Theorem chsscm 5147

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 ∈ Sℋ ∧ ∀f ∈ Cauchy (f:ℕ–→h → ∃x ∈ h f ⇝v x))}

Assertion

Ref	Expression
chsscm	⊢ Cℋ ⊆ C

Distinct variable group(s):   x,f,h   C,h

Proof of Theorem chsscm

Step	Hyp	Ref	Expression
1		impexp 276	. . . . . . . . . . . . . . . 16 ⊢ (((f:ℕ–→h ∧ f ⇝v x) → x ∈ h) ↔ (f:ℕ–→h → (f ⇝v x → x ∈ h)))
2		ancr 243	. . . . . . . . . . . . . . . . . 18 ⊢ ((f ⇝v x → x ∈ h) → (f ⇝v x → (x ∈ h ∧ f ⇝v x)))
3	2	adantld 307	. . . . . . . . . . . . . . . . 17 ⊢ ((f ⇝v x → x ∈ h) → ((x ∈ ℋ ∧ f ⇝v x) → (x ∈ h ∧ f ⇝v x)))
4	3	syl3 18	. . . . . . . . . . . . . . . 16 ⊢ ((f:ℕ–→h → (f ⇝v x → x ∈ h)) → (f:ℕ–→h → ((x ∈ ℋ ∧ f ⇝v x) → (x ∈ h ∧ f ⇝v x))))
5	1, 4	sylbi 174	. . . . . . . . . . . . . . 15 ⊢ (((f:ℕ–→h ∧ f ⇝v x) → x ∈ h) → (f:ℕ–→h → ((x ∈ ℋ ∧ f ⇝v x) → (x ∈ h ∧ f ⇝v x))))
6	5	com12 13	. . . . . . . . . . . . . 14 ⊢ (f:ℕ–→h → (((f:ℕ–→h ∧ f ⇝v x) → x ∈ h) → ((x ∈ ℋ ∧ f ⇝v x) → (x ∈ h ∧ f ⇝v x))))
7	6	19.20dv 946	. . . . . . . . . . . . 13 ⊢ (f:ℕ–→h → (∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h) → ∀x((x ∈ ℋ ∧ f ⇝v x) → (x ∈ h ∧ f ⇝v x))))
8	7	com12 13	. . . . . . . . . . . 12 ⊢ (∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h) → (f:ℕ–→h → ∀x((x ∈ ℋ ∧ f ⇝v x) → (x ∈ h ∧ f ⇝v x))))
9	8	imp 277	. . . . . . . . . . 11 ⊢ ((∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h) ∧ f:ℕ–→h) → ∀x((x ∈ ℋ ∧ f ⇝v x) → (x ∈ h ∧ f ⇝v x)))
10		19.22 722	. . . . . . . . . . 11 ⊢ (∀x((x ∈ ℋ ∧ f ⇝v x) → (x ∈ h ∧ f ⇝v x)) → (∃x(x ∈ ℋ ∧ f ⇝v x) → ∃x(x ∈ h ∧ f ⇝v x)))
11	9, 10	syl 12	. . . . . . . . . 10 ⊢ ((∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h) ∧ f:ℕ–→h) → (∃x(x ∈ ℋ ∧ f ⇝v x) → ∃x(x ∈ h ∧ f ⇝v x)))
12		df-rex 1206	. . . . . . . . . 10 ⊢ (∃x ∈ ℋ f ⇝v x ↔ ∃x(x ∈ ℋ ∧ f ⇝v x))
13		df-rex 1206	. . . . . . . . . 10 ⊢ (∃x ∈ h f ⇝v x ↔ ∃x(x ∈ h ∧ f ⇝v x))
14	11, 12, 13	3imtr4g 426	. . . . . . . . 9 ⊢ ((∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h) ∧ f:ℕ–→h) → (∃x ∈ ℋ f ⇝v x → ∃x ∈ h f ⇝v x))
15		ax-hcompl 5113	. . . . . . . . 9 ⊢ (f ∈ Cauchy → ∃x ∈ ℋ f ⇝v x)
16	14, 15	syl5 22	. . . . . . . 8 ⊢ ((∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h) ∧ f:ℕ–→h) → (f ∈ Cauchy → ∃x ∈ h f ⇝v x))
17	16	exp 291	. . . . . . 7 ⊢ (∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h) → (f:ℕ–→h → (f ∈ Cauchy → ∃x ∈ h f ⇝v x)))
18	17	com23 32	. . . . . 6 ⊢ (∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h) → (f ∈ Cauchy → (f:ℕ–→h → ∃x ∈ h f ⇝v x)))
19	18	19.20i 691	. . . . 5 ⊢ (∀f∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h) → ∀f(f ∈ Cauchy → (f:ℕ–→h → ∃x ∈ h f ⇝v x)))
20		df-ral 1205	. . . . 5 ⊢ (∀f ∈ Cauchy (f:ℕ–→h → ∃x ∈ h f ⇝v x) ↔ ∀f(f ∈ Cauchy → (f:ℕ–→h → ∃x ∈ h f ⇝v x)))
21	19, 20	sylibr 175	. . . 4 ⊢ (∀f∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h) → ∀f ∈ Cauchy (f:ℕ–→h → ∃x ∈ h f ⇝v x))
22	21	anim2i 270	. . 3 ⊢ ((h ∈ Sℋ ∧ ∀f∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h)) → (h ∈ Sℋ ∧ ∀f ∈ Cauchy (f:ℕ–→h → ∃x ∈ h f ⇝v x)))
23		closedsub 5128	. . 3 ⊢ (h ∈ Cℋ ↔ (h ∈ Sℋ ∧ ∀f∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h)))
24		cmh.1	. . . 4 ⊢ C = {h∣(h ∈ Sℋ ∧ ∀f ∈ Cauchy (f:ℕ–→h → ∃x ∈ h f ⇝v x))}
25	24	cleqabi 1176	. . 3 ⊢ (h ∈ C ↔ (h ∈ Sℋ ∧ ∀f ∈ Cauchy (f:ℕ–→h → ∃x ∈ h f ⇝v x)))
26	22, 23, 25	3imtr4 192	. 2 ⊢ (h ∈ Cℋ → h ∈ C)
27	26	ssriv 1508	1 ⊢ Cℋ ⊆ C

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ⊆ wss 1487   class class class wbr 2054  –→wf 2418  ℕcn 4093   ℋ chil 4958  Cauchyccau 4965   ⇝_v chli 4966   S_ℋ csh 4967   C_ℋ cch 4968

This theorem is referenced by:  chcmh 5148

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  ax-hcompl 5113

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-ral 1205  df-rex 1206  df-v 1349  df-in 1491  df-ss 1492  df-f 2434  df-ch 5127

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