Definition df-o1 15199

Description: Define the set of eventually bounded functions. We don't bother to build the full conception of big-O notation, because we can represent any big-O in terms of 𝑂(1) and division, and any little-O in terms of a limit and division. We could also use limsup for this, but it only works on integer sequences, while this will work for real sequences or integer sequences. (Contributed by Mario Carneiro, 15-Sep-2014.)

Assertion

Ref	Expression
df-o1	⊢ 𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}

Distinct variable group:   𝑥,𝑦,𝑓,𝑚

Detailed syntax breakdown of Definition df-o1

Step	Hyp	Ref	Expression
1		co1 15195	. 2 class 𝑂(1)
2		vy	. . . . . . . . . 10 setvar 𝑦
3	2	cv 1538	. . . . . . . . 9 class 𝑦
4		vf	. . . . . . . . . 10 setvar 𝑓
5	4	cv 1538	. . . . . . . . 9 class 𝑓
6	3, 5	cfv 6433	. . . . . . . 8 class (𝑓‘𝑦)
7		cabs 14945	. . . . . . . 8 class abs
8	6, 7	cfv 6433	. . . . . . 7 class (abs‘(𝑓‘𝑦))
9		vm	. . . . . . . 8 setvar 𝑚
10	9	cv 1538	. . . . . . 7 class 𝑚
11		cle 11010	. . . . . . 7 class ≤
12	8, 10, 11	wbr 5074	. . . . . 6 wff (abs‘(𝑓‘𝑦)) ≤ 𝑚
13	5	cdm 5589	. . . . . . 7 class dom 𝑓
14		vx	. . . . . . . . 9 setvar 𝑥
15	14	cv 1538	. . . . . . . 8 class 𝑥
16		cpnf 11006	. . . . . . . 8 class +∞
17		cico 13081	. . . . . . . 8 class [,)
18	15, 16, 17	co 7275	. . . . . . 7 class (𝑥[,)+∞)
19	13, 18	cin 3886	. . . . . 6 class (dom 𝑓 ∩ (𝑥[,)+∞))
20	12, 2, 19	wral 3064	. . . . 5 wff ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚
21		cr 10870	. . . . 5 class ℝ
22	20, 9, 21	wrex 3065	. . . 4 wff ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚
23	22, 14, 21	wrex 3065	. . 3 wff ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚
24		cc 10869	. . . 4 class ℂ
25		cpm 8616	. . . 4 class ↑pm
26	24, 21, 25	co 7275	. . 3 class (ℂ ↑pm ℝ)
27	23, 4, 26	crab 3068	. 2 class {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}
28	1, 27	wceq 1539	1 wff 𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}

This definition is referenced by:  elo1  15235