Theorem closedsub 10726

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 2299	. 2 |- (H e. CH -> H e. _V)
2		elisset 2299	. . 3 |- (H e. SH -> H e. _V)
3	2	adantr 425	. 2 |- ((H e. SH /\ A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H)) -> H e. _V)
4		eleq1 1957	. . . 4 |- (h = H -> (h e. SH <-> H e. SH))
5		feq3 4553	. . . . . . 7 |- (h = H -> (f:NN-->h <-> f:NN-->H))
6	5	anbi1d 679	. . . . . 6 |- (h = H -> ((f:NN-->h /\ f ~~>v x) <-> (f:NN-->H /\ f ~~>v x)))
7		eleq2 1958	. . . . . 6 |- (h = H -> (x e. h <-> x e. H))
8	6, 7	imbi12d 688	. . . . 5 |- (h = H -> (((f:NN-->h /\ f ~~>v x) -> x e. h) <-> ((f:NN-->H /\ f ~~>v x) -> x e. H)))
9	8	2albidv 1658	. . . 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 690	. . 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 10725	. . 3 |- CH = {h | (h e. SH /\ A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h))}
12	10, 11	elab2g 2406	. 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 743	1 |- (H e. CH <-> (H e. SH /\ A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  _Vcvv 2292   class class class wbr 3338  -->wf 3994  NNcn 6449   ~~>v chli 10428  SHcsh 10429  CHcch 10430

This theorem is referenced by:  chlimi 10737  chsscmi 10745  chcmhi 10746  helch 10749  hsn0elch 10753  occli 10814  chintcli 10928  osumlem7 11219  nlelchi 11631  hmopidmchi 11723

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

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-v 2294  df-in 2603  df-ss 2605  df-f 4010  df-ch 10725

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