Theorem closedsub 9014

Description: Closed subspace H of a Hilbert space. Definition of [Beran] p. 107.

Assertion

Ref	Expression
closedsub	|- (H e. CH <-> (H e. SH /\ A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H)))

Distinct variable group:   x,f,H

Proof of Theorem closedsub

Step	Hyp	Ref	Expression
1		elisset 1808	. 2 |- (H e. CH -> H e. V)
2		elisset 1808	. . 3 |- (H e. SH -> H e. V)
3	2	adantr 389	. 2 |- ((H e. SH /\ A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H)) -> H e. V)
4		eleq1 1526	. . . 4 |- (h = H -> (h e. SH <-> H e. SH))
5		feq3 3608	. . . . . . 7 |- (h = H -> (f:NN-->h <-> f:NN-->H))
6	5	anbi1d 615	. . . . . 6 |- (h = H -> ((f:NN-->h /\ f ~~>v x) <-> (f:NN-->H /\ f ~~>v x)))
7		eleq2 1527	. . . . . 6 |- (h = H -> (x e. h <-> x e. H))
8	6, 7	imbi12d 624	. . . . 5 |- (h = H -> (((f:NN-->h /\ f ~~>v x) -> x e. h) <-> ((f:NN-->H /\ f ~~>v x) -> x e. H)))
9	8	2albidv 1275	. . . 4 |- (h = H -> (A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h) <-> A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H)))
10	4, 9	anbi12d 626	. . 3 |- (h = H -> ((h e. SH /\ A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h)) <-> (H e. SH /\ A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H))))
11		df-ch 9013	. . 3 |- CH = {h | (h e. SH /\ A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h))}
12	10, 11	elab2g 1891	. 2 |- (H e. V -> (H e. CH <-> (H e. SH /\ A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H))))
13	1, 3, 12	pm5.21nii 677	1 |- (H e. CH <-> (H e. SH /\ A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  Vcvv 1802   class class class wbr 2609  -->wf 3168  NNcn 5268   ~~>v chli 8735  SHcsh 8736  CHcch 8737

This theorem is referenced by:  chlim 9025  chsscm 9033  chcmh 9034  helch 9037  hsn0elch 9041  occl 9097  chintcl 9210  osumlem7 9501  nlelch 9909  hmopidmch 9990

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

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-v 1803  df-in 2041  df-ss 2043  df-f 3184  df-ch 9013

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