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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 14848 | . . . . 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 5645 | . . . . 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 4003 | . . . 4 ⊢ ⇝𝑟 ⊆ ((ℂ ↑pm ℝ) × ℂ) |
4 | dmss 5773 | . . . 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 6030 | . . 3 ⊢ dom ((ℂ ↑pm ℝ) × ℂ) ⊆ (ℂ ↑pm ℝ) | |
7 | 5, 6 | sstri 3978 | . 2 ⊢ dom ⇝𝑟 ⊆ (ℂ ↑pm ℝ) |
8 | rlimrel 14852 | . . 3 ⊢ Rel ⇝𝑟 | |
9 | 8 | releldmi 5820 | . 2 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ dom ⇝𝑟 ) |
10 | 7, 9 | sseldi 3967 | 1 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ (ℂ ↑pm ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 class class class wbr 5068 {copab 5130 × cxp 5555 dom cdm 5557 ‘cfv 6357 (class class class)co 7158 ↑pm cpm 8409 ℂcc 10537 ℝcr 10538 < clt 10677 ≤ cle 10678 − cmin 10872 ℝ+crp 12392 abscabs 14595 ⇝𝑟 crli 14844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rlim 14848 |
This theorem is referenced by: rlimf 14860 rlimss 14861 rlimclim1 14904 |
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