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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 5031 | . . . 4 ⊢ (𝑥 = 𝑅 → ((abs‘(𝐵 − 𝐶)) < 𝑥 ↔ (abs‘(𝐵 − 𝐶)) < 𝑅)) | |
2 | 1 | imbi2d 344 | . . 3 ⊢ (𝑥 = 𝑅 → ((𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅))) |
3 | 2 | rexralbidv 3210 | . 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 14941 | . . . . . . 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 6878 | . . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ 𝑉 ↔ (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑉) |
10 | 7, 9 | sylib 221 | . . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑉) |
11 | 10 | fdmd 6509 | . . . . . . 7 ⊢ (𝜑 → dom (𝑧 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
12 | 11 | feq2d 6484 | . . . . . 6 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵):dom (𝑧 ∈ 𝐴 ↦ 𝐵)⟶ℂ ↔ (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)) |
13 | 6, 12 | mpbid 235 | . . . . 5 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
14 | 8 | fmpt 6878 | . . . . 5 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ ↔ (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
15 | 13, 14 | sylibr 237 | . . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ) |
16 | rlimss 14942 | . . . . . 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 3914 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
19 | rlimcl 14943 | . . . . 5 ⊢ ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 → 𝐶 ∈ ℂ) | |
20 | 4, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
21 | 15, 18, 20 | rlim2 14936 | . . 3 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥))) |
22 | 4, 21 | mpbid 235 | . 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 3526 | 1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ∀wral 3053 ∃wrex 3054 ⊆ wss 3841 class class class wbr 5027 ↦ cmpt 5107 dom cdm 5519 ⟶wf 6329 ‘cfv 6333 (class class class)co 7164 ℂcc 10606 ℝcr 10607 < clt 10746 ≤ cle 10747 − cmin 10941 ℝ+crp 12465 abscabs 14676 ⇝𝑟 crli 14925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-pm 8433 df-rlim 14929 |
This theorem is referenced by: rlimi2 14954 rlimclim1 14985 rlimuni 14990 rlimcld2 15018 rlimcn1 15028 rlimcn3 15030 rlimo1 15057 o1rlimmul 15059 rlimno1 15096 xrlimcnp 25698 rlimcxp 25703 chtppilimlem2 26202 dchrisumlem3 26219 |
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