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Mirrors > Home > MPE Home > Th. List > rlimi | Structured version Visualization version GIF version |
Description: Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 28-Feb-2015.) |
Ref | Expression |
---|---|
rlimi.1 | ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ 𝑉) |
rlimi.2 | ⊢ (𝜑 → 𝑅 ∈ ℝ+) |
rlimi.3 | ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶) |
Ref | Expression |
---|---|
rlimi | ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5142 | . . . 4 ⊢ (𝑥 = 𝑅 → ((abs‘(𝐵 − 𝐶)) < 𝑥 ↔ (abs‘(𝐵 − 𝐶)) < 𝑅)) | |
2 | 1 | imbi2d 340 | . . 3 ⊢ (𝑥 = 𝑅 → ((𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅))) |
3 | 2 | rexralbidv 3212 | . 2 ⊢ (𝑥 = 𝑅 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅))) |
4 | rlimi.3 | . . 3 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶) | |
5 | rlimf 15441 | . . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 → (𝑧 ∈ 𝐴 ↦ 𝐵):dom (𝑧 ∈ 𝐴 ↦ 𝐵)⟶ℂ) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵):dom (𝑧 ∈ 𝐴 ↦ 𝐵)⟶ℂ) |
7 | rlimi.1 | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ 𝑉) | |
8 | eqid 2724 | . . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 ↦ 𝐵) = (𝑧 ∈ 𝐴 ↦ 𝐵) | |
9 | 8 | fmpt 7101 | . . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ 𝑉 ↔ (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑉) |
10 | 7, 9 | sylib 217 | . . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑉) |
11 | 10 | fdmd 6718 | . . . . . . 7 ⊢ (𝜑 → dom (𝑧 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
12 | 11 | feq2d 6693 | . . . . . 6 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵):dom (𝑧 ∈ 𝐴 ↦ 𝐵)⟶ℂ ↔ (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)) |
13 | 6, 12 | mpbid 231 | . . . . 5 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
14 | 8 | fmpt 7101 | . . . . 5 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ ↔ (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
15 | 13, 14 | sylibr 233 | . . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ) |
16 | rlimss 15442 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 → dom (𝑧 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ) | |
17 | 4, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → dom (𝑧 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ) |
18 | 11, 17 | eqsstrrd 4013 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
19 | rlimcl 15443 | . . . . 5 ⊢ ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 → 𝐶 ∈ ℂ) | |
20 | 4, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
21 | 15, 18, 20 | rlim2 15436 | . . 3 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥))) |
22 | 4, 21 | mpbid 231 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)) |
23 | rlimi.2 | . 2 ⊢ (𝜑 → 𝑅 ∈ ℝ+) | |
24 | 3, 22, 23 | rspcdva 3605 | 1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 ⊆ wss 3940 class class class wbr 5138 ↦ cmpt 5221 dom cdm 5666 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ℂcc 11103 ℝcr 11104 < clt 11244 ≤ cle 11245 − cmin 11440 ℝ+crp 12970 abscabs 15177 ⇝𝑟 crli 15425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-pm 8818 df-rlim 15429 |
This theorem is referenced by: rlimi2 15454 rlimclim1 15485 rlimuni 15490 rlimcld2 15518 rlimcn1 15528 rlimcn3 15530 rlimo1 15557 o1rlimmul 15559 rlimno1 15596 xrlimcnp 26804 rlimcxp 26810 chtppilimlem2 27311 dchrisumlem3 27328 |
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