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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 14851 | . . . . 5 ⊢ Rel ⇝ | |
2 | 1 | brrelex1i 5610 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐹 ∈ V) |
3 | eqidd 2824 | . . . 4 ⊢ ((𝐹 ⇝ 𝐴 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
4 | 2, 3 | clim 14853 | . . 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 269 | . 2 ⊢ (𝐹 ⇝ 𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))) |
6 | 5 | simpld 497 | 1 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 Vcvv 3496 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 < clt 10677 − cmin 10872 ℤcz 11984 ℤ≥cuz 12246 ℝ+crp 12392 abscabs 14595 ⇝ cli 14843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-cnex 10595 ax-resscn 10596 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-neg 10875 df-z 11985 df-uz 12247 df-clim 14847 |
This theorem is referenced by: rlimclim 14905 climrlim2 14906 climuni 14911 fclim 14912 climeu 14914 climreu 14915 2clim 14931 climcn1lem 14961 climadd 14990 climmul 14991 climsub 14992 climaddc2 14994 climcau 15029 clim2div 15247 ntrivcvgtail 15258 ntrivcvgmullem 15259 mbflim 24271 ulmcau 24985 emcllem6 25580 dchrmusum2 26072 dchrvmasumiflem1 26079 dchrvmasumiflem2 26080 dchrisum0lem1b 26093 dchrmusumlem 26100 iprodefisum 32975 climrec 41891 climexp 41893 climsuse 41896 climneg 41898 climdivf 41900 climleltrp 41964 climuzlem 42031 climxlim2lem 42133 climxlim2 42134 sge0isum 42716 |
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