Theorem hlim 9056

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 3620	. . . 4 |- (f = F -> (f:NN-->H~ <-> F:NN-->H~))
4	3	anbi1d 617	. . 3 |- (f = F -> ((f:NN-->H~ /\ w e. H~) <-> (F:NN-->H~ /\ w e. H~)))
5		fveq1 3723	. . . . . . . . . 10 |- (f = F -> (f` z) = (F` z))
6	5	opreq1d 3975	. . . . . . . . 9 |- (f = F -> ((f` z) -h w) = ((F` z) -h w))
7	6	fveq2d 3728	. . . . . . . 8 |- (f = F -> (normh` ((f` z) -h w)) = (normh` ((F` z) -h w)))
8	7	breq1d 2629	. . . . . . 7 |- (f = F -> ((normh` ((f` z) -h w)) < x <-> (normh` ((F` z) -h w)) < x))
9	8	imbi2d 612	. . . . . 6 |- (f = F -> ((y <_ z -> (normh` ((f` z) -h w)) < x) <-> (y <_ z -> (normh` ((F` z) -h w)) < x)))
10	9	rexralbidv 1682	. . . . 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 612	. . . 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 1663	. . 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 628	. 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 1534	. . . 4 |- (w = A -> (w e. H~ <-> A e. H~))
15	14	anbi2d 616	. . 3 |- (w = A -> ((F:NN-->H~ /\ w e. H~) <-> (F:NN-->H~ /\ A e. H~)))
16		opreq2 3969	. . . . . . . . 9 |- (w = A -> ((F` z) -h w) = ((F` z) -h A))
17	16	fveq2d 3728	. . . . . . . 8 |- (w = A -> (normh` ((F` z) -h w)) = (normh` ((F` z) -h A)))
18	17	breq1d 2629	. . . . . . 7 |- (w = A -> ((normh` ((F` z) -h w)) < x <-> (normh` ((F` z) -h A)) < x))
19	18	imbi2d 612	. . . . . 6 |- (w = A -> ((y <_ z -> (normh` ((F` z) -h w)) < x) <-> (y <_ z -> (normh` ((F` z) -h A)) < x)))
20	19	rexralbidv 1682	. . . . 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 612	. . . 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 1663	. . 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 628	. 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 8841	. 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 2821	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 956   e. wcel 958  A.wral 1645  E.wrex 1646  Vcvv 1811   class class class wbr 2619  -->wf 3178  ` cfv 3182  (class class class)co 3963  RRcr 5233  0cc0 5234   <_ cle 5295  NNcn 5296   < clt 5486  H~chil 8788   -h cmv 8792  normhcno 8794   ~~>v chli 8796

This theorem is referenced by:  hlimseq 9057  hlimvec 9058  hlimconv 9059  hlim0 9105  occllem6 9178  osumlem4 9581

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-hlim 8841

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