Theorem chsscm 5147

Description: The hypothesis defines the set of complete subspaces of Hilbert space. A complete subspace is one in which every Cauchy sequence of vectors in the subspace converges to a member of the subspace (Definition of complete subspace in [Beran] p. 96). Any closed subspace of a Hilbert space is complete. Part of Remark 3.12 of [Beran] p. 107.

Hypothesis

Ref	Expression
cmh.1	|- C = {h | (h e. SH /\ A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x))}

Assertion

Ref	Expression
chsscm	|- CH (_ C

Distinct variable group(s):   x,f,h   C,h

Proof of Theorem chsscm

Step	Hyp	Ref	Expression
1		impexp 276	. . . . . . . . . . . . . . . 16 |- (((f:NN-->h /\ f ~~>v x) -> x e. h) <-> (f:NN-->h -> (f ~~>v x -> x e. h)))
2		ancr 243	. . . . . . . . . . . . . . . . . 18 |- ((f ~~>v x -> x e. h) -> (f ~~>v x -> (x e. h /\ f ~~>v x)))
3	2	adantld 307	. . . . . . . . . . . . . . . . 17 |- ((f ~~>v x -> x e. h) -> ((x e. H~ /\ f ~~>v x) -> (x e. h /\ f ~~>v x)))
4	3	syl3 18	. . . . . . . . . . . . . . . 16 |- ((f:NN-->h -> (f ~~>v x -> x e. h)) -> (f:NN-->h -> ((x e. H~ /\ f ~~>v x) -> (x e. h /\ f ~~>v x))))
5	1, 4	sylbi 174	. . . . . . . . . . . . . . 15 |- (((f:NN-->h /\ f ~~>v x) -> x e. h) -> (f:NN-->h -> ((x e. H~ /\ f ~~>v x) -> (x e. h /\ f ~~>v x))))
6	5	com12 13	. . . . . . . . . . . . . 14 |- (f:NN-->h -> (((f:NN-->h /\ f ~~>v x) -> x e. h) -> ((x e. H~ /\ f ~~>v x) -> (x e. h /\ f ~~>v x))))
7	6	19.20dv 946	. . . . . . . . . . . . 13 |- (f:NN-->h -> (A.x((f:NN-->h /\ f ~~>v x) -> x e. h) -> A.x((x e. H~ /\ f ~~>v x) -> (x e. h /\ f ~~>v x))))
8	7	com12 13	. . . . . . . . . . . 12 |- (A.x((f:NN-->h /\ f ~~>v x) -> x e. h) -> (f:NN-->h -> A.x((x e. H~ /\ f ~~>v x) -> (x e. h /\ f ~~>v x))))
9	8	imp 277	. . . . . . . . . . 11 |- ((A.x((f:NN-->h /\ f ~~>v x) -> x e. h) /\ f:NN-->h) -> A.x((x e. H~ /\ f ~~>v x) -> (x e. h /\ f ~~>v x)))
10		19.22 722	. . . . . . . . . . 11 |- (A.x((x e. H~ /\ f ~~>v x) -> (x e. h /\ f ~~>v x)) -> (E.x(x e. H~ /\ f ~~>v x) -> E.x(x e. h /\ f ~~>v x)))
11	9, 10	syl 12	. . . . . . . . . 10 |- ((A.x((f:NN-->h /\ f ~~>v x) -> x e. h) /\ f:NN-->h) -> (E.x(x e. H~ /\ f ~~>v x) -> E.x(x e. h /\ f ~~>v x)))
12		df-rex 1206	. . . . . . . . . 10 |- (E.x e. H~ f ~~>v x <-> E.x(x e. H~ /\ f ~~>v x))
13		df-rex 1206	. . . . . . . . . 10 |- (E.x e. h f ~~>v x <-> E.x(x e. h /\ f ~~>v x))
14	11, 12, 13	3imtr4g 426	. . . . . . . . 9 |- ((A.x((f:NN-->h /\ f ~~>v x) -> x e. h) /\ f:NN-->h) -> (E.x e. H~ f ~~>v x -> E.x e. h f ~~>v x))
15		ax-hcompl 5113	. . . . . . . . 9 |- (f e. Cauchy -> E.x e. H~ f ~~>v x)
16	14, 15	syl5 22	. . . . . . . 8 |- ((A.x((f:NN-->h /\ f ~~>v x) -> x e. h) /\ f:NN-->h) -> (f e. Cauchy -> E.x e. h f ~~>v x))
17	16	exp 291	. . . . . . 7 |- (A.x((f:NN-->h /\ f ~~>v x) -> x e. h) -> (f:NN-->h -> (f e. Cauchy -> E.x e. h f ~~>v x)))
18	17	com23 32	. . . . . 6 |- (A.x((f:NN-->h /\ f ~~>v x) -> x e. h) -> (f e. Cauchy -> (f:NN-->h -> E.x e. h f ~~>v x)))
19	18	19.20i 691	. . . . 5 |- (A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h) -> A.f(f e. Cauchy -> (f:NN-->h -> E.x e. h f ~~>v x)))
20		df-ral 1205	. . . . 5 |- (A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x) <-> A.f(f e. Cauchy -> (f:NN-->h -> E.x e. h f ~~>v x)))
21	19, 20	sylibr 175	. . . 4 |- (A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h) -> A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x))
22	21	anim2i 270	. . 3 |- ((h e. SH /\ A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h)) -> (h e. SH /\ A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x)))
23		closedsub 5128	. . 3 |- (h e. CH <-> (h e. SH /\ A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h)))
24		cmh.1	. . . 4 |- C = {h | (h e. SH /\ A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x))}
25	24	cleqabi 1176	. . 3 |- (h e. C <-> (h e. SH /\ A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x)))
26	22, 23, 25	3imtr4 192	. 2 |- (h e. CH -> h e. C)
27	26	ssriv 1508	1 |- CH (_ C

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202   (_ wss 1487   class class class wbr 2054  -->wf 2418  NNcn 4093  H~chil 4958  Cauchyccau 4965   ~~>v chli 4966  SHcsh 4967  CHcch 4968

This theorem is referenced by:  chcmh 5148

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074  ax-hcompl 5113

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-in 1491  df-ss 1492  df-f 2434  df-ch 5127

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