Definition df-o1 14221

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 e. ( CC ^pm RR ) | E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m }

Distinct variable group:   x, y, f, m

Detailed syntax breakdown of Definition df-o1

Step	Hyp	Ref	Expression
1		co1 14217	. 2 class O(1)
2		vy	. . . . . . . . . 10 setvar y
3	2	cv 1482	. . . . . . . . 9 class y
4		vf	. . . . . . . . . 10 setvar f
5	4	cv 1482	. . . . . . . . 9 class f
6	3, 5	cfv 5888	. . . . . . . 8 class ( f ` y )
7		cabs 13974	. . . . . . . 8 class abs
8	6, 7	cfv 5888	. . . . . . 7 class ( abs ` ( f ` y ) )
9		vm	. . . . . . . 8 setvar m
10	9	cv 1482	. . . . . . 7 class m
11		cle 10075	. . . . . . 7 class <_
12	8, 10, 11	wbr 4653	. . . . . 6 wff ( abs ` ( f ` y ) ) <_ m
13	5	cdm 5114	. . . . . . 7 class dom f
14		vx	. . . . . . . . 9 setvar x
15	14	cv 1482	. . . . . . . 8 class x
16		cpnf 10071	. . . . . . . 8 class +oo
17		cico 12177	. . . . . . . 8 class [,)
18	15, 16, 17	co 6650	. . . . . . 7 class ( x [,) +oo )
19	13, 18	cin 3573	. . . . . 6 class ( dom f i^i ( x [,) +oo ) )
20	12, 2, 19	wral 2912	. . . . 5 wff A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m
21		cr 9935	. . . . 5 class RR
22	20, 9, 21	wrex 2913	. . . 4 wff E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m
23	22, 14, 21	wrex 2913	. . 3 wff E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m
24		cc 9934	. . . 4 class CC
25		cpm 7858	. . . 4 class ^pm
26	24, 21, 25	co 6650	. . 3 class ( CC ^pm RR )
27	23, 4, 26	crab 2916	. 2 class { f e. ( CC ^pm RR ) | E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m }
28	1, 27	wceq 1483	1 wff O(1) = { f e. ( CC ^pm RR ) | E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m }

This definition is referenced by:  elo1  14257