MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-rlim Structured version   Visualization version   GIF version

Definition df-rlim 14419
Description: Define the limit relation for partial functions on the reals. See rlim 14425 for its relational expression. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
df-rlim 𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
Distinct variable group:   𝑥,𝑤,𝑦,𝑧,𝑓

Detailed syntax breakdown of Definition df-rlim
StepHypRef Expression
1 crli 14415 . 2 class 𝑟
2 vf . . . . . . 7 setvar 𝑓
32cv 1631 . . . . . 6 class 𝑓
4 cc 10126 . . . . . . 7 class
5 cr 10127 . . . . . . 7 class
6 cpm 8024 . . . . . . 7 class pm
74, 5, 6co 6813 . . . . . 6 class (ℂ ↑pm ℝ)
83, 7wcel 2139 . . . . 5 wff 𝑓 ∈ (ℂ ↑pm ℝ)
9 vx . . . . . . 7 setvar 𝑥
109cv 1631 . . . . . 6 class 𝑥
1110, 4wcel 2139 . . . . 5 wff 𝑥 ∈ ℂ
128, 11wa 383 . . . 4 wff (𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ)
13 vz . . . . . . . . . 10 setvar 𝑧
1413cv 1631 . . . . . . . . 9 class 𝑧
15 vw . . . . . . . . . 10 setvar 𝑤
1615cv 1631 . . . . . . . . 9 class 𝑤
17 cle 10267 . . . . . . . . 9 class
1814, 16, 17wbr 4804 . . . . . . . 8 wff 𝑧𝑤
1916, 3cfv 6049 . . . . . . . . . . 11 class (𝑓𝑤)
20 cmin 10458 . . . . . . . . . . 11 class
2119, 10, 20co 6813 . . . . . . . . . 10 class ((𝑓𝑤) − 𝑥)
22 cabs 14173 . . . . . . . . . 10 class abs
2321, 22cfv 6049 . . . . . . . . 9 class (abs‘((𝑓𝑤) − 𝑥))
24 vy . . . . . . . . . 10 setvar 𝑦
2524cv 1631 . . . . . . . . 9 class 𝑦
26 clt 10266 . . . . . . . . 9 class <
2723, 25, 26wbr 4804 . . . . . . . 8 wff (abs‘((𝑓𝑤) − 𝑥)) < 𝑦
2818, 27wi 4 . . . . . . 7 wff (𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦)
293cdm 5266 . . . . . . 7 class dom 𝑓
3028, 15, 29wral 3050 . . . . . 6 wff 𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦)
3130, 13, 5wrex 3051 . . . . 5 wff 𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦)
32 crp 12025 . . . . 5 class +
3331, 24, 32wral 3050 . . . 4 wff 𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦)
3412, 33wa 383 . . 3 wff ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))
3534, 2, 9copab 4864 . 2 class {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
361, 35wceq 1632 1 wff 𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
Colors of variables: wff setvar class
This definition is referenced by:  rlimrel  14423  rlim  14425  rlimpm  14430
  Copyright terms: Public domain W3C validator