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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 15525 | . . . . 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 5778 | . . . . 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 4030 | . . . 4 ⊢ ⇝𝑟 ⊆ ((ℂ ↑pm ℝ) × ℂ) |
| 4 | dmss 5913 | . . . 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 6191 | . . 3 ⊢ dom ((ℂ ↑pm ℝ) × ℂ) ⊆ (ℂ ↑pm ℝ) | |
| 7 | 5, 6 | sstri 3993 | . 2 ⊢ dom ⇝𝑟 ⊆ (ℂ ↑pm ℝ) |
| 8 | rlimrel 15529 | . . 3 ⊢ Rel ⇝𝑟 | |
| 9 | 8 | releldmi 5959 | . 2 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ dom ⇝𝑟 ) |
| 10 | 7, 9 | sselid 3981 | 1 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ (ℂ ↑pm ℝ)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 class class class wbr 5143 {copab 5205 × cxp 5683 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ↑pm cpm 8867 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-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rlim 15525 |
| This theorem is referenced by: rlimf 15537 rlimss 15538 rlimclim1 15581 |
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