Theorem hlimi 10689

Description: Express the predicate: The limit of vector sequence F in a Hilbert space is A, i.e. F converges to A. This means that for any real x, no matter how small, there always exists an integer y such that the norm of any later vector in the sequence minus the limit is less than x. Definition of converge in [Beran] p. 96.

Hypotheses

Ref	Expression
hlim.1	|- F e. _V
hlim.2	|- A e. _V

Assertion

Ref	Expression
hlimi	|- (F ~~>v A <-> ((F:NN-->~H /\ A e. ~H) /\ A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < x))))

Distinct variable groups:   x,y,z,F   x,A,y,z

Proof of Theorem hlimi

Step	Hyp	Ref	Expression
1		hlim.1	. 2 |- F e. _V
2		hlim.2	. 2 |- A e. _V
3		feq1 4551	. . . 4 |- (f = F -> (f:NN-->~H <-> F:NN-->~H))
4	3	anbi1d 679	. . 3 |- (f = F -> ((f:NN-->~H /\ w e. ~H) <-> (F:NN-->~H /\ w e. ~H)))
5		fveq1 4680	. . . . . . . . . 10 |- (f = F -> (f` z) = (F` z))
6	5	opreq1d 4897	. . . . . . . . 9 |- (f = F -> ((f` z) -h w) = ((F` z) -h w))
7	6	fveq2d 4685	. . . . . . . 8 |- (f = F -> (normh` ((f` z) -h w)) = (normh` ((F` z) -h w)))
8	7	breq1d 3348	. . . . . . 7 |- (f = F -> ((normh` ((f` z) -h w)) < x <-> (normh` ((F` z) -h w)) < x))
9	8	imbi2d 674	. . . . . 6 |- (f = F -> ((y <_ z -> (normh` ((f` z) -h w)) < x) <-> (y <_ z -> (normh` ((F` z) -h w)) < x)))
10	9	rexralbidv 2142	. . . . 5 |- (f = F -> (E.y e. NN A.z e. NN (y <_ z -> (normh` ((f` z) -h w)) < x) <-> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h w)) < x)))
11	10	imbi2d 674	. . . 4 |- (f = F -> ((0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((f` z) -h w)) < x)) <-> (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h w)) < x))))
12	11	ralbidv 2123	. . 3 |- (f = F -> (A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((f` z) -h w)) < x)) <-> A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h w)) < x))))
13	4, 12	anbi12d 690	. 2 |- (f = F -> (((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))) <-> ((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)))))
14		eleq1 1957	. . . 4 |- (w = A -> (w e. ~H <-> A e. ~H))
15	14	anbi2d 678	. . 3 |- (w = A -> ((F:NN-->~H /\ w e. ~H) <-> (F:NN-->~H /\ A e. ~H)))
16		opreq2 4890	. . . . . . . . 9 |- (w = A -> ((F` z) -h w) = ((F` z) -h A))
17	16	fveq2d 4685	. . . . . . . 8 |- (w = A -> (normh` ((F` z) -h w)) = (normh` ((F` z) -h A)))
18	17	breq1d 3348	. . . . . . 7 |- (w = A -> ((normh` ((F` z) -h w)) < x <-> (normh` ((F` z) -h A)) < x))
19	18	imbi2d 674	. . . . . 6 |- (w = A -> ((y <_ z -> (normh` ((F` z) -h w)) < x) <-> (y <_ z -> (normh` ((F` z) -h A)) < x)))
20	19	rexralbidv 2142	. . . . 5 |- (w = A -> (E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h w)) < x) <-> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < x)))
21	20	imbi2d 674	. . . 4 |- (w = A -> ((0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h w)) < x)) <-> (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < x))))
22	21	ralbidv 2123	. . 3 |- (w = A -> (A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h w)) < x)) <-> A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < x))))
23	15, 22	anbi12d 690	. 2 |- (w = A -> (((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))) <-> ((F:NN-->~H /\ A e. ~H) /\ A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < x)))))
24		df-hlim 10473	. 2 |- ~~>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)))}
25	1, 2, 13, 23, 24	brab 3571	1 |- (F ~~>v A <-> ((F:NN-->~H /\ A e. ~H) /\ A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < x))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   class class class wbr 3338  -->wf 3994  ` cfv 3998  (class class class)co 4884  RRcr 6385  0cc0 6386   <_ cle 6448  NNcn 6449   < clt 6653  ~Hchil 10420   -h cmv 10424  normhcno 10426   ~~>v chli 10428

This theorem is referenced by:  hlimseqi 10690  hlimveci 10691  hlimconvi 10692  hlim0 10738  occllem6 10811  osumlem4 11216

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-hlim 10473

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