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Mirrors > Home > MPE Home > Th. List > rlimcl | Structured version Visualization version GIF version |
Description: Closure of the limit of a sequence of complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
rlimcl | ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐴 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimf 14850 | . . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹:dom 𝐹⟶ℂ) | |
2 | rlimss 14851 | . . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → dom 𝐹 ⊆ ℝ) | |
3 | eqidd 2799 | . . . 4 ⊢ ((𝐹 ⇝𝑟 𝐴 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
4 | 1, 2, 3 | rlim 14844 | . . 3 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ⇝𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ ∀𝑥 ∈ dom 𝐹(𝑧 ≤ 𝑥 → (abs‘((𝐹‘𝑥) − 𝐴)) < 𝑦)))) |
5 | 4 | ibi 270 | . 2 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ ∀𝑥 ∈ dom 𝐹(𝑧 ≤ 𝑥 → (abs‘((𝐹‘𝑥) − 𝐴)) < 𝑦))) |
6 | 5 | simpld 498 | 1 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐴 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 class class class wbr 5030 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 < clt 10664 ≤ cle 10665 − cmin 10859 ℝ+crp 12377 abscabs 14585 ⇝𝑟 crli 14834 |
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-un 7441 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 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-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-pm 8392 df-rlim 14838 |
This theorem is referenced by: rlimi 14862 rlimclim1 14894 rlimuni 14899 rlimresb 14914 rlimcld2 14927 rlimabs 14957 rlimcj 14958 rlimre 14959 rlimim 14960 rlimo1 14965 rlimadd 14991 rlimsub 14992 rlimmul 14993 rlimdiv 14994 rlimsqzlem 14997 fsumrlim 15158 dchrisum0lem2a 26101 mulog2sumlem2 26119 mulog2sumlem3 26120 |
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