P. 305 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 30401-30500 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	norm3adifii 30401	Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℋ ⇒ ⊢ (abs‘((norm_ℎ‘(𝐴 −_ℎ 𝐶)) − (norm_ℎ‘(𝐵 −_ℎ 𝐶)))) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝐵))

Theorem	norm3lem 30402	Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℋ & ⊢ 𝐷 ∈ ℝ ⇒ ⊢ (((norm_ℎ‘(𝐴 −_ℎ 𝐶)) < (𝐷 / 2) ∧ (norm_ℎ‘(𝐶 −_ℎ 𝐵)) < (𝐷 / 2)) → (norm_ℎ‘(𝐴 −_ℎ 𝐵)) < 𝐷)

Theorem	norm3dif 30403	Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 20-Apr-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (norm_ℎ‘(𝐴 −_ℎ 𝐵)) ≤ ((norm_ℎ‘(𝐴 −_ℎ 𝐶)) + (norm_ℎ‘(𝐶 −_ℎ 𝐵))))

Theorem	norm3dif2 30404	Norm of differences around common element. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (norm_ℎ‘(𝐴 −_ℎ 𝐵)) ≤ ((norm_ℎ‘(𝐶 −_ℎ 𝐴)) + (norm_ℎ‘(𝐶 −_ℎ 𝐵))))

Theorem	norm3lemt 30405	Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℝ)) → (((norm_ℎ‘(𝐴 −_ℎ 𝐶)) < (𝐷 / 2) ∧ (norm_ℎ‘(𝐶 −_ℎ 𝐵)) < (𝐷 / 2)) → (norm_ℎ‘(𝐴 −_ℎ 𝐵)) < 𝐷))

Theorem	norm3adifi 30406	Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 3-Oct-1999.) (New usage is discouraged.)
⊢ 𝐶 ∈ ℋ ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (abs‘((norm_ℎ‘(𝐴 −_ℎ 𝐶)) − (norm_ℎ‘(𝐵 −_ℎ 𝐶)))) ≤ (norm_ℎ‘(𝐴 −_ℎ 𝐵)))

Theorem	normpari 30407	Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) + ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2)) = ((2 · ((norm_ℎ‘𝐴)↑2)) + (2 · ((norm_ℎ‘𝐵)↑2)))

Theorem	normpar 30408	Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) + ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2)) = ((2 · ((norm_ℎ‘𝐴)↑2)) + (2 · ((norm_ℎ‘𝐵)↑2))))

Theorem	normpar2i 30409	Corollary of parallelogram law for norms. Part of Lemma 3.6 of [Beran] p. 100. (Contributed by NM, 5-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℋ ⇒ ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) = (((2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) + (2 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))) − ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2))

Theorem	polid2i 30410	Generalized polarization identity. Generalization of Exercise 4(a) of [ReedSimon] p. 63. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℋ & ⊢ 𝐷 ∈ ℋ ⇒ ⊢ (𝐴 ·_ih 𝐵) = (((((𝐴 +_ℎ 𝐶) ·_ih (𝐷 +_ℎ 𝐵)) − ((𝐴 −_ℎ 𝐶) ·_ih (𝐷 −_ℎ 𝐵))) + (i · (((𝐴 +_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 +_ℎ (i ·_ℎ 𝐵))) − ((𝐴 −_ℎ (i ·_ℎ 𝐶)) ·_ih (𝐷 −_ℎ (i ·_ℎ 𝐵)))))) / 4)

Theorem	polidi 30411	Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of Axiom ax-his3 30337. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 ·_ih 𝐵) = (((((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2) − ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)) + (i · (((norm_ℎ‘(𝐴 +_ℎ (i ·_ℎ 𝐵)))↑2) − ((norm_ℎ‘(𝐴 −_ℎ (i ·_ℎ 𝐵)))↑2)))) / 4)

Theorem	polid 30412	Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of Axiom ax-his3 30337. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (((((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2) − ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)) + (i · (((norm_ℎ‘(𝐴 +_ℎ (i ·_ℎ 𝐵)))↑2) − ((norm_ℎ‘(𝐴 −_ℎ (i ·_ℎ 𝐵)))↑2)))) / 4))

20.2.3 Relate Hilbert space to normed complex vector spaces

Theorem	hilablo 30413	Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
⊢ +_ℎ ∈ AbelOp

Theorem	hilid 30414	The group identity element of Hilbert space vector addition is the zero vector. (Contributed by NM, 16-Apr-2007.) (New usage is discouraged.)
⊢ (GId‘ +_ℎ ) = 0_ℎ

Theorem	hilvc 30415	Hilbert space is a complex vector space. Vector addition is +_ℎ, and scalar product is ·_ℎ. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
⊢ ⟨ +_ℎ , ·_ℎ ⟩ ∈ CVec_OLD

Theorem	hilnormi 30416	Hilbert space norm in terms of vector space norm. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ ℋ = (BaseSet‘𝑈) & ⊢ ·_ih = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ norm_ℎ = (norm_CV‘𝑈)

Theorem	hilhhi 30417	Deduce the structure of Hilbert space from its components. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ ℋ = (BaseSet‘𝑈) & ⊢ +_ℎ = ( +_𝑣 ‘𝑈) & ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈) & ⊢ ·_ih = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩

Theorem	hhnv 30418	Hilbert space is a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ⇒ ⊢ 𝑈 ∈ NrmCVec

Theorem	hhva 30419	The group (addition) operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ⇒ ⊢ +_ℎ = ( +_𝑣 ‘𝑈)

Theorem	hhba 30420	The base set of Hilbert space. This theorem provides an independent proof of df-hba 30222 (see comments in that definition). (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ⇒ ⊢ ℋ = (BaseSet‘𝑈)

Theorem	hh0v 30421	The zero vector of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ⇒ ⊢ 0_ℎ = (0_vec‘𝑈)

Theorem	hhsm 30422	The scalar product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ⇒ ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)

Theorem	hhvs 30423	The vector subtraction operation of Hilbert space. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ⇒ ⊢ −_ℎ = ( −_𝑣 ‘𝑈)

Theorem	hhnm 30424	The norm function of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ⇒ ⊢ norm_ℎ = (norm_CV‘𝑈)

Theorem	hhims 30425	The induced metric of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ & ⊢ 𝐷 = (norm_ℎ ∘ −_ℎ ) ⇒ ⊢ 𝐷 = (IndMet‘𝑈)

Theorem	hhims2 30426	Hilbert space distance metric. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ 𝐷 = (norm_ℎ ∘ −_ℎ )

Theorem	hhmet 30427	The induced metric of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ 𝐷 ∈ (Met‘ ℋ)

Theorem	hhxmet 30428	The induced metric of Hilbert space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ 𝐷 ∈ (∞Met‘ ℋ)

Theorem	hhmetdval 30429	Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴𝐷𝐵) = (norm_ℎ‘(𝐴 −_ℎ 𝐵)))

Theorem	hhip 30430	The inner product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ⇒ ⊢ ·_ih = (·_𝑖OLD‘𝑈)

Theorem	hhph 30431	The Hilbert space of the Hilbert Space Explorer is an inner product space. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ⇒ ⊢ 𝑈 ∈ CPreHil_OLD

20.2.4 Bunjakovaskij-Cauchy-Schwarz inequality

Theorem	bcsiALT 30432	Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵))

Theorem	bcsiHIL 30433	Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Proved from ZFC version.) (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵))

Theorem	bcs 30434	Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (abs‘(𝐴 ·_ih 𝐵)) ≤ ((norm_ℎ‘𝐴) · (norm_ℎ‘𝐵)))

Theorem	bcs2 30435	Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 30433. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (norm_ℎ‘𝐴) ≤ 1) → (abs‘(𝐴 ·_ih 𝐵)) ≤ (norm_ℎ‘𝐵))

Theorem	bcs3 30436	Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 30433. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (norm_ℎ‘𝐵) ≤ 1) → (abs‘(𝐴 ·_ih 𝐵)) ≤ (norm_ℎ‘𝐴))

20.3 Cauchy sequences and completeness axiom

20.3.1 Cauchy sequences and limits

Theorem	hcau 30437*	Member of the set of Cauchy sequences on a Hilbert space. Definition for Cauchy sequence in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ (𝐹 ∈ Cauchy ↔ (𝐹:ℕ⟶ ℋ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ_≥‘𝑦)(norm_ℎ‘((𝐹‘𝑦) −_ℎ (𝐹‘𝑧))) < 𝑥))

Theorem	hcauseq 30438	A Cauchy sequences on a Hilbert space is a sequence. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ (𝐹 ∈ Cauchy → 𝐹:ℕ⟶ ℋ)

Theorem	hcaucvg 30439*	A Cauchy sequence on a Hilbert space converges. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ ((𝐹 ∈ Cauchy ∧ 𝐴 ∈ ℝ⁺) → ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ_≥‘𝑦)(norm_ℎ‘((𝐹‘𝑦) −_ℎ (𝐹‘𝑧))) < 𝐴)

Theorem	seq1hcau 30440*	A sequence on a Hilbert space is a Cauchy sequence if it converges. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ (𝐹:ℕ⟶ ℋ → (𝐹 ∈ Cauchy ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ_≥‘𝑦)(norm_ℎ‘((𝐹‘𝑦) −_ℎ (𝐹‘𝑧))) < 𝑥))

Theorem	hlimi 30441*	Express the predicate: The limit of vector sequence 𝐹 in a Hilbert space is 𝐴, i.e. 𝐹 converges to 𝐴. This means that for any real 𝑥, no matter how small, there always exists an integer 𝑦 such that the norm of any later vector in the sequence minus the limit is less than 𝑥. Definition of converge in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐹 ⇝_𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ_≥‘𝑦)(norm_ℎ‘((𝐹‘𝑧) −_ℎ 𝐴)) < 𝑥))

Theorem	hlimseqi 30442	A sequence with a limit on a Hilbert space is a sequence. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐹 ⇝_𝑣 𝐴 → 𝐹:ℕ⟶ ℋ)

Theorem	hlimveci 30443	Closure of the limit of a sequence on Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐹 ⇝_𝑣 𝐴 → 𝐴 ∈ ℋ)

Theorem	hlimconvi 30444*	Convergence of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ 𝐴 ∈ V ⇒ ⊢ ((𝐹 ⇝_𝑣 𝐴 ∧ 𝐵 ∈ ℝ⁺) → ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ_≥‘𝑦)(norm_ℎ‘((𝐹‘𝑧) −_ℎ 𝐴)) < 𝐵)

Theorem	hlim2 30445*	The limit of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝐹 ⇝_𝑣 𝐴 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ_≥‘𝑦)(norm_ℎ‘((𝐹‘𝑧) −_ℎ 𝐴)) < 𝑥))

Theorem	hlimadd 30446*	Limit of the sum of two sequences in a Hilbert vector space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ (𝜑 → 𝐹:ℕ⟶ ℋ) & ⊢ (𝜑 → 𝐺:ℕ⟶ ℋ) & ⊢ (𝜑 → 𝐹 ⇝_𝑣 𝐴) & ⊢ (𝜑 → 𝐺 ⇝_𝑣 𝐵) & ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛) +_ℎ (𝐺‘𝑛))) ⇒ ⊢ (𝜑 → 𝐻 ⇝_𝑣 (𝐴 +_ℎ 𝐵))

20.3.2 Derivation of the completeness axiom from ZF set theory

Theorem	hilmet 30447	The Hilbert space norm determines a metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
⊢ 𝐷 = (norm_ℎ ∘ −_ℎ ) ⇒ ⊢ 𝐷 ∈ (Met‘ ℋ)

Theorem	hilxmet 30448	The Hilbert space norm determines a metric space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
⊢ 𝐷 = (norm_ℎ ∘ −_ℎ ) ⇒ ⊢ 𝐷 ∈ (∞Met‘ ℋ)

Theorem	hilmetdval 30449	Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
⊢ 𝐷 = (norm_ℎ ∘ −_ℎ ) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴𝐷𝐵) = (norm_ℎ‘(𝐴 −_ℎ 𝐵)))

Theorem	hilims 30450	Hilbert space distance metric. (Contributed by NM, 13-Sep-2007.) (New usage is discouraged.)
⊢ ℋ = (BaseSet‘𝑈) & ⊢ +_ℎ = ( +_𝑣 ‘𝑈) & ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈) & ⊢ ·_ih = (·_𝑖OLD‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ 𝐷 = (norm_ℎ ∘ −_ℎ )

Theorem	hhcau 30451	The Cauchy sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ Cauchy = ((Cau‘𝐷) ∩ ( ℋ ↑_m ℕ))

Theorem	hhlm 30452	The limit sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ⇝_𝑣 = ((⇝_𝑡‘𝐽) ↾ ( ℋ ↑_m ℕ))

Theorem	hhcmpl 30453*	Lemma used for derivation of the completeness axiom ax-hcompl 30455 from ZFC Hilbert space theory. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝐹 ∈ (Cau‘𝐷) → ∃𝑥 ∈ ℋ 𝐹(⇝_𝑡‘𝐽)𝑥) ⇒ ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥)

Theorem	hilcompl 30454*	Lemma used for derivation of the completeness axiom ax-hcompl 30455 from ZFC Hilbert space theory. The first five hypotheses would be satisfied by the definitions described in ax-hilex 30252; the 6th would be satisfied by eqid 2733; the 7th by a given fixed Hilbert space; and the last by Theorem hlcompl 30168. (Contributed by NM, 13-Sep-2007.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ ℋ = (BaseSet‘𝑈) & ⊢ +_ℎ = ( +_𝑣 ‘𝑈) & ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈) & ⊢ ·_ih = (·_𝑖OLD‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑈 ∈ CHil_OLD & ⊢ (𝐹 ∈ (Cau‘𝐷) → ∃𝑥 ∈ ℋ 𝐹(⇝_𝑡‘𝐽)𝑥) ⇒ ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥)

20.3.3 Completeness postulate for a Hilbert space

Axiom	ax-hcompl 30455*	Completeness of a Hilbert space. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥)

20.3.4 Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces

Theorem	hhcms 30456	The Hilbert space induced metric determines a complete metric space. (Contributed by NM, 10-Apr-2008.) (Proof shortened by Mario Carneiro, 14-May-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ 𝐷 ∈ (CMet‘ ℋ)

Theorem	hhhl 30457	The Hilbert space structure is a complex Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ⇒ ⊢ 𝑈 ∈ CHil_OLD

Theorem	hilcms 30458	The Hilbert space norm determines a complete metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
⊢ 𝐷 = (norm_ℎ ∘ −_ℎ ) ⇒ ⊢ 𝐷 ∈ (CMet‘ ℋ)

Theorem	hilhl 30459	The Hilbert space of the Hilbert Space Explorer is a complex Hilbert space. (Contributed by Steve Rodriguez, 29-Apr-2007.) (New usage is discouraged.)
⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ CHil_OLD

20.4 Subspaces and projections

20.4.1 Subspaces

Definition	df-sh 30460	Define the set of subspaces of a Hilbert space. See issh 30461 for its membership relation. Basically, a subspace is a subset of a Hilbert space that acts like a vector space. From Definition of [Beran] p. 95. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ S_ℋ = {ℎ ∈ 𝒫 ℋ ∣ (0_ℎ ∈ ℎ ∧ ( +_ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·_ℎ “ (ℂ × ℎ)) ⊆ ℎ)}

Theorem	issh 30461	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. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝐻 ∈ S_ℋ ↔ ((𝐻 ⊆ ℋ ∧ 0_ℎ ∈ 𝐻) ∧ (( +_ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·_ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))

Theorem	issh2 30462*	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. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝐻 ∈ S_ℋ ↔ ((𝐻 ⊆ ℋ ∧ 0_ℎ ∈ 𝐻) ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +_ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·_ℎ 𝑦) ∈ 𝐻)))

Theorem	shss 30463	A subspace is a subset of Hilbert space. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝐻 ∈ S_ℋ → 𝐻 ⊆ ℋ)

Theorem	shel 30464	A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
⊢ ((𝐻 ∈ S_ℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ)

Theorem	shex 30465	The set of subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
⊢ S_ℋ ∈ V

Theorem	shssii 30466	A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ S_ℋ ⇒ ⊢ 𝐻 ⊆ ℋ

Theorem	sheli 30467	A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ S_ℋ ⇒ ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Theorem	shelii 30468	A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ S_ℋ & ⊢ 𝐴 ∈ 𝐻 ⇒ ⊢ 𝐴 ∈ ℋ

Theorem	sh0 30469	The zero vector belongs to any subspace of a Hilbert space. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝐻 ∈ S_ℋ → 0_ℎ ∈ 𝐻)

Theorem	shaddcl 30470	Closure of vector addition in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ S_ℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 +_ℎ 𝐵) ∈ 𝐻)

Theorem	shmulcl 30471	Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ S_ℋ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝐻) → (𝐴 ·_ℎ 𝐵) ∈ 𝐻)

Theorem	issh3 30472*	Subspace 𝐻 of a Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
⊢ (𝐻 ⊆ ℋ → (𝐻 ∈ S_ℋ ↔ (0_ℎ ∈ 𝐻 ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +_ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·_ℎ 𝑦) ∈ 𝐻))))

Theorem	shsubcl 30473	Closure of vector subtraction in a subspace of a Hilbert space. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ S_ℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 −_ℎ 𝐵) ∈ 𝐻)

20.4.2 Closed subspaces

Definition	df-ch 30474	Define the set of closed subspaces of a Hilbert space. A closed subspace is one in which the limit of every convergent sequence in the subspace belongs to the subspace. For its membership relation, see isch 30475. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 30476 and isch3 30494. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
⊢ C_ℋ = {ℎ ∈ S_ℋ ∣ ( ⇝_𝑣 “ (ℎ ↑_m ℕ)) ⊆ ℎ}

Theorem	isch 30475	Closed subspace 𝐻 of a Hilbert space. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝐻 ∈ C_ℋ ↔ (𝐻 ∈ S_ℋ ∧ ( ⇝_𝑣 “ (𝐻 ↑_m ℕ)) ⊆ 𝐻))

Theorem	isch2 30476*	Closed subspace 𝐻 of a Hilbert space. Definition of [Beran] p. 107. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝐻 ∈ C_ℋ ↔ (𝐻 ∈ S_ℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝_𝑣 𝑥) → 𝑥 ∈ 𝐻)))

Theorem	chsh 30477	A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ (𝐻 ∈ C_ℋ → 𝐻 ∈ S_ℋ )

Theorem	chsssh 30478	Closed subspaces are subspaces in a Hilbert space. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ C_ℋ ⊆ S_ℋ

Theorem	chex 30479	The set of closed subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
⊢ C_ℋ ∈ V

Theorem	chshii 30480	A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ C_ℋ ⇒ ⊢ 𝐻 ∈ S_ℋ

Theorem	ch0 30481	The zero vector belongs to any closed subspace of a Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)
⊢ (𝐻 ∈ C_ℋ → 0_ℎ ∈ 𝐻)

Theorem	chss 30482	A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)
⊢ (𝐻 ∈ C_ℋ → 𝐻 ⊆ ℋ)

Theorem	chel 30483	A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
⊢ ((𝐻 ∈ C_ℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ)

Theorem	chssii 30484	A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ C_ℋ ⇒ ⊢ 𝐻 ⊆ ℋ

Theorem	cheli 30485	A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ C_ℋ ⇒ ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Theorem	chelii 30486	A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
⊢ 𝐻 ∈ C_ℋ & ⊢ 𝐴 ∈ 𝐻 ⇒ ⊢ 𝐴 ∈ ℋ

Theorem	chlimi 30487	The limit property of a closed subspace of a Hilbert space. (Contributed by NM, 14-Sep-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ V ⇒ ⊢ ((𝐻 ∈ C_ℋ ∧ 𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝_𝑣 𝐴) → 𝐴 ∈ 𝐻)

Theorem	hlim0 30488	The zero sequence in Hilbert space converges to the zero vector. (Contributed by NM, 17-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ (ℕ × {0_ℎ}) ⇝_𝑣 0_ℎ

Theorem	hlimcaui 30489	If a sequence in Hilbert space subset converges to a limit, it is a Cauchy sequence. (Contributed by NM, 17-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ (𝐹 ⇝_𝑣 𝐴 → 𝐹 ∈ Cauchy)

Theorem	hlimf 30490	Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ ⇝_𝑣 :dom ⇝_𝑣 ⟶ ℋ

Theorem	hlimuni 30491	A Hilbert space sequence converges to at most one limit. (Contributed by NM, 19-Aug-1999.) (Revised by Mario Carneiro, 2-May-2015.) (New usage is discouraged.)
⊢ ((𝐹 ⇝_𝑣 𝐴 ∧ 𝐹 ⇝_𝑣 𝐵) → 𝐴 = 𝐵)

Theorem	hlimreui 30492*	The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ (∃𝑥 ∈ 𝐻 𝐹 ⇝_𝑣 𝑥 ↔ ∃!𝑥 ∈ 𝐻 𝐹 ⇝_𝑣 𝑥)

Theorem	hlimeui 30493*	The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ (∃𝑥 𝐹 ⇝_𝑣 𝑥 ↔ ∃!𝑥 𝐹 ⇝_𝑣 𝑥)

Theorem	isch3 30494*	A Hilbert subspace is closed iff it is complete. 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). Remark 3.12 of [Beran] p. 107. (Contributed by NM, 24-Dec-2001.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
⊢ (𝐻 ∈ C_ℋ ↔ (𝐻 ∈ S_ℋ ∧ ∀𝑓 ∈ Cauchy (𝑓:ℕ⟶𝐻 → ∃𝑥 ∈ 𝐻 𝑓 ⇝_𝑣 𝑥)))

Theorem	chcompl 30495*	Completeness of a closed subspace of Hilbert space. (Contributed by NM, 4-Oct-1999.) (New usage is discouraged.)
⊢ ((𝐻 ∈ C_ℋ ∧ 𝐹 ∈ Cauchy ∧ 𝐹:ℕ⟶𝐻) → ∃𝑥 ∈ 𝐻 𝐹 ⇝_𝑣 𝑥)

Theorem	helch 30496	The Hilbert lattice one (which is all of Hilbert space) belongs to the Hilbert lattice. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 6-Sep-1999.) (New usage is discouraged.)
⊢ ℋ ∈ C_ℋ

Theorem	ifchhv 30497	Prove if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∈ C_ℋ. (Contributed by David A. Wheeler, 8-Dec-2018.) (New usage is discouraged.)
⊢ if(𝐴 ∈ C_ℋ , 𝐴, ℋ) ∈ C_ℋ

Theorem	helsh 30498	Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
⊢ ℋ ∈ S_ℋ

Theorem	shsspwh 30499	Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
⊢ S_ℋ ⊆ 𝒫 ℋ

Theorem	chsspwh 30500	Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
⊢ C_ℋ ⊆ 𝒫 ℋ

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47852

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