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Theorem List for Metamath Proof Explorer - 33201-33300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremslmdass 33201 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 33202 Closure of scalar product for a semiring left module. (hvmulcl 31041 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 33203 Distributive law for scalar product. (ax-hvdistr1 31036 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 33204 Distributive law for scalar product. (ax-hvdistr1 31036 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 33205 Associative law for scalar product. (ax-hvmulass 31035 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 33206 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 33207 The ring unity 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 33208 Scalar product with ring unity. (ax-hvmulid 31034 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 33209 The zero vector is a vector. (ax-hv0cl 31031 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 33210 Left identity law for the zero vector. (hvaddlid 31051 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 33211 Right identity law for the zero vector. (ax-hvaddid 31032 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 33212 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 31038 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 33213 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 31052 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 33214* 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 33215* 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))
 
21.3.9.16  Simple groups
 
Theoremprmsimpcyc 33216 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))
 
21.3.9.17  Rings - misc additions
 
Theoremcringmul32d 33217 Commutative/associative law that swaps the last two factors in a triple product in a commutative ring. See also mul32 11424. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 · 𝑌) · 𝑍) = ((𝑋 · 𝑍) · 𝑌))
 
Theoremringdid 33218 Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
 
Theoremringdird 33219 Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
 
Theoremringdi22 33220 Expand the product of two sums in a ring. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑇𝐵)       (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))
 
Theoremurpropd 33221* Sufficient condition for ring unities to be equal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝐵 = (Base‘𝑆)    &   (𝜑𝑆𝑉)    &   (𝜑𝑇𝑊)    &   (𝜑𝐵 = (Base‘𝑇))    &   (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → (𝑥(.r𝑆)𝑦) = (𝑥(.r𝑇)𝑦))       (𝜑 → (1r𝑆) = (1r𝑇))
 
Theoremsubrgmcld 33222 A subring is closed under multiplication. (Contributed by Thierry Arnoux, 6-Jul-2025.)
· = (.r𝑅)    &   (𝜑𝐴 ∈ (SubRing‘𝑅))    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)       (𝜑 → (𝑋 · 𝑌) ∈ 𝐴)
 
Theoremress1r 33223 1r is unaffected by restriction. This is a bit more generic than subrg1 20598. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑆 = (𝑅s 𝐴)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 1𝐴𝐴𝐵) → 1 = (1r𝑆))
 
Theoremringinvval 33224* 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 33225 Cancellation law for common factor in ratio. (divcan5 11966 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 33226 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 33227* 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))
 
Theoremunitnz 33228 In a nonzero ring, a unit cannot be zero. (Contributed by Thierry Arnoux, 25-Apr-2025.)
𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ NzRing)    &   (𝜑𝑋𝑈)       (𝜑𝑋0 )
 
Theoremisunit2 33229* Alternate definition of being a unit. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       (𝑋𝑈 ↔ (𝑋𝐵 ∧ (∃𝑢𝐵 (𝑋 · 𝑢) = 1 ∧ ∃𝑣𝐵 (𝑣 · 𝑋) = 1 )))
 
Theoremisunit3 33230* Alternate definition of being a unit. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑅 ∈ Ring)       (𝜑 → (𝑋𝑈 ↔ ∃𝑦𝐵 ((𝑋 · 𝑦) = 1 ∧ (𝑦 · 𝑋) = 1 )))
 
21.3.9.18  Subrings generated by a set
 
Theoremelrgspnlem1 33231* Lemma for elrgspn 33235. (Contributed by Thierry Arnoux, 5-Oct-2025.)
𝐵 = (Base‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &    · = (.g𝑅)    &   𝑁 = (RingSpan‘𝑅)    &   𝐹 = {𝑓 ∈ (ℤ ↑m Word 𝐴) ∣ 𝑓 finSupp 0}    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐴𝐵)    &   𝑆 = ran (𝑔𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔𝑤) · (𝑀 Σg 𝑤)))))       (𝜑𝑆 ∈ (SubGrp‘𝑅))
 
Theoremelrgspnlem2 33232* Lemma for elrgspn 33235. (Contributed by Thierry Arnoux, 5-Oct-2025.)
𝐵 = (Base‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &    · = (.g𝑅)    &   𝑁 = (RingSpan‘𝑅)    &   𝐹 = {𝑓 ∈ (ℤ ↑m Word 𝐴) ∣ 𝑓 finSupp 0}    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐴𝐵)    &   𝑆 = ran (𝑔𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔𝑤) · (𝑀 Σg 𝑤)))))       (𝜑𝑆 ∈ (SubRing‘𝑅))
 
Theoremelrgspnlem3 33233* Lemma for elrgspn 33235. (Contributed by Thierry Arnoux, 5-Oct-2025.)
𝐵 = (Base‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &    · = (.g𝑅)    &   𝑁 = (RingSpan‘𝑅)    &   𝐹 = {𝑓 ∈ (ℤ ↑m Word 𝐴) ∣ 𝑓 finSupp 0}    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐴𝐵)    &   𝑆 = ran (𝑔𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔𝑤) · (𝑀 Σg 𝑤)))))       (𝜑𝐴𝑆)
 
Theoremelrgspnlem4 33234* Lemma for elrgspn 33235. (Contributed by Thierry Arnoux, 5-Oct-2025.)
𝐵 = (Base‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &    · = (.g𝑅)    &   𝑁 = (RingSpan‘𝑅)    &   𝐹 = {𝑓 ∈ (ℤ ↑m Word 𝐴) ∣ 𝑓 finSupp 0}    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐴𝐵)    &   𝑆 = ran (𝑔𝐹 ↦ (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔𝑤) · (𝑀 Σg 𝑤)))))       (𝜑 → (𝑁𝐴) = 𝑆)
 
Theoremelrgspn 33235* Membership in the subring generated by the subset 𝐴. An element 𝑋 lies in that subring if and only if 𝑋 is a linear combination with integer coefficients of products of elements of 𝐴. (Contributed by Thierry Arnoux, 5-Oct-2025.)
𝐵 = (Base‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &    · = (.g𝑅)    &   𝑁 = (RingSpan‘𝑅)    &   𝐹 = {𝑓 ∈ (ℤ ↑m Word 𝐴) ∣ 𝑓 finSupp 0}    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐴𝐵)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑋 ∈ (𝑁𝐴) ↔ ∃𝑔𝐹 𝑋 = (𝑅 Σg (𝑤 ∈ Word 𝐴 ↦ ((𝑔𝑤) · (𝑀 Σg 𝑤))))))
 
21.3.9.19  The zero ring
 
Theoremirrednzr 33236 A ring with an irreducible element cannot be the zero ring. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐼 = (Irred‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐼)       (𝜑𝑅 ∈ NzRing)
 
Theorem0ringsubrg 33237 A subring of a zero ring is a zero ring. (Contributed by Thierry Arnoux, 5-Feb-2025.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑 → (♯‘𝐵) = 1)    &   (𝜑𝑆 ∈ (SubRing‘𝑅))       (𝜑 → (♯‘𝑆) = 1)
 
Theorem0ringcring 33238 The zero ring is commutative. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑 → (♯‘𝐵) = 1)       (𝜑𝑅 ∈ CRing)
 
21.3.9.20  Localization of rings
 
Syntaxcerl 33239 Syntax for ring localization equivalence class operation.
class ~RL
 
Syntaxcrloc 33240 Syntax for ring localization operation.
class RLocal
 
Definitiondf-erl 33241* Define the operation giving the equivalence relation used in the localization of a ring 𝑟 by a set 𝑠. Two pairs 𝑎 = ⟨𝑥, 𝑦 and 𝑏 = ⟨𝑧, 𝑤 are equivalent if there exists 𝑡𝑠 such that 𝑡 · (𝑥 · 𝑤𝑧 · 𝑦) = 0. This corresponds to the usual comparison of fractions 𝑥 / 𝑦 and 𝑧 / 𝑤. (Contributed by Thierry Arnoux, 28-Apr-2025.)
~RL = (𝑟 ∈ V, 𝑠 ∈ V ↦ (.r𝑟) / 𝑥((Base‘𝑟) × 𝑠) / 𝑤{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑤𝑏𝑤) ∧ ∃𝑡𝑠 (𝑡𝑥(((1st𝑎)𝑥(2nd𝑏))(-g𝑟)((1st𝑏)𝑥(2nd𝑎)))) = (0g𝑟))})
 
Definitiondf-rloc 33242* Define the operation giving the localization of a ring 𝑟 by a given set 𝑠. The localized ring 𝑟 RLocal 𝑠 is the set of equivalence classes of pairs of elements in 𝑟 over the relation 𝑟 ~RL 𝑠 with addition and multiplication defined naturally. (Contributed by Thierry Arnoux, 27-Apr-2025.)
RLocal = (𝑟 ∈ V, 𝑠 ∈ V ↦ (.r𝑟) / 𝑥((Base‘𝑟) × 𝑠) / 𝑤((({⟨(Base‘ndx), 𝑤⟩, ⟨(+g‘ndx), (𝑎𝑤, 𝑏𝑤 ↦ ⟨(((1st𝑎)𝑥(2nd𝑏))(+g𝑟)((1st𝑏)𝑥(2nd𝑎))), ((2nd𝑎)𝑥(2nd𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎𝑤, 𝑏𝑤 ↦ ⟨((1st𝑎)𝑥(1st𝑏)), ((2nd𝑎)𝑥(2nd𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑟)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑟)), 𝑎𝑤 ↦ ⟨(𝑘( ·𝑠𝑟)(1st𝑎)), (2nd𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑟) ×t ((TopSet‘𝑟) ↾t 𝑠))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑤𝑏𝑤) ∧ ((1st𝑎)𝑥(2nd𝑏))(le‘𝑟)((1st𝑏)𝑥(2nd𝑎)))}⟩, ⟨(dist‘ndx), (𝑎𝑤, 𝑏𝑤 ↦ (((1st𝑎)𝑥(2nd𝑏))(dist‘𝑟)((1st𝑏)𝑥(2nd𝑎))))⟩}) /s (𝑟 ~RL 𝑠)))
 
Theoremreldmrloc 33243 Ring localization is a proper operator, so it can be used with ovprc1 7469. (Contributed by Thierry Arnoux, 10-May-2025.)
Rel dom RLocal
 
Theoremerlval 33244* Value of the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    = (-g𝑅)    &   𝑊 = (𝐵 × 𝑆)    &    = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ∃𝑡𝑆 (𝑡 · (((1st𝑎) · (2nd𝑏)) ((1st𝑏) · (2nd𝑎)))) = 0 )}    &   (𝜑𝑆𝐵)       (𝜑 → (𝑅 ~RL 𝑆) = )
 
Theoremrlocval 33245* Expand the value of the ring localization operation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    = (-g𝑅)    &    + = (+g𝑅)    &    = (le‘𝑅)    &   𝐹 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐹)    &   𝐶 = ( ·𝑠𝑅)    &   𝑊 = (𝐵 × 𝑆)    &    = (𝑅 ~RL 𝑆)    &   𝐽 = (TopSet‘𝑅)    &   𝐷 = (dist‘𝑅)    &    = (𝑎𝑊, 𝑏𝑊 ↦ ⟨(((1st𝑎) · (2nd𝑏)) + ((1st𝑏) · (2nd𝑎))), ((2nd𝑎) · (2nd𝑏))⟩)    &    = (𝑎𝑊, 𝑏𝑊 ↦ ⟨((1st𝑎) · (1st𝑏)), ((2nd𝑎) · (2nd𝑏))⟩)    &    × = (𝑘𝐾, 𝑎𝑊 ↦ ⟨(𝑘𝐶(1st𝑎)), (2nd𝑎)⟩)    &    = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ((1st𝑎) · (2nd𝑏)) ((1st𝑏) · (2nd𝑎)))}    &   𝐸 = (𝑎𝑊, 𝑏𝑊 ↦ (((1st𝑎) · (2nd𝑏))𝐷((1st𝑏) · (2nd𝑎))))    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝐵)       (𝜑 → (𝑅 RLocal 𝑆) = ((({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), ⟩} ∪ {⟨(Scalar‘ndx), 𝐹⟩, ⟨( ·𝑠 ‘ndx), × ⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), (𝐽 ×t (𝐽t 𝑆))⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐸⟩}) /s ))
 
Theoremerlcl1 33246 Closure for the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    = (𝑅 ~RL 𝑆)    &   (𝜑𝑆𝐵)    &   (𝜑𝑈 𝑉)       (𝜑𝑈 ∈ (𝐵 × 𝑆))
 
Theoremerlcl2 33247 Closure for the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    = (𝑅 ~RL 𝑆)    &   (𝜑𝑆𝐵)    &   (𝜑𝑈 𝑉)       (𝜑𝑉 ∈ (𝐵 × 𝑆))
 
Theoremerldi 33248* Main property of the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    = (𝑅 ~RL 𝑆)    &   (𝜑𝑆𝐵)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    = (-g𝑅)    &   (𝜑𝑈 𝑉)       (𝜑 → ∃𝑡𝑆 (𝑡 · (((1st𝑈) · (2nd𝑉)) ((1st𝑉) · (2nd𝑈)))) = 0 )
 
Theoremerlbrd 33249 Deduce the ring localization equivalence relation. If for some 𝑇𝑆 we have 𝑇 · (𝐸 · 𝐻𝐹 · 𝐺) = 0, then pairs 𝐸, 𝐺 and 𝐹, 𝐻 are equivalent under the localization relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    = (𝑅 ~RL 𝑆)    &   (𝜑𝑆𝐵)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    = (-g𝑅)    &   (𝜑𝑈 = ⟨𝐸, 𝐺⟩)    &   (𝜑𝑉 = ⟨𝐹, 𝐻⟩)    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝑆)    &   (𝜑𝐻𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑 → (𝑇 · ((𝐸 · 𝐻) (𝐹 · 𝐺))) = 0 )       (𝜑𝑈 𝑉)
 
Theoremerlbr2d 33250 Deduce the ring localization equivalence relation. Pairs 𝐸, 𝐺 and 𝑇 · 𝐸, 𝑇 · 𝐺 for 𝑇𝑆 are equivalent under the localization relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    = (𝑅 ~RL 𝑆)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &    · = (.r𝑅)    &   (𝜑𝑈 = ⟨𝐸, 𝐺⟩)    &   (𝜑𝑉 = ⟨𝐹, 𝐻⟩)    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝑆)    &   (𝜑𝐻𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝐹 = (𝑇 · 𝐸))    &   (𝜑𝐻 = (𝑇 · 𝐺))       (𝜑𝑈 𝑉)
 
Theoremerler 33251 The relation used to build the ring localization is an equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    · = (.r𝑅)    &    = (-g𝑅)    &   𝑊 = (𝐵 × 𝑆)    &    = (𝑅 ~RL 𝑆)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))       (𝜑 Er 𝑊)
 
Theoremelrlocbasi 33252* Membership in the basis of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)
(𝜑𝑋 ∈ ((𝐵 × 𝑆) / ))       (𝜑 → ∃𝑎𝐵𝑏𝑆 𝑋 = [⟨𝑎, 𝑏⟩] )
 
Theoremrlocbas 33253 The base set of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    = (-g𝑅)    &   𝑊 = (𝐵 × 𝑆)    &   𝐿 = (𝑅 RLocal 𝑆)    &    = (𝑅 ~RL 𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝐵)       (𝜑 → (𝑊 / ) = (Base‘𝐿))
 
Theoremrlocaddval 33254 Value of the addition in the ring localization, given two representatives. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    + = (+g𝑅)    &   𝐿 = (𝑅 RLocal 𝑆)    &    = (𝑅 ~RL 𝑆)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝑆)    &   (𝜑𝐻𝑆)    &    = (+g𝐿)       (𝜑 → ([⟨𝐸, 𝐺⟩] [⟨𝐹, 𝐻⟩] ) = [⟨((𝐸 · 𝐻) + (𝐹 · 𝐺)), (𝐺 · 𝐻)⟩] )
 
Theoremrlocmulval 33255 Value of the addition in the ring localization, given two representatives. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    + = (+g𝑅)    &   𝐿 = (𝑅 RLocal 𝑆)    &    = (𝑅 ~RL 𝑆)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝑆)    &   (𝜑𝐻𝑆)    &    = (.r𝐿)       (𝜑 → ([⟨𝐸, 𝐺⟩] [⟨𝐹, 𝐻⟩] ) = [⟨(𝐸 · 𝐹), (𝐺 · 𝐻)⟩] )
 
Theoremrloccring 33256 The ring localization 𝐿 of a commutative ring 𝑅 by a multiplicatively closed set 𝑆 is itself a commutative ring. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    + = (+g𝑅)    &   𝐿 = (𝑅 RLocal 𝑆)    &    = (𝑅 ~RL 𝑆)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))       (𝜑𝐿 ∈ CRing)
 
Theoremrloc0g 33257 The zero of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)
0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐿 = (𝑅 RLocal 𝑆)    &    = (𝑅 ~RL 𝑆)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   𝑂 = [⟨ 0 , 1 ⟩]        (𝜑𝑂 = (0g𝐿))
 
Theoremrloc1r 33258 The multiplicative identity of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)
0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐿 = (𝑅 RLocal 𝑆)    &    = (𝑅 ~RL 𝑆)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   𝐼 = [⟨ 1 , 1 ⟩]        (𝜑𝐼 = (1r𝐿))
 
Theoremrlocf1 33259* The embedding 𝐹 of a ring 𝑅 into its localization 𝐿. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)    &   𝐿 = (𝑅 RLocal 𝑆)    &    = (𝑅 ~RL 𝑆)    &   𝐹 = (𝑥𝐵 ↦ [⟨𝑥, 1 ⟩] )    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   (𝜑𝑆 ⊆ (RLReg‘𝑅))       (𝜑 → (𝐹:𝐵1-1→((𝐵 × 𝑆) / ) ∧ 𝐹 ∈ (𝑅 RingHom 𝐿)))
 
21.3.9.21  Integral Domains
 
Theoremdomnmuln0rd 33260 In a domain, factors of a nonzero product are nonzero. (Contributed by Thierry Arnoux, 8-Jun-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ Domn)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑋 · 𝑌) ≠ 0 )       (𝜑 → (𝑋0𝑌0 ))
 
Theoremdomnprodn0 33261 In a domain, a finite product of nonzero terms is nonzero. (Contributed by Thierry Arnoux, 6-Jun-2025.)
𝐵 = (Base‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ Domn)    &   (𝜑𝐹 ∈ Word (𝐵 ∖ { 0 }))       (𝜑 → (𝑀 Σg 𝐹) ≠ 0 )
 
Theoremidomrcan 33262 Right-cancellation law for integral domains. (Contributed by Thierry Arnoux, 22-Mar-2025.) (Proof shortened by SN, 21-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍 ∈ (𝐵 ∖ { 0 }))    &   (𝜑𝑅 ∈ IDomn)    &   (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍))       (𝜑𝑋 = 𝑌)
 
TheoremdomnlcanOLD 33263 Obsolete version of domnlcan 20737 as of 21-Jun-2025. (Contributed by Thierry Arnoux, 22-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑋 ∈ (𝐵 ∖ { 0 }))    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑅 ∈ Domn)    &   (𝜑 → (𝑋 · 𝑌) = (𝑋 · 𝑍))       (𝜑𝑌 = 𝑍)
 
TheoremdomnlcanbOLD 33264 Obsolete version of domnlcanb 20736 as of 21-Jun-2025. (Contributed by Thierry Arnoux, 8-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑋 ∈ (𝐵 ∖ { 0 }))    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑅 ∈ Domn)       (𝜑 → ((𝑋 · 𝑌) = (𝑋 · 𝑍) ↔ 𝑌 = 𝑍))
 
TheoremidomrcanOLD 33265 Obsolete version of idomrcan 33262 as of 21-Jun-2025. (Contributed by Thierry Arnoux, 22-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑋 ∈ (𝐵 ∖ { 0 }))    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑅 ∈ IDomn)    &   (𝜑 → (𝑌 · 𝑋) = (𝑍 · 𝑋))       (𝜑𝑌 = 𝑍)
 
Theorem1rrg 33266 The multiplicative identity is a left-regular element. (Contributed by Thierry Arnoux, 6-May-2025.)
1 = (1r𝑅)    &   𝐸 = (RLReg‘𝑅)    &   (𝜑𝑅 ∈ Ring)       (𝜑1𝐸)
 
Theoremrrgsubm 33267 The left regular elements of a ring form a submonoid of the multiplicative group. (Contributed by Thierry Arnoux, 10-May-2025.)
𝐸 = (RLReg‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   (𝜑𝑅 ∈ Ring)       (𝜑𝐸 ∈ (SubMnd‘𝑀))
 
Theoremsubrdom 33268 A subring of a domain is a domain. (Contributed by Thierry Arnoux, 18-May-2025.)
(𝜑𝑅 ∈ Domn)    &   (𝜑𝑆 ∈ (SubRing‘𝑅))       (𝜑 → (𝑅s 𝑆) ∈ Domn)
 
Theoremsubridom 33269 A subring of an integral domain is an integral domain. (Contributed by Thierry Arnoux, 18-May-2025.)
(𝜑𝑅 ∈ IDomn)    &   (𝜑𝑆 ∈ (SubRing‘𝑅))       (𝜑 → (𝑅s 𝑆) ∈ IDomn)
 
Theoremsubrfld 33270 A subring of a field is an integral domain. (Contributed by Thierry Arnoux, 18-May-2025.)
(𝜑𝑅 ∈ Field)    &   (𝜑𝑆 ∈ (SubRing‘𝑅))       (𝜑 → (𝑅s 𝑆) ∈ IDomn)
 
21.3.9.22  Euclidean Domains
 
Syntaxceuf 33271 Declare the syntax for the Euclidean function index extractor.
class EuclF
 
Definitiondf-euf 33272 Define the Euclidean function. (Contributed by Thierry Arnoux, 22-Mar-2025.) Use its index-independent form eufid 33274 instead. (New usage is discouraged.)
EuclF = Slot 21
 
Theoremeufndx 33273 Index value of the Euclidean function slot. Use ndxarg 17229. (Contributed by Thierry Arnoux, 22-Mar-2025.) (New usage is discouraged.)
(EuclF‘ndx) = 21
 
Theoremeufid 33274 Utility theorem: index-independent form of df-euf 33272. (Contributed by Thierry Arnoux, 22-Mar-2025.)
EuclF = Slot (EuclF‘ndx)
 
Syntaxcedom 33275 Declare the syntax for the Euclidean Domain.
class EDomn
 
Definitiondf-edom 33276* Define Euclidean Domains. (Contributed by Thierry Arnoux, 22-Mar-2025.)
EDomn = {𝑑 ∈ IDomn ∣ [(EuclF‘𝑑) / 𝑒][(Base‘𝑑) / 𝑣](Fun 𝑒 ∧ (𝑒 “ (𝑣 ∖ {(0g𝑑)})) ⊆ (0[,)+∞) ∧ ∀𝑎𝑣𝑏 ∈ (𝑣 ∖ {(0g𝑑)})∃𝑞𝑣𝑟𝑣 (𝑎 = ((𝑏(.r𝑑)𝑞)(+g𝑑)𝑟) ∧ (𝑟 = (0g𝑑) ∨ (𝑒𝑟) < (𝑒𝑏))))}
 
21.3.9.23  Division Rings
 
Theoremringinveu 33277 If a ring unit element 𝑋 admits both a left inverse 𝑌 and a right inverse 𝑍, they are equal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    · = (.r𝑅)    &   𝑈 = (Unit‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑 → (𝑌 · 𝑋) = 1 )    &   (𝜑 → (𝑋 · 𝑍) = 1 )       (𝜑𝑍 = 𝑌)
 
Theoremisdrng4 33278* A division ring is a ring in which 1 ≠ 0 and every nonzero element has a left and right inverse. (Contributed by Thierry Arnoux, 2-Mar-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    · = (.r𝑅)    &   𝑈 = (Unit‘𝑅)    &   (𝜑𝑅 ∈ Ring)       (𝜑 → (𝑅 ∈ DivRing ↔ ( 10 ∧ ∀𝑥 ∈ (𝐵 ∖ { 0 })∃𝑦𝐵 ((𝑥 · 𝑦) = 1 ∧ (𝑦 · 𝑥) = 1 ))))
 
Theoremrndrhmcl 33279 The image of a division ring by a ring homomorphism is a division ring. (Contributed by Thierry Arnoux, 25-Feb-2025.)
𝑅 = (𝑁s ran 𝐹)    &    0 = (0g𝑁)    &   (𝜑𝐹 ∈ (𝑀 RingHom 𝑁))    &   (𝜑 → ran 𝐹 ≠ { 0 })    &   (𝜑𝑀 ∈ DivRing)       (𝜑𝑅 ∈ DivRing)
 
21.3.9.24  Subfields
 
Theoremsdrgdvcl 33280 A sub-division-ring is closed under the ring division operation. (Contributed by Thierry Arnoux, 15-Jan-2025.)
/ = (/r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝐴 ∈ (SubDRing‘𝑅))    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   (𝜑𝑌0 )       (𝜑 → (𝑋 / 𝑌) ∈ 𝐴)
 
Theoremsdrginvcl 33281 A sub-division-ring is closed under the ring inverse operation. (Contributed by Thierry Arnoux, 15-Jan-2025.)
𝐼 = (invr𝑅)    &    0 = (0g𝑅)       ((𝐴 ∈ (SubDRing‘𝑅) ∧ 𝑋𝐴𝑋0 ) → (𝐼𝑋) ∈ 𝐴)
 
Theoremprimefldchr 33282 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))
 
21.3.9.25  Field of fractions
 
Syntaxcfrac 33283 Syntax for the field of fractions of a given integral domain.
class Frac
 
Definitiondf-frac 33284 Define the field of fractions of a given integral domain. (Contributed by Thierry Arnoux, 26-Apr-2025.)
Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟)))
 
Theoremfracval 33285 Value of the field of fractions. (Contributed by Thierry Arnoux, 5-May-2025.)
( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅))
 
Theoremfracbas 33286 The base of the field of fractions. (Contributed by Thierry Arnoux, 10-May-2025.)
𝐵 = (Base‘𝑅)    &   𝐸 = (RLReg‘𝑅)    &   𝐹 = ( Frac ‘𝑅)    &    = (𝑅 ~RL 𝐸)       ((𝐵 × 𝐸) / ) = (Base‘𝐹)
 
Theoremfracerl 33287 Rewrite the ring localization equivalence relation in the case of a field of fractions. (Contributed by Thierry Arnoux, 5-May-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    = (𝑅 ~RL (RLReg‘𝑅))    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐸𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐹 ∈ (RLReg‘𝑅))    &   (𝜑𝐻 ∈ (RLReg‘𝑅))       (𝜑 → (⟨𝐸, 𝐹𝐺, 𝐻⟩ ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹)))
 
Theoremfracf1 33288* The embedding of a commutative ring 𝑅 into its field of fractions. (Contributed by Thierry Arnoux, 10-May-2025.)
𝐵 = (Base‘𝑅)    &   𝐸 = (RLReg‘𝑅)    &    1 = (1r𝑅)    &   (𝜑𝑅 ∈ CRing)    &    = (𝑅 ~RL 𝐸)    &   𝐹 = (𝑥𝐵 ↦ [⟨𝑥, 1 ⟩] )       (𝜑 → (𝐹:𝐵1-1→((𝐵 × 𝐸) / ) ∧ 𝐹 ∈ (𝑅 RingHom ( Frac ‘𝑅))))
 
Theoremfracfld 33289 The field of fractions of an integral domain is a field. (Contributed by Thierry Arnoux, 4-May-2025.)
(𝜑𝑅 ∈ IDomn)       (𝜑 → ( Frac ‘𝑅) ∈ Field)
 
Theoremidomsubr 33290* Every integral domain is isomorphic with a subring of some field. (Proposed by Gerard Lang, 10-May-2025.) (Contributed by Thierry Arnoux, 10-May-2025.)
(𝜑𝑅 ∈ IDomn)       (𝜑 → ∃𝑓 ∈ Field ∃𝑠 ∈ (SubRing‘𝑓)𝑅𝑟 (𝑓s 𝑠))
 
21.3.9.26  Field extensions generated by a set
 
Syntaxcfldgen 33291 Syntax for a function generating sub-fields.
class fldGen
 
Definitiondf-fldgen 33292* Define a function generating the smallest sub-division-ring of a given ring containing a given set. If the base structure is a division ring, then this is also a division ring (see fldgensdrg 33295). If the base structure is a field, this is a subfield (see fldgenfld 33301 and fldsdrgfld 20815). In general this will be used in the context of fields, hence the name fldGen. (Contributed by Saveliy Skresanov and Thierry Arnoux, 9-Jan-2025.)
fldGen = (𝑓 ∈ V, 𝑠 ∈ V ↦ {𝑎 ∈ (SubDRing‘𝑓) ∣ 𝑠𝑎})
 
Theoremfldgenval 33293* Value of the field generating function: (𝐹 fldGen 𝑆) is the smallest sub-division-ring of 𝐹 containing 𝑆. (Contributed by Thierry Arnoux, 11-Jan-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑𝐹 ∈ DivRing)    &   (𝜑𝑆𝐵)       (𝜑 → (𝐹 fldGen 𝑆) = {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎})
 
Theoremfldgenssid 33294 The field generated by a set of elements contains those elements. See lspssid 21000. (Contributed by Thierry Arnoux, 15-Jan-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑𝐹 ∈ DivRing)    &   (𝜑𝑆𝐵)       (𝜑𝑆 ⊆ (𝐹 fldGen 𝑆))
 
Theoremfldgensdrg 33295 A generated subfield is a sub-division-ring. (Contributed by Thierry Arnoux, 11-Jan-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑𝐹 ∈ DivRing)    &   (𝜑𝑆𝐵)       (𝜑 → (𝐹 fldGen 𝑆) ∈ (SubDRing‘𝐹))
 
Theoremfldgenssv 33296 A generated subfield is a subset of the field's base. (Contributed by Thierry Arnoux, 25-Feb-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑𝐹 ∈ DivRing)    &   (𝜑𝑆𝐵)       (𝜑 → (𝐹 fldGen 𝑆) ⊆ 𝐵)
 
Theoremfldgenss 33297 Generated subfields preserve subset ordering. ( see lspss 20999 and spanss 31376) (Contributed by Thierry Arnoux, 15-Jan-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑𝐹 ∈ DivRing)    &   (𝜑𝑆𝐵)    &   (𝜑𝑇𝑆)       (𝜑 → (𝐹 fldGen 𝑇) ⊆ (𝐹 fldGen 𝑆))
 
Theoremfldgenidfld 33298 The subfield generated by a subfield is the subfield itself. (Contributed by Thierry Arnoux, 15-Jan-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑𝐹 ∈ DivRing)    &   (𝜑𝑆 ∈ (SubDRing‘𝐹))       (𝜑 → (𝐹 fldGen 𝑆) = 𝑆)
 
Theoremfldgenssp 33299 The field generated by a set of elements in a division ring is contained in any sub-division-ring which contains those elements. (Contributed by Thierry Arnoux, 25-Feb-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑𝐹 ∈ DivRing)    &   (𝜑𝑆 ∈ (SubDRing‘𝐹))    &   (𝜑𝑇𝑆)       (𝜑 → (𝐹 fldGen 𝑇) ⊆ 𝑆)
 
Theoremfldgenid 33300 The subfield of a field 𝐹 generated by the whole base set of 𝐹 is 𝐹 itself. (Contributed by Thierry Arnoux, 11-Jan-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑𝐹 ∈ DivRing)       (𝜑 → (𝐹 fldGen 𝐵) = 𝐵)
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