Definition df-o1 15463

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 15459	. 2 class 𝑂(1)
2		vy	. . . . . . . . . 10 setvar 𝑦
3	2	cv 1539	. . . . . . . . 9 class 𝑦
4		vf	. . . . . . . . . 10 setvar 𝑓
5	4	cv 1539	. . . . . . . . 9 class 𝑓
6	3, 5	cfv 6514	. . . . . . . 8 class (𝑓‘𝑦)
7		cabs 15207	. . . . . . . 8 class abs
8	6, 7	cfv 6514	. . . . . . 7 class (abs‘(𝑓‘𝑦))
9		vm	. . . . . . . 8 setvar 𝑚
10	9	cv 1539	. . . . . . 7 class 𝑚
11		cle 11216	. . . . . . 7 class ≤
12	8, 10, 11	wbr 5110	. . . . . 6 wff (abs‘(𝑓‘𝑦)) ≤ 𝑚
13	5	cdm 5641	. . . . . . 7 class dom 𝑓
14		vx	. . . . . . . . 9 setvar 𝑥
15	14	cv 1539	. . . . . . . 8 class 𝑥
16		cpnf 11212	. . . . . . . 8 class +∞
17		cico 13315	. . . . . . . 8 class [,)
18	15, 16, 17	co 7390	. . . . . . 7 class (𝑥[,)+∞)
19	13, 18	cin 3916	. . . . . 6 class (dom 𝑓 ∩ (𝑥[,)+∞))
20	12, 2, 19	wral 3045	. . . . 5 wff ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚
21		cr 11074	. . . . 5 class ℝ
22	20, 9, 21	wrex 3054	. . . 4 wff ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚
23	22, 14, 21	wrex 3054	. . 3 wff ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚
24		cc 11073	. . . 4 class ℂ
25		cpm 8803	. . . 4 class ↑pm
26	24, 21, 25	co 7390	. . 3 class (ℂ ↑pm ℝ)
27	23, 4, 26	crab 3408	. 2 class {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}
28	1, 27	wceq 1540	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  15499