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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 5078 | . . . 4 ⊢ (𝑥 = 𝑅 → ((abs‘(𝐵 − 𝐶)) < 𝑥 ↔ (abs‘(𝐵 − 𝐶)) < 𝑅)) | |
2 | 1 | imbi2d 341 | . . 3 ⊢ (𝑥 = 𝑅 → ((𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅))) |
3 | 2 | rexralbidv 3230 | . 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 15210 | . . . . . . 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 2738 | . . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 ↦ 𝐵) = (𝑧 ∈ 𝐴 ↦ 𝐵) | |
9 | 8 | fmpt 6984 | . . . . . . . . 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 6611 | . . . . . . 7 ⊢ (𝜑 → dom (𝑧 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
12 | 11 | feq2d 6586 | . . . . . 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 6984 | . . . . 5 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ ↔ (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
15 | 13, 14 | sylibr 233 | . . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ) |
16 | rlimss 15211 | . . . . . 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 3960 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
19 | rlimcl 15212 | . . . . 5 ⊢ ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 → 𝐶 ∈ ℂ) | |
20 | 4, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
21 | 15, 18, 20 | rlim2 15205 | . . 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 3562 | 1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 < clt 11009 ≤ cle 11010 − cmin 11205 ℝ+crp 12730 abscabs 14945 ⇝𝑟 crli 15194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-pm 8618 df-rlim 15198 |
This theorem is referenced by: rlimi2 15223 rlimclim1 15254 rlimuni 15259 rlimcld2 15287 rlimcn1 15297 rlimcn3 15299 rlimo1 15326 o1rlimmul 15328 rlimno1 15365 xrlimcnp 26118 rlimcxp 26123 chtppilimlem2 26622 dchrisumlem3 26639 |
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