Theorem shex 5115

Description: The set of subspaces of a Hilbert space exists (is a set).

Assertion

Ref	Expression
shex	⊢ Sℋ ∈ V

Proof of Theorem shex

Step	Hyp	Ref	Expression
1		df-sh 5114	. 2 ⊢ Sℋ = {h∣((h ⊆ ℋ ∧ 0v ∈ h) ∧ (∀x ∈ h ∀y ∈ h (x +v y) ∈ h ∧ ∀x ∈ ℂ ∀y ∈ h (x ·s y) ∈ h))}
2		df-pw 1799	. . . 4 ⊢ ℘ ℋ = {h∣h ⊆ ℋ }
3		ax-hilex 4983	. . . . 5 ⊢ ℋ ∈ V
4	3	pwex 1806	. . . 4 ⊢ ℘ ℋ ∈ V
5	2, 4	eqeltrr 1160	. . 3 ⊢ {h∣h ⊆ ℋ } ∈ V
6		pm3.26 256	. . . . 5 ⊢ ((h ⊆ ℋ ∧ 0v ∈ h) → h ⊆ ℋ )
7	6	adantr 306	. . . 4 ⊢ (((h ⊆ ℋ ∧ 0v ∈ h) ∧ (∀x ∈ h ∀y ∈ h (x +v y) ∈ h ∧ ∀x ∈ ℂ ∀y ∈ h (x ·s y) ∈ h)) → h ⊆ ℋ )
8	7	ss2abi 1552	. . 3 ⊢ {h∣((h ⊆ ℋ ∧ 0v ∈ h) ∧ (∀x ∈ h ∀y ∈ h (x +v y) ∈ h ∧ ∀x ∈ ℂ ∀y ∈ h (x ·s y) ∈ h))} ⊆ {h∣h ⊆ ℋ }
9	5, 8	ssexi 1701	. 2 ⊢ {h∣((h ⊆ ℋ ∧ 0v ∈ h) ∧ (∀x ∈ h ∀y ∈ h (x +v y) ∈ h ∧ ∀x ∈ ℂ ∀y ∈ h (x ·s y) ∈ h))} ∈ V
10	1, 9	eqeltr 1159	1 ⊢ Sℋ ∈ V

Syntax hints:   ∧ wa 196  {cab 1090   ∈ wcel 1092  ∀wral 1201  Vcvv 1348   ⊆ wss 1487  ℘cpw 1798  (class class class)co 3001  ℂcc 4026   ℋ chil 4958   +_v cva 4959   ·_s csm 4960  0_vc0v 4961   S_ℋ csh 4967

This theorem is referenced by:  chex 5130

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-or 197  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-pw 1799  df-sh 5114

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