Theorem closedsub 9093

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 1817	. 2 |- (H e. CH -> H e. V)
2		elisset 1817	. . 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 1534	. . . 4 |- (h = H -> (h e. SH <-> H e. SH))
5		feq3 3622	. . . . . . 7 |- (h = H -> (f:NN-->h <-> f:NN-->H))
6	5	anbi1d 617	. . . . . 6 |- (h = H -> ((f:NN-->h /\ f ~~>v x) <-> (f:NN-->H /\ f ~~>v x)))
7		eleq2 1535	. . . . . 6 |- (h = H -> (x e. h <-> x e. H))
8	6, 7	imbi12d 626	. . . . 5 |- (h = H -> (((f:NN-->h /\ f ~~>v x) -> x e. h) <-> ((f:NN-->H /\ f ~~>v x) -> x e. H)))
9	8	2albidv 1280	. . . 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 628	. . 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 9092	. . 3 |- CH = {h | (h e. SH /\ A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h))}
12	10, 11	elab2g 1900	. 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 679	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 954   = wceq 956   e. wcel 958  Vcvv 1811   class class class wbr 2619  -->wf 3178  NNcn 5296   ~~>v chli 8796  SHcsh 8797  CHcch 8798

This theorem is referenced by:  chlim 9104  chsscm 9112  chcmh 9113  helch 9116  hsn0elch 9120  occl 9181  chintcl 9295  osumlem7 9584  nlelch 9994  hmopidmch 10079

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-12 968  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

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053  df-f 3194  df-ch 9092

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