981 climsub 14982 climaddc2 14984 climcau 15019 clim2div 15237 ntrivcvgtail 15248 ntrivcvgmullem 15249 mbflim 24272 ulmcau 24990 emcllem6 25586 dchrmusum2 26078 dchrvmasumiflem1 26085 dchrvmasumiflem2 26086 dchrisum0lem1b 26099 dchrmusumlem 26106 iprodefisum 33086 climrec 42245 climexp 42247 climsuse 42250 climneg 42252 climdivf 42254 climleltrp 42318 climuzlem 42385 climxlim2lem 42487 climxlim2 42488 sge0isum 43066 |
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