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Mirrors > Home > MPE Home > Th. List > rlimpm | Structured version Visualization version GIF version |
Description: Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.) |
Ref | Expression |
---|---|
rlimpm | ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ (ℂ ↑pm ℝ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rlim 15521 | . . . . 5 ⊢ ⇝𝑟 = {〈𝑓, 𝑥〉 ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧 ≤ 𝑤 → (abs‘((𝑓‘𝑤) − 𝑥)) < 𝑦))} | |
2 | opabssxp 5780 | . . . . 5 ⊢ {〈𝑓, 𝑥〉 ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧 ≤ 𝑤 → (abs‘((𝑓‘𝑤) − 𝑥)) < 𝑦))} ⊆ ((ℂ ↑pm ℝ) × ℂ) | |
3 | 1, 2 | eqsstri 4029 | . . . 4 ⊢ ⇝𝑟 ⊆ ((ℂ ↑pm ℝ) × ℂ) |
4 | dmss 5915 | . . . 4 ⊢ ( ⇝𝑟 ⊆ ((ℂ ↑pm ℝ) × ℂ) → dom ⇝𝑟 ⊆ dom ((ℂ ↑pm ℝ) × ℂ)) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ dom ⇝𝑟 ⊆ dom ((ℂ ↑pm ℝ) × ℂ) |
6 | dmxpss 6192 | . . 3 ⊢ dom ((ℂ ↑pm ℝ) × ℂ) ⊆ (ℂ ↑pm ℝ) | |
7 | 5, 6 | sstri 4004 | . 2 ⊢ dom ⇝𝑟 ⊆ (ℂ ↑pm ℝ) |
8 | rlimrel 15525 | . . 3 ⊢ Rel ⇝𝑟 | |
9 | 8 | releldmi 5961 | . 2 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ dom ⇝𝑟 ) |
10 | 7, 9 | sselid 3992 | 1 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ (ℂ ↑pm ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 ⊆ wss 3962 class class class wbr 5147 {copab 5209 × cxp 5686 dom cdm 5688 ‘cfv 6562 (class class class)co 7430 ↑pm cpm 8865 ℂcc 11150 ℝcr 11151 < clt 11292 ≤ cle 11293 − cmin 11489 ℝ+crp 13031 abscabs 15269 ⇝𝑟 crli 15517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 df-dm 5698 df-rlim 15521 |
This theorem is referenced by: rlimf 15533 rlimss 15534 rlimclim1 15577 |
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