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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 15537 | . . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹:dom 𝐹⟶ℂ) | |
| 2 | rlimss 15538 | . . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → dom 𝐹 ⊆ ℝ) | |
| 3 | eqidd 2738 | . . . 4 ⊢ ((𝐹 ⇝𝑟 𝐴 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
| 4 | 1, 2, 3 | rlim 15531 | . . 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 267 | . 2 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ ∀𝑥 ∈ dom 𝐹(𝑧 ≤ 𝑥 → (abs‘((𝐹‘𝑥) − 𝐴)) < 𝑦))) |
| 6 | 5 | simpld 494 | 1 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐴 ∈ ℂ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 < clt 11295 ≤ cle 11296 − cmin 11492 ℝ+crp 13034 abscabs 15273 ⇝𝑟 crli 15521 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pm 8869 df-rlim 15525 |
| This theorem is referenced by: rlimi 15549 rlimclim1 15581 rlimuni 15586 rlimresb 15601 rlimcld2 15614 rlimabs 15645 rlimcj 15646 rlimre 15647 rlimim 15648 rlimo1 15653 rlimadd 15679 rlimsub 15680 rlimmul 15681 rlimdiv 15682 rlimsqzlem 15685 fsumrlim 15847 dchrisum0lem2a 27561 mulog2sumlem2 27579 mulog2sumlem3 27580 |
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