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Mirrors > Home > MPE Home > Th. List > rlimi2 | Structured version Visualization version GIF version |
Description: Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 12-May-2016.) |
Ref | Expression |
---|---|
rlimi.1 | ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ 𝑉) |
rlimi.2 | ⊢ (𝜑 → 𝑅 ∈ ℝ+) |
rlimi.3 | ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶) |
rlimi.4 | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
Ref | Expression |
---|---|
rlimi2 | ⊢ (𝜑 → ∃𝑦 ∈ (𝐷[,)+∞)∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimi.1 | . . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ 𝑉) | |
2 | rlimi.2 | . . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ+) | |
3 | rlimi.3 | . . 3 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶) | |
4 | 1, 2, 3 | rlimi 14872 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅)) |
5 | eqid 2823 | . . . . . 6 ⊢ (𝑧 ∈ 𝐴 ↦ 𝐵) = (𝑧 ∈ 𝐴 ↦ 𝐵) | |
6 | 5 | fnmpt 6490 | . . . . 5 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑧 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
7 | fndm 6457 | . . . . 5 ⊢ ((𝑧 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 → dom (𝑧 ∈ 𝐴 ↦ 𝐵) = 𝐴) | |
8 | 1, 6, 7 | 3syl 18 | . . . 4 ⊢ (𝜑 → dom (𝑧 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
9 | rlimss 14861 | . . . . 5 ⊢ ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 → dom (𝑧 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ) | |
10 | 3, 9 | syl 17 | . . . 4 ⊢ (𝜑 → dom (𝑧 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ) |
11 | 8, 10 | eqsstrrd 4008 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
12 | rlimi.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
13 | rexico 14715 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐷 ∈ ℝ) → (∃𝑦 ∈ (𝐷[,)+∞)∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅) ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅))) | |
14 | 11, 12, 13 | syl2anc 586 | . 2 ⊢ (𝜑 → (∃𝑦 ∈ (𝐷[,)+∞)∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅) ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅))) |
15 | 4, 14 | mpbird 259 | 1 ⊢ (𝜑 → ∃𝑦 ∈ (𝐷[,)+∞)∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 class class class wbr 5068 ↦ cmpt 5148 dom cdm 5557 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 +∞cpnf 10674 < clt 10677 ≤ cle 10678 − cmin 10872 ℝ+crp 12392 [,)cico 12743 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-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-ico 12747 df-rlim 14848 |
This theorem is referenced by: (None) |
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