Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5061 | . . . 4 ⊢ (𝑥 = 𝑅 → ((abs‘(𝐵 − 𝐶)) < 𝑥 ↔ (abs‘(𝐵 − 𝐶)) < 𝑅)) | |
2 | 1 | imbi2d 342 | . . 3 ⊢ (𝑥 = 𝑅 → ((𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅))) |
3 | 2 | rexralbidv 3298 | . 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 14846 | . . . . . . 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 2818 | . . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 ↦ 𝐵) = (𝑧 ∈ 𝐴 ↦ 𝐵) | |
9 | 8 | fmpt 6866 | . . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ 𝑉 ↔ (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑉) |
10 | 7, 9 | sylib 219 | . . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑉) |
11 | 10 | fdmd 6516 | . . . . . . 7 ⊢ (𝜑 → dom (𝑧 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
12 | 11 | feq2d 6493 | . . . . . 6 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵):dom (𝑧 ∈ 𝐴 ↦ 𝐵)⟶ℂ ↔ (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)) |
13 | 6, 12 | mpbid 233 | . . . . 5 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
14 | 8 | fmpt 6866 | . . . . 5 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ ↔ (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
15 | 13, 14 | sylibr 235 | . . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ) |
16 | rlimss 14847 | . . . . . 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 4003 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
19 | rlimcl 14848 | . . . . 5 ⊢ ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 → 𝐶 ∈ ℂ) | |
20 | 4, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
21 | 15, 18, 20 | rlim2 14841 | . . 3 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥))) |
22 | 4, 21 | mpbid 233 | . 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 3622 | 1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ⊆ wss 3933 class class class wbr 5057 ↦ cmpt 5137 dom cdm 5548 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 < clt 10663 ≤ cle 10664 − cmin 10858 ℝ+crp 12377 abscabs 14581 ⇝𝑟 crli 14830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-pm 8398 df-rlim 14834 |
This theorem is referenced by: rlimi2 14859 rlimclim1 14890 rlimuni 14895 rlimcld2 14923 rlimcn1 14933 rlimcn2 14935 rlimo1 14961 o1rlimmul 14963 rlimno1 14998 xrlimcnp 25473 rlimcxp 25478 chtppilimlem2 25977 dchrisumlem3 25994 |
Copyright terms: Public domain | W3C validator |