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Theorem List for Metamath Proof Explorer - 31201-31300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremarchiabllem2b 31201* Lemma for archiabl 31203. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &    + = (+g𝑊)    &   (𝜑 → (oppg𝑊) ∈ oGrp)    &   ((𝜑𝑎𝐵0 < 𝑎) → ∃𝑏𝐵 ( 0 < 𝑏𝑏 < 𝑎))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
 
Theoremarchiabllem2 31202* Archimedean ordered groups with no minimal positive value are abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &    + = (+g𝑊)    &   (𝜑 → (oppg𝑊) ∈ oGrp)    &   ((𝜑𝑎𝐵0 < 𝑎) → ∃𝑏𝐵 ( 0 < 𝑏𝑏 < 𝑎))       (𝜑𝑊 ∈ Abel)
 
Theoremarchiabl 31203 Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)
 
20.3.9.12  Semiring left modules
 
Syntaxcslmd 31204 Extend class notation with class of all semimodules.
class SLMod
 
Definitiondf-slmd 31205* Define the class of all (left) modules over semirings, i.e. semimodules, which are generalizations of left modules. A semimodule is a commutative monoid (=vectors) together with a semiring (=scalars) and a left scalar product connecting them. (0[,]+∞) for example is not a full fledged left module, but is a semimodule. Definition of [Golan] p. 149. (Contributed by Thierry Arnoux, 21-Mar-2018.)
SLMod = {𝑔 ∈ CMnd ∣ [(Base‘𝑔) / 𝑣][(+g𝑔) / 𝑎][( ·𝑠𝑔) / 𝑠][(Scalar‘𝑔) / 𝑓][(Base‘𝑓) / 𝑘][(+g𝑓) / 𝑝][(.r𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞𝑘𝑟𝑘𝑥𝑣𝑤𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r𝑓)𝑠𝑤) = 𝑤 ∧ ((0g𝑓)𝑠𝑤) = (0g𝑔))))}
 
Theoremisslmd 31206* The predicate "is a semimodule". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)    &    × = (.r𝐹)    &    1 = (1r𝐹)    &   𝑂 = (0g𝐹)       (𝑊 ∈ SLMod ↔ (𝑊 ∈ CMnd ∧ 𝐹 ∈ SRing ∧ ∀𝑞𝐾𝑟𝐾𝑥𝑉𝑤𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤 ∧ (𝑂 · 𝑤) = 0 ))))
 
Theoremslmdlema 31207 Lemma for properties of a semimodule. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)    &    × = (.r𝐹)    &    1 = (1r𝐹)    &   𝑂 = (0g𝐹)       ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑌𝑉)) → (((𝑅 · 𝑌) ∈ 𝑉 ∧ (𝑅 · (𝑌 + 𝑋)) = ((𝑅 · 𝑌) + (𝑅 · 𝑋)) ∧ ((𝑄 𝑅) · 𝑌) = ((𝑄 · 𝑌) + (𝑅 · 𝑌))) ∧ (((𝑄 × 𝑅) · 𝑌) = (𝑄 · (𝑅 · 𝑌)) ∧ ( 1 · 𝑌) = 𝑌 ∧ (𝑂 · 𝑌) = 0 )))
 
Theoremlmodslmd 31208 Left semimodules generalize the notion of left modules. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑊 ∈ LMod → 𝑊 ∈ SLMod)
 
Theoremslmdcmn 31209 A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑊 ∈ SLMod → 𝑊 ∈ CMnd)
 
Theoremslmdmnd 31210 A semimodule is a monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑊 ∈ SLMod → 𝑊 ∈ Mnd)
 
Theoremslmdsrg 31211 The scalar component of a semimodule is a semiring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
 
Theoremslmdbn0 31212 The base set of a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.) (Proof shortened by AV, 10-Jan-2023.)
𝐵 = (Base‘𝑊)       (𝑊 ∈ SLMod → 𝐵 ≠ ∅)
 
Theoremslmdacl 31213 Closure of ring addition for a semimodule. (Contributed by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    + = (+g𝐹)       ((𝑊 ∈ SLMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
 
Theoremslmdmcl 31214 Closure of ring multiplication for a semimodule. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = (.r𝐹)       ((𝑊 ∈ SLMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 · 𝑌) ∈ 𝐾)
 
Theoremslmdsn0 31215 The set of scalars in a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.) (Proof shortened by AV, 10-Jan-2023.)
𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       (𝑊 ∈ SLMod → 𝐵 ≠ ∅)
 
Theoremslmdvacl 31216 Closure of vector addition for a semiring left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉𝑌𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
 
Theoremslmdass 31217 Semiring left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ SLMod ∧ (𝑋𝑉𝑌𝑉𝑍𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
 
Theoremslmdvscl 31218 Closure of scalar product for a semiring left module. (hvmulcl 29126 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ SLMod ∧ 𝑅𝐾𝑋𝑉) → (𝑅 · 𝑋) ∈ 𝑉)
 
Theoremslmdvsdi 31219 Distributive law for scalar product. (ax-hvdistr1 29121 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))
 
Theoremslmdvsdir 31220 Distributive law for scalar product. (ax-hvdistr1 29121 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)       ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
 
Theoremslmdvsass 31221 Associative law for scalar product. (ax-hvmulass 29120 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    × = (.r𝐹)       ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
 
Theoremslmd0cl 31222 The ring zero in a semimodule belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝐹)       (𝑊 ∈ SLMod → 0𝐾)
 
Theoremslmd1cl 31223 The ring unit in a semiring left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    1 = (1r𝐹)       (𝑊 ∈ SLMod → 1𝐾)
 
Theoremslmdvs1 31224 Scalar product with ring unit. (ax-hvmulid 29119 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &    1 = (1r𝐹)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → ( 1 · 𝑋) = 𝑋)
 
Theoremslmd0vcl 31225 The zero vector is a vector. (ax-hv0cl 29116 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ SLMod → 0𝑉)
 
Theoremslmd0vlid 31226 Left identity law for the zero vector. (hvaddid2 29136 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → ( 0 + 𝑋) = 𝑋)
 
Theoremslmd0vrid 31227 Right identity law for the zero vector. (ax-hvaddid 29117 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑋 + 0 ) = 𝑋)
 
Theoremslmd0vs 31228 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 29123 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑂 = (0g𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑂 · 𝑋) = 0 )
 
Theoremslmdvs0 31229 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 29137 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋 · 0 ) = 0 )
 
Theoremgsumvsca1 31230* Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.)
𝐵 = (Base‘𝑊)    &   𝐺 = (Scalar‘𝑊)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   (𝜑𝐾 ⊆ (Base‘𝐺))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑊 ∈ SLMod)    &   (𝜑𝑃𝐾)    &   ((𝜑𝑘𝐴) → 𝑄𝐵)       (𝜑 → (𝑊 Σg (𝑘𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘𝐴𝑄))))
 
Theoremgsumvsca2 31231* Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.) (Proof shortened by AV, 12-Dec-2019.)
𝐵 = (Base‘𝑊)    &   𝐺 = (Scalar‘𝑊)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   (𝜑𝐾 ⊆ (Base‘𝐺))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑊 ∈ SLMod)    &   (𝜑𝑄𝐵)    &   ((𝜑𝑘𝐴) → 𝑃𝐾)       (𝜑 → (𝑊 Σg (𝑘𝐴 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘𝐴𝑃)) · 𝑄))
 
20.3.9.13  Simple groups
 
Theoremprmsimpcyc 31232 A group of prime order is cyclic if and only if it is simple. This is the first family of finite simple groups. (Contributed by Thierry Arnoux, 21-Sep-2023.)
𝐵 = (Base‘𝐺)       ((♯‘𝐵) ∈ ℙ → (𝐺 ∈ SimpGrp ↔ 𝐺 ∈ CycGrp))
 
20.3.9.14  Rings - misc additions
 
Theoremrngurd 31233* Deduce the unit of a ring from its properties. (Contributed by Thierry Arnoux, 6-Sep-2016.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑1𝐵)    &   ((𝜑𝑥𝐵) → ( 1 · 𝑥) = 𝑥)    &   ((𝜑𝑥𝐵) → (𝑥 · 1 ) = 𝑥)       (𝜑1 = (1r𝑅))
 
Theoremdvdschrmulg 31234 In a ring, any multiple of the characteristics annihilates all elements. (Contributed by Thierry Arnoux, 6-Sep-2016.)
𝐶 = (chr‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.g𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝐶𝑁𝐴𝐵) → (𝑁 · 𝐴) = 0 )
 
Theoremfreshmansdream 31235 For a prime number 𝑃, if 𝑋 and 𝑌 are members of a commutative ring 𝑅 of characteristic 𝑃, then ((𝑋 + 𝑌)↑𝑃) = ((𝑋𝑃) + (𝑌𝑃)). This theorem is sometimes referred to as "the freshman's dream" . (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.g‘(mulGrp‘𝑅))    &   𝑃 = (chr‘𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑃 (𝑋 + 𝑌)) = ((𝑃 𝑋) + (𝑃 𝑌)))
 
Theoremfrobrhm 31236* In a commutative ring with prime characteristic, the Frobenius function 𝐹 is a ring endomorphism, thus named the Frobenius endomorphism. (Contributed by Thierry Arnoux, 31-May-2024.)
𝐵 = (Base‘𝑅)    &   𝑃 = (chr‘𝑅)    &    = (.g‘(mulGrp‘𝑅))    &   𝐹 = (𝑥𝐵 ↦ (𝑃 𝑥))    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑃 ∈ ℙ)       (𝜑𝐹 ∈ (𝑅 RingHom 𝑅))
 
Theoremress1r 31237 1r is unaffected by restriction. This is a bit more generic than subrg1 19843. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑆 = (𝑅s 𝐴)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 1𝐴𝐴𝐵) → 1 = (1r𝑆))
 
Theoremdvrdir 31238 Distributive law for the division operation of a ring. (Contributed by Thierry Arnoux, 30-Oct-2017.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    + = (+g𝑅)    &    / = (/r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝑈)) → ((𝑋 + 𝑌) / 𝑍) = ((𝑋 / 𝑍) + (𝑌 / 𝑍)))
 
Theoremrdivmuldivd 31239 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 30-Oct-2017.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    + = (+g𝑅)    &    / = (/r𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝑈)    &   (𝜑𝑍𝐵)    &   (𝜑𝑊𝑈)       (𝜑 → ((𝑋 / 𝑌) · (𝑍 / 𝑊)) = ((𝑋 · 𝑍) / (𝑌 · 𝑊)))
 
Theoremringinvval 31240* The ring inverse expressed in terms of multiplication. (Contributed by Thierry Arnoux, 23-Oct-2017.)
𝐵 = (Base‘𝑅)    &    = (.r𝑅)    &    1 = (1r𝑅)    &   𝑁 = (invr𝑅)    &   𝑈 = (Unit‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋) = (𝑦𝑈 (𝑦 𝑋) = 1 ))
 
Theoremdvrcan5 31241 Cancellation law for common factor in ratio. (divcan5 11564 analog.) (Contributed by Thierry Arnoux, 26-Oct-2016.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝑈𝑍𝑈)) → ((𝑋 · 𝑍) / (𝑌 · 𝑍)) = (𝑋 / 𝑌))
 
Theoremsubrgchr 31242 If 𝐴 is a subring of 𝑅, then they have the same characteristic. (Contributed by Thierry Arnoux, 24-Feb-2018.)
(𝐴 ∈ (SubRing‘𝑅) → (chr‘(𝑅s 𝐴)) = (chr‘𝑅))
 
Theoremrmfsupp2 31243* A mapping of a multiplication of a constant with a function into a ring is finitely supported if the function is finitely supported. (Contributed by Thierry Arnoux, 3-Jun-2023.)
𝑅 = (Base‘𝑀)    &   (𝜑𝑀 ∈ Ring)    &   (𝜑𝑉𝑋)    &   ((𝜑𝑣𝑉) → 𝐶𝑅)    &   (𝜑𝐴:𝑉𝑅)    &   (𝜑𝐴 finSupp (0g𝑀))       (𝜑 → (𝑣𝑉 ↦ ((𝐴𝑣)(.r𝑀)𝐶)) finSupp (0g𝑀))
 
20.3.9.15  Subfields
 
Theoremprimefldchr 31244 The characteristic of a prime field is the same as the characteristic of the main field. (Contributed by Thierry Arnoux, 21-Aug-2023.)
𝑃 = (𝑅s (SubDRing‘𝑅))       (𝑅 ∈ DivRing → (chr‘𝑃) = (chr‘𝑅))
 
20.3.9.16  Totally ordered rings and fields
 
Syntaxcorng 31245 Extend class notation with the class of all ordered rings.
class oRing
 
Syntaxcofld 31246 Extend class notation with the class of all ordered fields.
class oField
 
Definitiondf-orng 31247* Define class of all ordered rings. An ordered ring is a ring with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.)
oRing = {𝑟 ∈ (Ring ∩ oGrp) ∣ [(Base‘𝑟) / 𝑣][(0g𝑟) / 𝑧][(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))}
 
Definitiondf-ofld 31248 Define class of all ordered fields. An ordered field is a field with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)
oField = (Field ∩ oRing)
 
Theoremisorng 31249* An ordered ring is a ring with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    = (le‘𝑅)       (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
 
Theoremorngring 31250 An ordered ring is a ring. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝑅 ∈ oRing → 𝑅 ∈ Ring)
 
Theoremorngogrp 31251 An ordered ring is an ordered group. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝑅 ∈ oRing → 𝑅 ∈ oGrp)
 
Theoremisofld 31252 An ordered field is a field with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
 
Theoremorngmul 31253 In an ordered ring, the ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 20-Jan-2018.)
𝐵 = (Base‘𝑅)    &    = (le‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 (𝑋 · 𝑌))
 
Theoremorngsqr 31254 In an ordered ring, all squares are positive. (Contributed by Thierry Arnoux, 20-Jan-2018.)
𝐵 = (Base‘𝑅)    &    = (le‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ oRing ∧ 𝑋𝐵) → 0 (𝑋 · 𝑋))
 
Theoremornglmulle 31255 In an ordered ring, multiplication with a positive does not change comparison. (Contributed by Thierry Arnoux, 10-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ oRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    = (le‘𝑅)    &   (𝜑𝑋 𝑌)    &   (𝜑0 𝑍)       (𝜑 → (𝑍 · 𝑋) (𝑍 · 𝑌))
 
Theoremorngrmulle 31256 In an ordered ring, multiplication with a positive does not change comparison. (Contributed by Thierry Arnoux, 10-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ oRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    = (le‘𝑅)    &   (𝜑𝑋 𝑌)    &   (𝜑0 𝑍)       (𝜑 → (𝑋 · 𝑍) (𝑌 · 𝑍))
 
Theoremornglmullt 31257 In an ordered ring, multiplication with a positive does not change strict comparison. (Contributed by Thierry Arnoux, 9-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ oRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    < = (lt‘𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋 < 𝑌)    &   (𝜑0 < 𝑍)       (𝜑 → (𝑍 · 𝑋) < (𝑍 · 𝑌))
 
Theoremorngrmullt 31258 In an ordered ring, multiplication with a positive does not change strict comparison. (Contributed by Thierry Arnoux, 9-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ oRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    < = (lt‘𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋 < 𝑌)    &   (𝜑0 < 𝑍)       (𝜑 → (𝑋 · 𝑍) < (𝑌 · 𝑍))
 
Theoremorngmullt 31259 In an ordered ring, the strict ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   (𝜑𝑅 ∈ oRing)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑0 < 𝑋)    &   (𝜑0 < 𝑌)       (𝜑0 < (𝑋 · 𝑌))
 
Theoremofldfld 31260 An ordered field is a field. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ oField → 𝐹 ∈ Field)
 
Theoremofldtos 31261 An ordered field is a totally ordered set. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ oField → 𝐹 ∈ Toset)
 
Theoremorng0le1 31262 In an ordered ring, the ring unit is positive. (Contributed by Thierry Arnoux, 21-Jan-2018.)
0 = (0g𝐹)    &    1 = (1r𝐹)    &    = (le‘𝐹)       (𝐹 ∈ oRing → 0 1 )
 
Theoremofldlt1 31263 In an ordered field, the ring unit is strictly positive. (Contributed by Thierry Arnoux, 21-Jan-2018.)
0 = (0g𝐹)    &    1 = (1r𝐹)    &    < = (lt‘𝐹)       (𝐹 ∈ oField → 0 < 1 )
 
Theoremofldchr 31264 The characteristic of an ordered field is zero. (Contributed by Thierry Arnoux, 21-Jan-2018.) (Proof shortened by AV, 6-Oct-2020.)
(𝐹 ∈ oField → (chr‘𝐹) = 0)
 
Theoremsuborng 31265 Every subring of an ordered ring is also an ordered ring. (Contributed by Thierry Arnoux, 21-Jan-2018.)
((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oRing)
 
Theoremsubofld 31266 Every subfield of an ordered field is also an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
((𝐹 ∈ oField ∧ (𝐹s 𝐴) ∈ Field) → (𝐹s 𝐴) ∈ oField)
 
Theoremisarchiofld 31267* Axiom of Archimedes : a characterization of the Archimedean property for ordered fields. (Contributed by Thierry Arnoux, 9-Apr-2018.)
𝐵 = (Base‘𝑊)    &   𝐻 = (ℤRHom‘𝑊)    &    < = (lt‘𝑊)       (𝑊 ∈ oField → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑛 ∈ ℕ 𝑥 < (𝐻𝑛)))
 
20.3.9.17  Ring homomorphisms - misc additions
 
Theoremrhmdvdsr 31268 A ring homomorphism preserves the divisibility relation. (Contributed by Thierry Arnoux, 22-Oct-2017.)
𝑋 = (Base‘𝑅)    &    = (∥r𝑅)    &    / = (∥r𝑆)       (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 𝐵) → (𝐹𝐴) / (𝐹𝐵))
 
Theoremrhmopp 31269 A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.)
(𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr𝑅) RingHom (oppr𝑆)))
 
Theoremelrhmunit 31270 Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹𝐴) ∈ (Unit‘𝑆))
 
Theoremrhmdvd 31271 A ring homomorphism preserves ratios. (Contributed by Thierry Arnoux, 22-Oct-2017.)
𝑈 = (Unit‘𝑆)    &   𝑋 = (Base‘𝑅)    &    / = (/r𝑆)    &    · = (.r𝑅)       ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋) ∧ ((𝐹𝐵) ∈ 𝑈 ∧ (𝐹𝐶) ∈ 𝑈)) → ((𝐹𝐴) / (𝐹𝐵)) = ((𝐹‘(𝐴 · 𝐶)) / (𝐹‘(𝐵 · 𝐶))))
 
Theoremrhmunitinv 31272 Ring homomorphisms preserve the inverse of unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘((invr𝑅)‘𝐴)) = ((invr𝑆)‘(𝐹𝐴)))
 
Theoremkerunit 31273 If a unit element lies in the kernel of a ring homomorphism, then 0 = 1, i.e. the target ring is the zero ring. (Contributed by Thierry Arnoux, 24-Oct-2017.)
𝑈 = (Unit‘𝑅)    &    0 = (0g𝑆)    &    1 = (1r𝑆)       ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑈 ∩ (𝐹 “ { 0 })) ≠ ∅) → 1 = 0 )
 
20.3.9.18  Scalar restriction operation
 
Syntaxcresv 31274 Extend class notation with the scalar restriction operation.
class v
 
Definitiondf-resv 31275* Define an operator to restrict the scalar field component of an extended structure. (Contributed by Thierry Arnoux, 5-Sep-2018.)
v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)))
 
Theoremreldmresv 31276 The scalar restriction is a proper operator, so it can be used with ovprc1 7274. (Contributed by Thierry Arnoux, 6-Sep-2018.)
Rel dom ↾v
 
Theoremresvval 31277 Value of structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       ((𝑊𝑋𝐴𝑌) → 𝑅 = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)))
 
Theoremresvid2 31278 General behavior of trivial restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = 𝑊)
 
Theoremresvval2 31279 Value of nontrivial structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩))
 
Theoremresvsca 31280 Base set of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       (𝐴𝑉 → (𝐹s 𝐴) = (Scalar‘𝑅))
 
Theoremresvlem 31281 Other elements of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐶 = (𝐸𝑊)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 ≠ 5       (𝐴𝑉𝐶 = (𝐸𝑅))
 
Theoremresvbas 31282 Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉𝐵 = (Base‘𝐻))
 
Theoremresvplusg 31283 +g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    + = (+g𝐺)       (𝐴𝑉+ = (+g𝐻))
 
Theoremresvvsca 31284 ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    · = ( ·𝑠𝐺)       (𝐴𝑉· = ( ·𝑠𝐻))
 
Theoremresvmulr 31285 ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    · = (.r𝐺)       (𝐴𝑉· = (.r𝐻))
 
Theoremresv0g 31286 0g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    0 = (0g𝐺)       (𝐴𝑉0 = (0g𝐻))
 
Theoremresv1r 31287 1r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    1 = (1r𝐺)       (𝐴𝑉1 = (1r𝐻))
 
Theoremresvcmn 31288 Scalar restriction preserves commutative monoids. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)       (𝐴𝑉 → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd))
 
20.3.9.19  The commutative ring of gaussian integers
 
Theoremgzcrng 31289 The gaussian integers form a commutative ring. (Contributed by Thierry Arnoux, 18-Mar-2018.)
(ℂflds ℤ[i]) ∈ CRing
 
20.3.9.20  The archimedean ordered field of real numbers
 
Theoremreofld 31290 The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
fld ∈ oField
 
Theoremnn0omnd 31291 The nonnegative integers form an ordered monoid. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(ℂflds0) ∈ oMnd
 
Theoremrearchi 31292 The field of the real numbers is Archimedean. See also arch 12117. (Contributed by Thierry Arnoux, 9-Apr-2018.)
fld ∈ Archi
 
Theoremnn0archi 31293 The monoid of the nonnegative integers is Archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
(ℂflds0) ∈ Archi
 
Theoremxrge0slmod 31294 The extended nonnegative real numbers form a semiring left module. One could also have used subringAlg to get the same structure. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐺 = (ℝ*𝑠s (0[,]+∞))    &   𝑊 = (𝐺v (0[,)+∞))       𝑊 ∈ SLMod
 
20.3.9.21  The quotient map and quotient modules
 
Theoremqusker 31295* The kernel of a quotient map. (Contributed by Thierry Arnoux, 20-May-2023.)
𝑉 = (Base‘𝑀)    &   𝐹 = (𝑥𝑉 ↦ [𝑥](𝑀 ~QG 𝐺))    &   𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))    &    0 = (0g𝑁)       (𝐺 ∈ (NrmSGrp‘𝑀) → (𝐹 “ { 0 }) = 𝐺)
 
Theoremeqgvscpbl 31296 The left coset equivalence relation is compatible with the scalar multiplication operation. (Contributed by Thierry Arnoux, 18-May-2023.)
𝐵 = (Base‘𝑀)    &    = (𝑀 ~QG 𝐺)    &   𝑆 = (Base‘(Scalar‘𝑀))    &    · = ( ·𝑠𝑀)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝐺 ∈ (LSubSp‘𝑀))    &   (𝜑𝐾𝑆)       (𝜑 → (𝑋 𝑌 → (𝐾 · 𝑋) (𝐾 · 𝑌)))
 
Theoremqusvscpbl 31297* The quotient map distributes over the scalar multiplication. (Contributed by Thierry Arnoux, 18-May-2023.)
𝐵 = (Base‘𝑀)    &    = (𝑀 ~QG 𝐺)    &   𝑆 = (Base‘(Scalar‘𝑀))    &    · = ( ·𝑠𝑀)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝐺 ∈ (LSubSp‘𝑀))    &   (𝜑𝐾𝑆)    &   𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))    &    = ( ·𝑠𝑁)    &   𝐹 = (𝑥𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))    &   (𝜑𝑈𝐵)    &   (𝜑𝑉𝐵)       (𝜑 → ((𝐹𝑈) = (𝐹𝑉) → (𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉))))
 
Theoremqusscaval 31298 Value of the scalar multiplication operation on the quotient structure. (Contributed by Thierry Arnoux, 18-May-2023.)
𝐵 = (Base‘𝑀)    &    = (𝑀 ~QG 𝐺)    &   𝑆 = (Base‘(Scalar‘𝑀))    &    · = ( ·𝑠𝑀)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝐺 ∈ (LSubSp‘𝑀))    &   (𝜑𝐾𝑆)    &   𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))    &    = ( ·𝑠𝑁)       ((𝜑𝐾𝑆𝑋𝐵) → (𝐾 [𝑋](𝑀 ~QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺))
 
Theoremimaslmod 31299* The image structure of a left module is a left module. (Contributed by Thierry Arnoux, 15-May-2023.)
(𝜑𝑁 = (𝐹s 𝑀))    &   𝑉 = (Base‘𝑀)    &   𝑆 = (Base‘(Scalar‘𝑀))    &    + = (+g𝑀)    &    · = ( ·𝑠𝑀)    &    0 = (0g𝑀)    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   ((𝜑 ∧ (𝑘𝑆𝑎𝑉𝑏𝑉)) → ((𝐹𝑎) = (𝐹𝑏) → (𝐹‘(𝑘 · 𝑎)) = (𝐹‘(𝑘 · 𝑏))))    &   (𝜑𝑀 ∈ LMod)       (𝜑𝑁 ∈ LMod)
 
Theoremquslmod 31300 If 𝐺 is a submodule in 𝑀, then 𝑁 = 𝑀 / 𝐺 is a left module, called the quotient module of 𝑀 by 𝐺. (Contributed by Thierry Arnoux, 18-May-2023.)
𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))    &   𝑉 = (Base‘𝑀)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝐺 ∈ (LSubSp‘𝑀))       (𝜑𝑁 ∈ LMod)
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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 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