Theorem hlim 8995

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
hlim	|- (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 hlim

Step	Hyp	Ref	Expression
1		hlim.1	. 2 |- F e. V
2		hlim.2	. 2 |- A e. V
3		feq1 3612	. . . 4 |- (f = F -> (f:NN-->H~ <-> F:NN-->H~))
4	3	anbi1d 616	. . 3 |- (f = F -> ((f:NN-->H~ /\ w e. H~) <-> (F:NN-->H~ /\ w e. H~)))
5		fveq1 3714	. . . . . . . . . 10 |- (f = F -> (f` z) = (F` z))
6	5	opreq1d 3966	. . . . . . . . 9 |- (f = F -> ((f` z) -h w) = ((F` z) -h w))
7	6	fveq2d 3719	. . . . . . . 8 |- (f = F -> (normh` ((f` z) -h w)) = (normh` ((F` z) -h w)))
8	7	breq1d 2624	. . . . . . 7 |- (f = F -> ((normh` ((f` z) -h w)) < x <-> (normh` ((F` z) -h w)) < x))
9	8	imbi2d 611	. . . . . 6 |- (f = F -> ((y <_ z -> (normh` ((f` z) -h w)) < x) <-> (y <_ z -> (normh` ((F` z) -h w)) < x)))
10	9	rexralbidv 1679	. . . . 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 611	. . . 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 1660	. . 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 627	. 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 1531	. . . 4 |- (w = A -> (w e. H~ <-> A e. H~))
15	14	anbi2d 615	. . 3 |- (w = A -> ((F:NN-->H~ /\ w e. H~) <-> (F:NN-->H~ /\ A e. H~)))
16		opreq2 3960	. . . . . . . . 9 |- (w = A -> ((F` z) -h w) = ((F` z) -h A))
17	16	fveq2d 3719	. . . . . . . 8 |- (w = A -> (normh` ((F` z) -h w)) = (normh` ((F` z) -h A)))
18	17	breq1d 2624	. . . . . . 7 |- (w = A -> ((normh` ((F` z) -h w)) < x <-> (normh` ((F` z) -h A)) < x))
19	18	imbi2d 611	. . . . . 6 |- (w = A -> ((y <_ z -> (normh` ((F` z) -h w)) < x) <-> (y <_ z -> (normh` ((F` z) -h A)) < x)))
20	19	rexralbidv 1679	. . . . 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 611	. . . 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 1660	. . 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 627	. 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 8780	. 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 2816	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 146   /\ wa 223   = wceq 954   e. wcel 956  A.wral 1642  E.wrex 1643  Vcvv 1807   class class class wbr 2614  -->wf 3173  ` cfv 3177  (class class class)co 3954  RRcr 5213  0cc0 5214   <_ cle 5275  NNcn 5276   < clt 5466  H~chil 8727   -h cmv 8731  normhcno 8733   ~~>v chli 8735

This theorem is referenced by:  hlimseq 8996  hlimvec 8997  hlimconv 8998  hlim0 9044  occllem6 9117  osumlem4 9521

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-opr 3956  df-hlim 8780

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