Definition df-cnop 28699

Description: Define the set of continuous operators on Hilbert space. For every "epsilon" ( y) there is a "delta" ( z) such that... (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
df-cnop	|- ContOp = { t e. ( ~H ^m ~H ) | A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) }

Distinct variable group:   w, t, x, y, z

Detailed syntax breakdown of Definition df-cnop

Step	Hyp	Ref	Expression
1		ccop 27803	. 2 class ContOp
2		vw	. . . . . . . . . . . 12 setvar w
3	2	cv 1482	. . . . . . . . . . 11 class w
4		vx	. . . . . . . . . . . 12 setvar x
5	4	cv 1482	. . . . . . . . . . 11 class x
6		cmv 27782	. . . . . . . . . . 11 class -h
7	3, 5, 6	co 6650	. . . . . . . . . 10 class ( w -h x )
8		cno 27780	. . . . . . . . . 10 class normh
9	7, 8	cfv 5888	. . . . . . . . 9 class ( normh ` ( w -h x ) )
10		vz	. . . . . . . . . 10 setvar z
11	10	cv 1482	. . . . . . . . 9 class z
12		clt 10074	. . . . . . . . 9 class <
13	9, 11, 12	wbr 4653	. . . . . . . 8 wff ( normh ` ( w -h x ) ) < z
14		vt	. . . . . . . . . . . . 13 setvar t
15	14	cv 1482	. . . . . . . . . . . 12 class t
16	3, 15	cfv 5888	. . . . . . . . . . 11 class ( t ` w )
17	5, 15	cfv 5888	. . . . . . . . . . 11 class ( t ` x )
18	16, 17, 6	co 6650	. . . . . . . . . 10 class ( ( t ` w ) -h ( t ` x ) )
19	18, 8	cfv 5888	. . . . . . . . 9 class ( normh ` ( ( t ` w ) -h ( t ` x ) ) )
20		vy	. . . . . . . . . 10 setvar y
21	20	cv 1482	. . . . . . . . 9 class y
22	19, 21, 12	wbr 4653	. . . . . . . 8 wff ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y
23	13, 22	wi 4	. . . . . . 7 wff ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y )
24		chil 27776	. . . . . . 7 class ~H
25	23, 2, 24	wral 2912	. . . . . 6 wff A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y )
26		crp 11832	. . . . . 6 class RR+
27	25, 10, 26	wrex 2913	. . . . 5 wff E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y )
28	27, 20, 26	wral 2912	. . . 4 wff A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y )
29	28, 4, 24	wral 2912	. . 3 wff A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y )
30		cmap 7857	. . . 4 class ^m
31	24, 24, 30	co 6650	. . 3 class ( ~H ^m ~H )
32	29, 14, 31	crab 2916	. 2 class { t e. ( ~H ^m ~H ) | A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) }
33	1, 32	wceq 1483	1 wff ContOp = { t e. ( ~H ^m ~H ) | A. x e. ~H A. y e. RR+ E. z e. RR+ A. w e. ~H ( ( normh ` ( w -h x ) ) < z -> ( normh ` ( ( t ` w ) -h ( t ` x ) ) ) < y ) }

This definition is referenced by:  elcnop  28716  hhcno  28763