Theorem chsscm 9033

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 groups:   x,f,h   C,h

Proof of Theorem chsscm

Step	Hyp	Ref	Expression
1		impexp 347	. . . . . . . . . . . . . . 15 |- (((f:NN-->h /\ f ~~>v x) -> x e. h) <-> (f:NN-->h -> (f ~~>v x -> x e. h)))
2		ancr 295	. . . . . . . . . . . . . . . . 17 |- ((f ~~>v x -> x e. h) -> (f ~~>v x -> (x e. h /\ f ~~>v x)))
3	2	adantld 390	. . . . . . . . . . . . . . . 16 |- ((f ~~>v x -> x e. h) -> ((x e. H~ /\ f ~~>v x) -> (x e. h /\ f ~~>v x)))
4	3	imim2i 17	. . . . . . . . . . . . . . 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))))
5	1, 4	sylbi 199	. . . . . . . . . . . . . 14 |- (((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 11	. . . . . . . . . . . . 13 |- (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 1284	. . . . . . . . . . . 12 |- (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	impcom 351	. . . . . . . . . . 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)))
9		19.22 1035	. . . . . . . . . . 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)))
10	8, 9	syl 10	. . . . . . . . . 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)))
11		df-rex 1642	. . . . . . . . . 10 |- (E.x e. H~ f ~~>v x <-> E.x(x e. H~ /\ f ~~>v x))
12		df-rex 1642	. . . . . . . . . 10 |- (E.x e. h f ~~>v x <-> E.x(x e. h /\ f ~~>v x))
13	10, 11, 12	3imtr4g 551	. . . . . . . . 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))
14		ax-hcompl 8992	. . . . . . . . 9 |- (f e. Cauchy -> E.x e. H~ f ~~>v x)
15	13, 14	syl5 21	. . . . . . . 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))
16	15	ex 373	. . . . . . 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)))
17	16	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)))
18	17	19.20i 989	. . . . 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)))
19		df-ral 1641	. . . . 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)))
20	18, 19	sylibr 200	. . . 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))
21	20	anim2i 335	. . 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)))
22		closedsub 9014	. . 3 |- (h e. CH <-> (h e. SH /\ A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h)))
23		cmh.1	. . . 4 |- C = {h | (h e. SH /\ A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x))}
24	23	abeq2i 1562	. . 3 |- (h e. C <-> (h e. SH /\ A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x)))
25	21, 22, 24	3imtr4 219	. 2 |- (h e. CH -> h e. C)
26	25	ssriv 2059	1 |- CH (_ C

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  {cab 1456  A.wral 1637  E.wrex 1638   (_ wss 2037   class class class wbr 2609  -->wf 3168  NNcn 5268  H~chil 8727  Cauchyccau 8734   ~~>v chli 8735  SHcsh 8736  CHcch 8737

This theorem is referenced by:  chcmh 9034

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-hcompl 8992

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-rex 1642  df-v 1803  df-in 2041  df-ss 2043  df-f 3184  df-ch 9013

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