Theorem chsscmi 10745

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
chsscmi	|- CH C_ C

Distinct variable groups:   x,f,h   C,h

Proof of Theorem chsscmi

Step	Hyp	Ref	Expression
1		impexp 374	. . . . . . . . . . . . . . 15 |- (((f:NN-->h /\ f ~~>v x) -> x e. h) <-> (f:NN-->h -> (f ~~>v x -> x e. h)))
2		ancr 319	. . . . . . . . . . . . . . . . 17 |- ((f ~~>v x -> x e. h) -> (f ~~>v x -> (x e. h /\ f ~~>v x)))
3	2	adantld 426	. . . . . . . . . . . . . . . 16 |- ((f ~~>v x -> x e. h) -> ((x e. ~H /\ f ~~>v x) -> (x e. h /\ f ~~>v x)))
4	3	imim2i 11	. . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . 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 14	. . . . . . . . . . . . 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	alimdv 1668	. . . . . . . . . . . 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 378	. . . . . . . . . . 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		exim 1386	. . . . . . . . . . 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 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)))
11		df-rex 2110	. . . . . . . . . 10 |- (E.x e. ~H f ~~>v x <-> E.x(x e. ~H /\ f ~~>v x))
12		df-rex 2110	. . . . . . . . . 10 |- (E.x e. h f ~~>v x <-> E.x(x e. h /\ f ~~>v x))
13	10, 11, 12	3imtr4g 612	. . . . . . . . 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 10704	. . . . . . . . 9 |- (f e. Cauchy -> E.x e. ~H f ~~>v x)
15	13, 14	syl5 20	. . . . . . . 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 402	. . . . . . 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 36	. . . . . 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	alimi 1338	. . . . 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 2109	. . . . 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 217	. . . 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 362	. . 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 10726	. . 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 2001	. . 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	3imtr4i 236	. 2 |- (h e. CH -> h e. C)
26	25	ssriv 2621	1 |- CH C_ C

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106   C_ wss 2593   class class class wbr 3338  -->wf 3994  NNcn 6449  ~Hchil 10420  Cauchyccau 10427   ~~>v chli 10428  SHcsh 10429  CHcch 10430

This theorem is referenced by:  chcmhi 10746

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-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-hcompl 10704

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-in 2603  df-ss 2605  df-f 4010  df-ch 10725

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