Theorem closedsub 5128

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

Assertion

Ref	Expression
closedsub	⊢ (H ∈ Cℋ ↔ (H ∈ Sℋ ∧ ∀f∀x((f:ℕ–→H ∧ f ⇝v x) → x ∈ H)))

Distinct variable group(s):   x,f,H

Proof of Theorem closedsub

Step	Hyp	Ref	Expression
1		elisset 1354	. 2 ⊢ (H ∈ Cℋ → H ∈ V)
2		elisset 1354	. . 3 ⊢ (H ∈ Sℋ → H ∈ V)
3	2	adantr 306	. 2 ⊢ ((H ∈ Sℋ ∧ ∀f∀x((f:ℕ–→H ∧ f ⇝v x) → x ∈ H)) → H ∈ V)
4		eleq1 1149	. . . 4 ⊢ (h = H → (h ∈ Sℋ ↔ H ∈ Sℋ ))
5		feq3 2750	. . . . . . 7 ⊢ (h = H → (f:ℕ–→h ↔ f:ℕ–→H))
6	5	anbi1d 469	. . . . . 6 ⊢ (h = H → ((f:ℕ–→h ∧ f ⇝v x) ↔ (f:ℕ–→H ∧ f ⇝v x)))
7		eleq2 1150	. . . . . 6 ⊢ (h = H → (x ∈ h ↔ x ∈ H))
8	6, 7	imbi12d 474	. . . . 5 ⊢ (h = H → (((f:ℕ–→h ∧ f ⇝v x) → x ∈ h) ↔ ((f:ℕ–→H ∧ f ⇝v x) → x ∈ H)))
9	8	bi2aldv 937	. . . 4 ⊢ (h = H → (∀f∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h) ↔ ∀f∀x((f:ℕ–→H ∧ f ⇝v x) → x ∈ H)))
10	4, 9	anbi12d 476	. . 3 ⊢ (h = H → ((h ∈ Sℋ ∧ ∀f∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h)) ↔ (H ∈ Sℋ ∧ ∀f∀x((f:ℕ–→H ∧ f ⇝v x) → x ∈ H))))
11		df-ch 5127	. . 3 ⊢ Cℋ = {h∣(h ∈ Sℋ ∧ ∀f∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h))}
12	10, 11	elab2g 1418	. 2 ⊢ (H ∈ V → (H ∈ Cℋ ↔ (H ∈ Sℋ ∧ ∀f∀x((f:ℕ–→H ∧ f ⇝v x) → x ∈ H))))
13	1, 3, 12	pm5.21nii 504	1 ⊢ (H ∈ Cℋ ↔ (H ∈ Sℋ ∧ ∀f∀x((f:ℕ–→H ∧ f ⇝v x) → x ∈ H)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   ∈ wel 803   = wceq 1091   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054  –→wf 2418  ℕcn 4093   ⇝_v chli 4966   S_ℋ csh 4967   C_ℋ cch 4968

This theorem is referenced by:  chlim 5139  chsscm 5147  chcmh 5148  helch 5151  hsn0elch 5155  occl 5188  chintcl 5296  osumlem7 5536

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-f 2434  df-ch 5127

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