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Theorem List for Metamath Proof Explorer - 28201-28300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-gdiv 28201* Define a function that maps a group operation to the group's division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
/𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
 
Theoremisgrpo 28202* The predicate "is a group operation." Note that 𝑋 is the base set of the group. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺𝐴 → (𝐺 ∈ GrpOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))))
 
Theoremisgrpoi 28203* Properties that determine a group operation. Read 𝑁 as 𝑁(𝑥). (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
𝑋 ∈ V    &   𝐺:(𝑋 × 𝑋)⟶𝑋    &   ((𝑥𝑋𝑦𝑋𝑧𝑋) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))    &   𝑈𝑋    &   (𝑥𝑋 → (𝑈𝐺𝑥) = 𝑥)    &   (𝑥𝑋𝑁𝑋)    &   (𝑥𝑋 → (𝑁𝐺𝑥) = 𝑈)       𝐺 ∈ GrpOp
 
Theoremgrpofo 28204 A group operation maps onto the group's underlying set. (Contributed by NM, 30-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto𝑋)
 
Theoremgrpocl 28205 Closure law for a group operation. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
 
Theoremgrpolidinv 28206* A group has a left identity element, and every member has a left inverse. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
 
Theoremgrpon0 28207 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ GrpOp → 𝑋 ≠ ∅)
 
Theoremgrpoass 28208 A group operation is associative. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))
 
Theoremgrpoidinvlem1 28209 Lemma for grpoidinv 28213. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑈𝐺𝐴) = 𝑈)
 
Theoremgrpoidinvlem2 28210 Lemma for grpoidinv 28213. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺𝑌))
 
Theoremgrpoidinvlem3 28211* Lemma for grpoidinv 28213. (Contributed by NM, 11-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   (𝜑 ↔ ∀𝑥𝑋 (𝑈𝐺𝑥) = 𝑥)    &   (𝜓 ↔ ∀𝑥𝑋𝑧𝑋 (𝑧𝐺𝑥) = 𝑈)       ((((𝐺 ∈ GrpOp ∧ 𝑈𝑋) ∧ (𝜑𝜓)) ∧ 𝐴𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈))
 
Theoremgrpoidinvlem4 28212* Lemma for grpoidinv 28213. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))
 
Theoremgrpoidinv 28213* A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
 
Theoremgrpoideu 28214* The left identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ GrpOp → ∃!𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
 
Theoremgrporndm 28215 A group's range in terms of its domain. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
(𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺)
 
Theorem0ngrp 28216 The empty set is not a group. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
¬ ∅ ∈ GrpOp
 
Theoremgidval 28217* The value of the identity element of a group. (Contributed by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺𝑉 → (GId‘𝐺) = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
 
Theoremgrpoidval 28218* Lemma for grpoidcl 28219 and others. (Contributed by NM, 5-Feb-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       (𝐺 ∈ GrpOp → 𝑈 = (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥))
 
Theoremgrpoidcl 28219 The identity element of a group belongs to the group. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       (𝐺 ∈ GrpOp → 𝑈𝑋)
 
Theoremgrpoidinv2 28220* A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
 
Theoremgrpolid 28221 The identity element of a group is a left identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑈𝐺𝐴) = 𝐴)
 
Theoremgrporid 28222 The identity element of a group is a right identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺𝑈) = 𝐴)
 
Theoremgrporcan 28223 Right cancellation law for groups. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))
 
Theoremgrpoinveu 28224* The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
 
Theoremgrpoid 28225 Two ways of saying that an element of a group is the identity element. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴))
 
Theoremgrporn 28226 The range of a group operation. Useful for satisfying group base set hypotheses of the form 𝑋 = ran 𝐺. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
𝐺 ∈ GrpOp    &   dom 𝐺 = (𝑋 × 𝑋)       𝑋 = ran 𝐺
 
Theoremgrpoinvfval 28227* The inverse function of a group. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
 
Theoremgrpoinvval 28228* The inverse of a group element. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
 
Theoremgrpoinvcl 28229 A group element's inverse is a group element. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
 
Theoremgrpoinv 28230 The properties of a group element's inverse. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈))
 
Theoremgrpolinv 28231 The left inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
 
Theoremgrporinv 28232 The right inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
 
Theoremgrpoinvid1 28233 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐴𝐺𝐵) = 𝑈))
 
Theoremgrpoinvid2 28234 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐵𝐺𝐴) = 𝑈))
 
Theoremgrpolcan 28235 Left cancellation law for groups. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))
 
Theoremgrpo2inv 28236 Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁‘(𝑁𝐴)) = 𝐴)
 
Theoremgrpoinvf 28237 Mapping of the inverse function of a group. (Contributed by NM, 29-Mar-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)       (𝐺 ∈ GrpOp → 𝑁:𝑋1-1-onto𝑋)
 
Theoremgrpoinvop 28238 The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺𝐵)) = ((𝑁𝐵)𝐺(𝑁𝐴)))
 
Theoremgrpodivfval 28239* Group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)    &   𝐷 = ( /𝑔𝐺)       (𝐺 ∈ GrpOp → 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
 
Theoremgrpodivval 28240 Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁𝐵)))
 
Theoremgrpodivinv 28241 Group division by an inverse. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝑁𝐵)) = (𝐴𝐺𝐵))
 
Theoremgrpoinvdiv 28242 Inverse of a group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝐵𝐷𝐴))
 
Theoremgrpodivf 28243 Mapping for group division. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       (𝐺 ∈ GrpOp → 𝐷:(𝑋 × 𝑋)⟶𝑋)
 
Theoremgrpodivcl 28244 Closure of group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)
 
Theoremgrpodivdiv 28245 Double group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷(𝐵𝐷𝐶)) = (𝐴𝐺(𝐶𝐷𝐵)))
 
Theoremgrpomuldivass 28246 Associative-type law for multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = (𝐴𝐺(𝐵𝐷𝐶)))
 
Theoremgrpodivid 28247 Division of a group member by itself. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 𝑈)
 
Theoremgrponpcan 28248 Cancellation law for group division. (npcan 10884 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
 
18.1.2  Abelian groups
 
Syntaxcablo 28249 Extend class notation with the class of all Abelian group operations.
class AbelOp
 
Definitiondf-ablo 28250* Define the class of all Abelian group operations. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
 
Theoremisablo 28251* The predicate "is an Abelian (commutative) group operation." (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
 
Theoremablogrpo 28252 An Abelian group operation is a group operation. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
(𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
 
Theoremablocom 28253 An Abelian group operation is commutative. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
 
Theoremablo32 28254 Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))
 
Theoremablo4 28255 Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
 
Theoremisabloi 28256* Properties that determine an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
𝐺 ∈ GrpOp    &   dom 𝐺 = (𝑋 × 𝑋)    &   ((𝑥𝑋𝑦𝑋) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))       𝐺 ∈ AbelOp
 
Theoremablomuldiv 28257 Law for group multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
 
Theoremablodivdiv 28258 Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷(𝐵𝐷𝐶)) = ((𝐴𝐷𝐵)𝐺𝐶))
 
Theoremablodivdiv4 28259 Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = (𝐴𝐷(𝐵𝐺𝐶)))
 
Theoremablodiv32 28260 Swap the second and third terms in a double division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐷𝐵))
 
Theoremablonncan 28261 Cancellation law for group division. (nncan 10904 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = 𝐵)
 
Theoremablonnncan1 28262 Cancellation law for group division. (nnncan1 10911 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷(𝐴𝐷𝐶)) = (𝐶𝐷𝐵))
 
18.2  Complex vector spaces
 
18.2.1  Definition and basic properties
 
Syntaxcvc 28263 Extend class notation with the class of all complex vector spaces.
class CVecOLD
 
Definitiondf-vc 28264* Define the class of all complex vector spaces. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
CVecOLD = {⟨𝑔, 𝑠⟩ ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))}
 
Theoremvcrel 28265 The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
Rel CVecOLD
 
TheoremvciOLD 28266* Obsolete version of cvsi 23663. The properties of a complex vector space, which is an Abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of complex numbers. The variable 𝑊 was chosen because V is already used for the universal class. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       (𝑊 ∈ CVecOLD → (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))))
 
Theoremvcsm 28267 Functionality of th scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       (𝑊 ∈ CVecOLD𝑆:(ℂ × 𝑋)⟶𝑋)
 
Theoremvccl 28268 Closure of the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
 
TheoremvcidOLD 28269 Identity element for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) Obsolete theorem, use clmvs1 23626 together with cvsclm 23659 instead. (New usage is discouraged.) (Proof modification is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)
 
Theoremvcdi 28270 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))
 
Theoremvcdir 28271 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
 
Theoremvcass 28272 Associative law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))
 
Theoremvc2OLD 28273 A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) Obsolete theorem, use clmvs2 23627 together with cvsclm 23659 instead. (New usage is discouraged.) (Proof modification is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))
 
Theoremvcablo 28274 Vector addition is an Abelian group operation. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)       (𝑊 ∈ CVecOLD𝐺 ∈ AbelOp)
 
Theoremvcgrp 28275 Vector addition is a group operation. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)       (𝑊 ∈ CVecOLD𝐺 ∈ GrpOp)
 
Theoremvclcan 28276 Left cancellation law for vector addition. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))
 
Theoremvczcl 28277 The zero vector is a vector. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (𝑊 ∈ CVecOLD𝑍𝑋)
 
Theoremvc0rid 28278 The zero vector is a right identity element. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)
 
Theoremvc0 28279 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       ((𝑊 ∈ CVecOLD𝐴𝑋) → (0𝑆𝐴) = 𝑍)
 
Theoremvcz 28280 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       ((𝑊 ∈ CVecOLD𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)
 
Theoremvcm 28281 Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 25-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺    &   𝑀 = (inv‘𝐺)       ((𝑊 ∈ CVecOLD𝐴𝑋) → (-1𝑆𝐴) = (𝑀𝐴))
 
Theoremisvclem 28282* Lemma for isvcOLD 28284. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (⟨𝐺, 𝑆⟩ ∈ CVecOLD ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))))
 
Theoremvcex 28283 The components of a complex vector space are sets. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
(⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))
 
TheoremisvcOLD 28284* The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) Obsolete version of iscvsp 23661. (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = ran 𝐺       (⟨𝐺, 𝑆⟩ ∈ CVecOLD ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))))
 
TheoremisvciOLD 28285* Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) Obsolete version of iscvsi 23662. (New usage is discouraged.) (Proof modification is discouraged.)
𝐺 ∈ AbelOp    &   dom 𝐺 = (𝑋 × 𝑋)    &   𝑆:(ℂ × 𝑋)⟶𝑋    &   (𝑥𝑋 → (1𝑆𝑥) = 𝑥)    &   ((𝑦 ∈ ℂ ∧ 𝑥𝑋𝑧𝑋) → (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))    &   ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥𝑋) → ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))    &   ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥𝑋) → ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))    &   𝑊 = ⟨𝐺, 𝑆       𝑊 ∈ CVecOLD
 
18.2.2  Examples of complex vector spaces
 
TheoremcnaddabloOLD 28286 Obsolete version of cnaddabl 18920. Complex number addition is an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
+ ∈ AbelOp
 
TheoremcnidOLD 28287 Obsolete version of cnaddid 18921. The group identity element of complex number addition is zero. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
0 = (GId‘ + )
 
TheoremcncvcOLD 28288 Obsolete version of cncvs 23678. The set of complex numbers is a complex vector space. The vector operation is +, and the scalar product is ·. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
⟨ + , · ⟩ ∈ CVecOLD
 
18.3  Normed complex vector spaces
 
18.3.1  Definition and basic properties
 
Syntaxcnv 28289 Extend class notation with the class of all normed complex vector spaces.
class NrmCVec
 
Syntaxcpv 28290 Extend class notation with vector addition in a normed complex vector space. In the literature, the subscript "v" is omitted, but we need it to avoid ambiguity with complex number addition + caddc 10529.
class +𝑣
 
Syntaxcba 28291 Extend class notation with the base set of a normed complex vector space. (Note that BaseSet is capitalized because, once it is fixed for a particular vector space 𝑈, it is not a function, unlike e.g., normCV. This is our typical convention.) (New usage is discouraged.)
class BaseSet
 
Syntaxcns 28292 Extend class notation with scalar multiplication in a normed complex vector space. In the literature scalar multiplication is usually indicated by juxtaposition, but we need an explicit symbol to prevent ambiguity.
class ·𝑠OLD
 
Syntaxcn0v 28293 Extend class notation with zero vector in a normed complex vector space.
class 0vec
 
Syntaxcnsb 28294 Extend class notation with vector subtraction in a normed complex vector space.
class 𝑣
 
Syntaxcnmcv 28295 Extend class notation with the norm function in a normed complex vector space. In the literature, the norm of 𝐴 is usually written "|| 𝐴 ||", but we use function notation to take advantage of our existing theorems about functions.
class normCV
 
Syntaxcims 28296 Extend class notation with the class of the induced metrics on normed complex vector spaces.
class IndMet
 
Definitiondf-nv 28297* Define the class of all normed complex vector spaces. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
NrmCVec = {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ (⟨𝑔, 𝑠⟩ ∈ CVecOLD𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛𝑥) + (𝑛𝑦))))}
 
Theoremnvss 28298 Structure of the class of all normed complex vectors spaces. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)
NrmCVec ⊆ (CVecOLD × V)
 
Theoremnvvcop 28299 A normed complex vector space is a vector space. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)
(⟨𝑊, 𝑁⟩ ∈ NrmCVec → 𝑊 ∈ CVecOLD)
 
Definitiondf-va 28300 Define vector addition on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
+𝑣 = (1st ∘ 1st )
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 44801-44804
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