Theorem sh 5116

Description: Subspace H of a Hilbert space. A subspace is a subset of Hilbert space which contains the zero vector and is closed under vector addition and scalar multiplication. Definition of [Beran] p. 95.

Assertion

Ref	Expression
sh	⊢ (H ∈ Sℋ ↔ ((H ⊆ ℋ ∧ 0v ∈ H) ∧ (∀x ∈ H ∀y ∈ H (x +v y) ∈ H ∧ ∀x ∈ ℂ ∀y ∈ H (x ·s y) ∈ H)))

Distinct variable group(s):   x,y,H

Proof of Theorem sh

Step	Hyp	Ref	Expression
1		elisset 1354	. 2 ⊢ (H ∈ Sℋ → H ∈ V)
2		ax-hilex 4983	. . . 4 ⊢ ℋ ∈ V
3	2	ssex 1700	. . 3 ⊢ (H ⊆ ℋ → H ∈ V)
4	3	ad2antll 320	. 2 ⊢ (((H ⊆ ℋ ∧ 0v ∈ H) ∧ (∀x ∈ H ∀y ∈ H (x +v y) ∈ H ∧ ∀x ∈ ℂ ∀y ∈ H (x ·s y) ∈ H)) → H ∈ V)
5		sseq1 1521	. . . . 5 ⊢ (h = H → (h ⊆ ℋ ↔ H ⊆ ℋ ))
6		eleq2 1150	. . . . 5 ⊢ (h = H → (0v ∈ h ↔ 0v ∈ H))
7	5, 6	anbi12d 476	. . . 4 ⊢ (h = H → ((h ⊆ ℋ ∧ 0v ∈ h) ↔ (H ⊆ ℋ ∧ 0v ∈ H)))
8		eleq2 1150	. . . . . . 7 ⊢ (h = H → ((x +v y) ∈ h ↔ (x +v y) ∈ H))
9	8	raleqd 1327	. . . . . 6 ⊢ (h = H → (∀y ∈ h (x +v y) ∈ h ↔ ∀y ∈ H (x +v y) ∈ H))
10	9	raleqd 1327	. . . . 5 ⊢ (h = H → (∀x ∈ h ∀y ∈ h (x +v y) ∈ h ↔ ∀x ∈ H ∀y ∈ H (x +v y) ∈ H))
11		eleq2 1150	. . . . . . 7 ⊢ (h = H → ((x ·s y) ∈ h ↔ (x ·s y) ∈ H))
12	11	raleqd 1327	. . . . . 6 ⊢ (h = H → (∀y ∈ h (x ·s y) ∈ h ↔ ∀y ∈ H (x ·s y) ∈ H))
13	12	biraldv 1219	. . . . 5 ⊢ (h = H → (∀x ∈ ℂ ∀y ∈ h (x ·s y) ∈ h ↔ ∀x ∈ ℂ ∀y ∈ H (x ·s y) ∈ H))
14	10, 13	anbi12d 476	. . . 4 ⊢ (h = H → ((∀x ∈ h ∀y ∈ h (x +v y) ∈ h ∧ ∀x ∈ ℂ ∀y ∈ h (x ·s y) ∈ h) ↔ (∀x ∈ H ∀y ∈ H (x +v y) ∈ H ∧ ∀x ∈ ℂ ∀y ∈ H (x ·s y) ∈ H)))
15	7, 14	anbi12d 476	. . 3 ⊢ (h = H → (((h ⊆ ℋ ∧ 0v ∈ h) ∧ (∀x ∈ h ∀y ∈ h (x +v y) ∈ h ∧ ∀x ∈ ℂ ∀y ∈ h (x ·s y) ∈ h)) ↔ ((H ⊆ ℋ ∧ 0v ∈ H) ∧ (∀x ∈ H ∀y ∈ H (x +v y) ∈ H ∧ ∀x ∈ ℂ ∀y ∈ H (x ·s y) ∈ H))))
16		df-sh 5114	. . 3 ⊢ Sℋ = {h∣((h ⊆ ℋ ∧ 0v ∈ h) ∧ (∀x ∈ h ∀y ∈ h (x +v y) ∈ h ∧ ∀x ∈ ℂ ∀y ∈ h (x ·s y) ∈ h))}
17	15, 16	elab2g 1418	. 2 ⊢ (H ∈ V → (H ∈ Sℋ ↔ ((H ⊆ ℋ ∧ 0v ∈ H) ∧ (∀x ∈ H ∀y ∈ H (x +v y) ∈ H ∧ ∀x ∈ ℂ ∀y ∈ H (x ·s y) ∈ H))))
18	1, 4, 17	pm5.21nii 504	1 ⊢ (H ∈ Sℋ ↔ ((H ⊆ ℋ ∧ 0v ∈ H) ∧ (∀x ∈ H ∀y ∈ H (x +v y) ∈ H ∧ ∀x ∈ ℂ ∀y ∈ H (x ·s y) ∈ H)))

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348   ⊆ wss 1487  (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:  shss 5117  sh0 5122  shaddclt 5123  shmulclt 5124  sh2 5126  helch 5151  hsn0elch 5155  ocsh 5164  shscl 5282  shintcl 5294

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-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-hilex 4983

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-v 1349  df-in 1491  df-ss 1492  df-sh 5114

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