Definition df-hlim 9116

Description: Define the limit relation for Hilbert space. See hlimi 9332 for its relational expression. Note that f:NN-->H~ is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of converge in [Beran] p. 96.

Assertion

Ref	Expression
df-hlim	|- ~~>v = {<.f, w>. | ((f:NN-->H~ /\ w e. H~) /\ A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((f` z) -h w)) < x)))}

Distinct variable group:   x,y,z,f,w

Detailed syntax breakdown of Definition df-hlim

Step	Hyp	Ref	Expression
1		chli 9071	. 2 class ~~>v
2		cn 5450	. . . . . 6 class NN
3		chil 9063	. . . . . 6 class H~
4		vf	. . . . . . 7 set f
5	4	cv 991	. . . . . 6 class f
6	2, 3, 5	wf 3259	. . . . 5 wff f:NN-->H~
7		vw	. . . . . . 7 set w
8	7	cv 991	. . . . . 6 class w
9	8, 3	wcel 994	. . . . 5 wff w e. H~
10	6, 9	wa 221	. . . 4 wff (f:NN-->H~ /\ w e. H~)
11		cc0 5388	. . . . . . 7 class 0
12		vx	. . . . . . . 8 set x
13	12	cv 991	. . . . . . 7 class x
14		clt 5640	. . . . . . 7 class <
15	11, 13, 14	wbr 2692	. . . . . 6 wff 0 < x
16		vy	. . . . . . . . . . 11 set y
17	16	cv 991	. . . . . . . . . 10 class y
18		vz	. . . . . . . . . . 11 set z
19	18	cv 991	. . . . . . . . . 10 class z
20		cle 5449	. . . . . . . . . 10 class <_
21	17, 19, 20	wbr 2692	. . . . . . . . 9 wff y <_ z
22	19, 5	cfv 3263	. . . . . . . . . . . 12 class (f` z)
23		cmv 9067	. . . . . . . . . . . 12 class -h
24	22, 8, 23	co 4021	. . . . . . . . . . 11 class ((f` z) -h w)
25		cno 9069	. . . . . . . . . . 11 class normh
26	24, 25	cfv 3263	. . . . . . . . . 10 class (normh` ((f` z) -h w))
27	26, 13, 14	wbr 2692	. . . . . . . . 9 wff (normh` ((f` z) -h w)) < x
28	21, 27	wi 3	. . . . . . . 8 wff (y <_ z -> (normh` ((f` z) -h w)) < x)
29	28, 18, 2	wral 1691	. . . . . . 7 wff A.z e. NN (y <_ z -> (normh` ((f` z) -h w)) < x)
30	29, 16, 2	wrex 1692	. . . . . 6 wff E.y e. NN A.z e. NN (y <_ z -> (normh` ((f` z) -h w)) < x)
31	15, 30	wi 3	. . . . 5 wff (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((f` z) -h w)) < x))
32		cr 5387	. . . . 5 class RR
33	31, 12, 32	wral 1691	. . . 4 wff A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((f` z) -h w)) < x))
34	10, 33	wa 221	. . 3 wff ((f:NN-->H~ /\ w e. H~) /\ A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((f` z) -h w)) < x)))
35	34, 4, 7	copab 2740	. 2 class {<.f, w>. | ((f:NN-->H~ /\ w e. H~) /\ A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((f` z) -h w)) < x)))}
36	1, 35	wceq 992	1 wff ~~>v = {<.f, w>. | ((f:NN-->H~ /\ w e. H~) /\ A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((f` z) -h w)) < x)))}

This definition is referenced by:  h2hlm 9125  hlimi 9332  hlim2 9336

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