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Mirrors > Home > ILE Home > Th. List > climcl | 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 11291 | . . . . 5 ⊢ Rel ⇝ | |
2 | 1 | brrelex1i 4671 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐹 ∈ V) |
3 | eqidd 2178 | . . . 4 ⊢ ((𝐹 ⇝ 𝐴 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
4 | 2, 3 | clim 11292 | . . 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 176 | . 2 ⊢ (𝐹 ⇝ 𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))) |
6 | 5 | simpld 112 | 1 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 Vcvv 2739 class class class wbr 4005 ‘cfv 5218 (class class class)co 5878 ℂcc 7812 < clt 7995 − cmin 8131 ℤcz 9256 ℤ≥cuz 9531 ℝ+crp 9656 abscabs 11009 ⇝ cli 11289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-cnex 7905 ax-resscn 7906 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-ov 5881 df-neg 8134 df-z 9257 df-uz 9532 df-clim 11290 |
This theorem is referenced by: climuni 11304 fclim 11305 climeu 11307 climreu 11308 2clim 11312 climcn1lem 11330 climrecl 11335 climadd 11337 climmul 11338 climsub 11339 climaddc2 11341 climcau 11358 geoisum1c 11531 clim2divap 11551 ntrivcvgap 11559 |
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