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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 14232 | . . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹:dom 𝐹⟶ℂ) | |
2 | rlimss 14233 | . . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → dom 𝐹 ⊆ ℝ) | |
3 | eqidd 2623 | . . . 4 ⊢ ((𝐹 ⇝𝑟 𝐴 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
4 | 1, 2, 3 | rlim 14226 | . . 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 256 | . 2 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐴 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ ∀𝑥 ∈ dom 𝐹(𝑧 ≤ 𝑥 → (abs‘((𝐹‘𝑥) − 𝐴)) < 𝑦))) |
6 | 5 | simpld 475 | 1 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐴 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 class class class wbr 4653 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 < clt 10074 ≤ cle 10075 − cmin 10266 ℝ+crp 11832 abscabs 13974 ⇝𝑟 crli 14216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-pm 7860 df-rlim 14220 |
This theorem is referenced by: rlimi 14244 rlimclim1 14276 rlimuni 14281 rlimresb 14296 rlimcld2 14309 rlimabs 14339 rlimcj 14340 rlimre 14341 rlimim 14342 rlimo1 14347 rlimadd 14373 rlimsub 14374 rlimmul 14375 rlimdiv 14376 rlimsqzlem 14379 fsumrlim 14543 dchrisum0lem2a 25206 mulog2sumlem2 25224 mulog2sumlem3 25225 |
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