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Theorem List for Metamath Proof Explorer - 32901-33000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcyc3co2 32901 Represent a 3-cycle as a composition of two 2-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    &    Β· = (+gβ€˜π‘†)    β‡’   (πœ‘ β†’ (πΆβ€˜βŸ¨β€œπΌπ½πΎβ€βŸ©) = ((πΆβ€˜βŸ¨β€œπΌπΎβ€βŸ©) Β· (πΆβ€˜βŸ¨β€œπΌπ½β€βŸ©)))
 
Theoremcycpmconjvlem 32902 Lemma for cycpmconjv 32903. (Contributed by Thierry Arnoux, 9-Oct-2023.)
(πœ‘ β†’ 𝐹:𝐷–1-1-onto→𝐷)    &   (πœ‘ β†’ 𝐡 βŠ† 𝐷)    β‡’   (πœ‘ β†’ ((𝐹 β†Ύ (𝐷 βˆ– 𝐡)) ∘ ◑𝐹) = ( I β†Ύ (𝐷 βˆ– ran (𝐹 β†Ύ 𝐡))))
 
Theoremcycpmconjv 32903 A formula for computing conjugacy classes of cyclic permutations. Formula in property (b) of [Lang] p. 32. (Contributed by Thierry Arnoux, 9-Oct-2023.)
𝑆 = (SymGrpβ€˜π·)    &   π‘€ = (toCycβ€˜π·)    &    + = (+gβ€˜π‘†)    &    βˆ’ = (-gβ€˜π‘†)    &   π΅ = (Baseβ€˜π‘†)    β‡’   ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐡 ∧ π‘Š ∈ dom 𝑀) β†’ ((𝐺 + (π‘€β€˜π‘Š)) βˆ’ 𝐺) = (π‘€β€˜(𝐺 ∘ π‘Š)))
 
Theoremcycpmrn 32904 The range of the word used to build a cycle is the cycle's orbit, i.e., the set of points it moves. (Contributed by Thierry Arnoux, 20-Nov-2023.)
𝑀 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   (πœ‘ β†’ 1 < (β™―β€˜π‘Š))    β‡’   (πœ‘ β†’ ran π‘Š = dom ((π‘€β€˜π‘Š) βˆ– I ))
 
Theoremtocyccntz 32905* All elements of a (finite) set of cycles commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 27-Nov-2023.)
𝑆 = (SymGrpβ€˜π·)    &   π‘ = (Cntzβ€˜π‘†)    &   π‘€ = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ Disj π‘₯ ∈ 𝐴 ran π‘₯)    &   (πœ‘ β†’ 𝐴 βŠ† dom 𝑀)    β‡’   (πœ‘ β†’ (𝑀 β€œ 𝐴) βŠ† (π‘β€˜(𝑀 β€œ 𝐴)))
 
21.3.9.10  The Alternating Group
 
Theoremevpmval 32906 Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝐴 = (pmEvenβ€˜π·)    β‡’   (𝐷 ∈ 𝑉 β†’ 𝐴 = (β—‘(pmSgnβ€˜π·) β€œ {1}))
 
Theoremcnmsgn0g 32907 The neutral element of the sign subgroup of the complex numbers. (Contributed by Thierry Arnoux, 1-Nov-2023.)
π‘ˆ = ((mulGrpβ€˜β„‚fld) β†Ύs {1, -1})    β‡’   1 = (0gβ€˜π‘ˆ)
 
Theoremevpmsubg 32908 The alternating group is a subgroup of the symmetric group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝑆 = (SymGrpβ€˜π·)    &   π΄ = (pmEvenβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ 𝐴 ∈ (SubGrpβ€˜π‘†))
 
Theoremevpmid 32909 The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝑆 = (SymGrpβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ ( I β†Ύ 𝐷) ∈ (pmEvenβ€˜π·))
 
Theoremaltgnsg 32910 The alternating group (pmEvenβ€˜π·) is a normal subgroup of the symmetric group. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝑆 = (SymGrpβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ (pmEvenβ€˜π·) ∈ (NrmSGrpβ€˜π‘†))
 
Theoremcyc3evpm 32911 3-Cycles are even permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = ((toCycβ€˜π·) β€œ (β—‘β™― β€œ {3}))    &   π΄ = (pmEvenβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ 𝐢 βŠ† 𝐴)
 
Theoremcyc3genpmlem 32912* Lemma for cyc3genpm 32913. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {3}))    &   π΄ = (pmEvenβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &    Β· = (+gβ€˜π‘†)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐿 ∈ 𝐷)    &   (πœ‘ β†’ 𝐸 = (π‘€β€˜βŸ¨β€œπΌπ½β€βŸ©))    &   (πœ‘ β†’ 𝐹 = (π‘€β€˜βŸ¨β€œπΎπΏβ€βŸ©))    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐾 β‰  𝐿)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ Word 𝐢(𝐸 Β· 𝐹) = (𝑆 Ξ£g 𝑐))
 
Theoremcyc3genpm 32913* The alternating group 𝐴 is generated by 3-cycles. Property (a) of [Lang] p. 32 . (Contributed by Thierry Arnoux, 27-Sep-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {3}))    &   π΄ = (pmEvenβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ (𝑄 ∈ 𝐴 ↔ βˆƒπ‘€ ∈ Word 𝐢𝑄 = (𝑆 Ξ£g 𝑀)))
 
Theoremcycpmgcl 32914 Cyclic permutations are permutations, similar to cycpmcl 32877, but where the set of cyclic permutations of length 𝑃 is expressed in terms of a preimage. (Contributed by Thierry Arnoux, 13-Oct-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {𝑃}))    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    β‡’   ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ (0...𝑁)) β†’ 𝐢 βŠ† 𝐡)
 
Theoremcycpmconjslem1 32915 Lemma for cycpmconjs 32917. (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {𝑃}))    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   (πœ‘ β†’ (β™―β€˜π‘Š) = 𝑃)    β‡’   (πœ‘ β†’ ((β—‘π‘Š ∘ (π‘€β€˜π‘Š)) ∘ π‘Š) = (( I β†Ύ (0..^𝑃)) cyclShift 1))
 
Theoremcycpmconjslem2 32916* Lemma for cycpmconjs 32917. (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {𝑃}))    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    &    + = (+gβ€˜π‘†)    &    βˆ’ = (-gβ€˜π‘†)    &   (πœ‘ β†’ 𝑃 ∈ (0...𝑁))    &   (πœ‘ β†’ 𝐷 ∈ Fin)    &   (πœ‘ β†’ 𝑄 ∈ 𝐢)    β‡’   (πœ‘ β†’ βˆƒπ‘ž(π‘ž:(0..^𝑁)–1-1-onto→𝐷 ∧ ((β—‘π‘ž ∘ 𝑄) ∘ π‘ž) = ((( I β†Ύ (0..^𝑃)) cyclShift 1) βˆͺ ( I β†Ύ (𝑃..^𝑁)))))
 
Theoremcycpmconjs 32917* All cycles of the same length are conjugate in the symmetric group. (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {𝑃}))    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    &    + = (+gβ€˜π‘†)    &    βˆ’ = (-gβ€˜π‘†)    &   (πœ‘ β†’ 𝑃 ∈ (0...𝑁))    &   (πœ‘ β†’ 𝐷 ∈ Fin)    &   (πœ‘ β†’ 𝑄 ∈ 𝐢)    &   (πœ‘ β†’ 𝑇 ∈ 𝐢)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ 𝐡 𝑄 = ((𝑝 + 𝑇) βˆ’ 𝑝))
 
Theoremcyc3conja 32918* All 3-cycles are conjugate in the alternating group An for n>= 5. Property (b) of [Lang] p. 32. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {3}))    &   π΄ = (pmEvenβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &    + = (+gβ€˜π‘†)    &    βˆ’ = (-gβ€˜π‘†)    &   (πœ‘ β†’ 5 ≀ 𝑁)    &   (πœ‘ β†’ 𝐷 ∈ Fin)    &   (πœ‘ β†’ 𝑄 ∈ 𝐢)    &   (πœ‘ β†’ 𝑇 ∈ 𝐢)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ 𝐴 𝑄 = ((𝑝 + 𝑇) βˆ’ 𝑝))
 
21.3.9.11  Signum in an ordered monoid
 
Syntaxcsgns 32919 Extend class notation to include the Signum function.
class sgns
 
Definitiondf-sgns 32920* Signum function for a structure. See also df-sgn 15061 for the version for extended reals. (Contributed by Thierry Arnoux, 10-Sep-2018.)
sgns = (π‘Ÿ ∈ V ↦ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ if(π‘₯ = (0gβ€˜π‘Ÿ), 0, if((0gβ€˜π‘Ÿ)(ltβ€˜π‘Ÿ)π‘₯, 1, -1))))
 
Theoremsgnsv 32921* The sign mapping. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    < = (ltβ€˜π‘…)    &   π‘† = (sgnsβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ 𝑆 = (π‘₯ ∈ 𝐡 ↦ if(π‘₯ = 0 , 0, if( 0 < π‘₯, 1, -1))))
 
Theoremsgnsval 32922 The sign value. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    < = (ltβ€˜π‘…)    &   π‘† = (sgnsβ€˜π‘…)    β‡’   ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐡) β†’ (π‘†β€˜π‘‹) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
 
Theoremsgnsf 32923 The sign function. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    < = (ltβ€˜π‘…)    &   π‘† = (sgnsβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ 𝑆:𝐡⟢{-1, 0, 1})
 
21.3.9.12  The Archimedean property for generic ordered algebraic structures
 
Syntaxcinftm 32924 Class notation for the infinitesimal relation.
class β‹˜
 
Syntaxcarchi 32925 Class notation for the Archimedean property.
class Archi
 
Definitiondf-inftm 32926* Define the relation "π‘₯ is infinitesimal with respect to 𝑦 " for a structure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-2018.)
β‹˜ = (𝑀 ∈ V ↦ {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜π‘€) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) ∧ ((0gβ€˜π‘€)(ltβ€˜π‘€)π‘₯ ∧ βˆ€π‘› ∈ β„• (𝑛(.gβ€˜π‘€)π‘₯)(ltβ€˜π‘€)𝑦))})
 
Definitiondf-archi 32927 A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Archi = {𝑀 ∣ (β‹˜β€˜π‘€) = βˆ…}
 
Theoreminftmrel 32928 The infinitesimal relation for a structure π‘Š. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    β‡’   (π‘Š ∈ 𝑉 β†’ (β‹˜β€˜π‘Š) βŠ† (𝐡 Γ— 𝐡))
 
Theoremisinftm 32929* Express π‘₯ is infinitesimal with respect to 𝑦 for a structure π‘Š. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    β‡’   ((π‘Š ∈ 𝑉 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋(β‹˜β€˜π‘Š)π‘Œ ↔ ( 0 < 𝑋 ∧ βˆ€π‘› ∈ β„• (𝑛 Β· 𝑋) < π‘Œ)))
 
Theoremisarchi 32930* Express the predicate "π‘Š is Archimedean ". (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    < = (β‹˜β€˜π‘Š)    β‡’   (π‘Š ∈ 𝑉 β†’ (π‘Š ∈ Archi ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ π‘₯ < 𝑦))
 
Theorempnfinf 32931 Plus infinity is an infinite for the completed real line, as any real number is infinitesimal compared to it. (Contributed by Thierry Arnoux, 1-Feb-2018.)
(𝐴 ∈ ℝ+ β†’ 𝐴(β‹˜β€˜β„*𝑠)+∞)
 
Theoremxrnarchi 32932 The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.)
Β¬ ℝ*𝑠 ∈ Archi
 
Theoremisarchi2 32933* Alternative way to express the predicate "π‘Š is Archimedean ", for Tosets. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    β‡’   ((π‘Š ∈ Toset ∧ π‘Š ∈ Mnd) β†’ (π‘Š ∈ Archi ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ( 0 < π‘₯ β†’ βˆƒπ‘› ∈ β„• 𝑦 ≀ (𝑛 Β· π‘₯))))
 
Theoremsubmarchi 32934 A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
(((π‘Š ∈ Toset ∧ π‘Š ∈ Archi) ∧ 𝐴 ∈ (SubMndβ€˜π‘Š)) β†’ (π‘Š β†Ύs 𝐴) ∈ Archi)
 
Theoremisarchi3 32935* This is the usual definition of the Archimedean property for an ordered group. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    β‡’   (π‘Š ∈ oGrp β†’ (π‘Š ∈ Archi ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ( 0 < π‘₯ β†’ βˆƒπ‘› ∈ β„• 𝑦 < (𝑛 Β· π‘₯))))
 
Theoremarchirng 32936* Property of Archimedean ordered groups, framing positive π‘Œ between multiples of 𝑋. (Contributed by Thierry Arnoux, 12-Apr-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 0 < 𝑋)    &   (πœ‘ β†’ 0 < π‘Œ)    β‡’   (πœ‘ β†’ βˆƒπ‘› ∈ β„•0 ((𝑛 Β· 𝑋) < π‘Œ ∧ π‘Œ ≀ ((𝑛 + 1) Β· 𝑋)))
 
Theoremarchirngz 32937* Property of Archimedean left and right ordered groups. (Contributed by Thierry Arnoux, 6-May-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 0 < 𝑋)    &   (πœ‘ β†’ (oppgβ€˜π‘Š) ∈ oGrp)    β‡’   (πœ‘ β†’ βˆƒπ‘› ∈ β„€ ((𝑛 Β· 𝑋) < π‘Œ ∧ π‘Œ ≀ ((𝑛 + 1) Β· 𝑋)))
 
Theoremarchiexdiv 32938* In an Archimedean group, given two positive elements, there exists a "divisor" 𝑛. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    β‡’   (((π‘Š ∈ oGrp ∧ π‘Š ∈ Archi) ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 0 < 𝑋) β†’ βˆƒπ‘› ∈ β„• π‘Œ < (𝑛 Β· 𝑋))
 
Theoremarchiabllem1a 32939* Lemma for archiabl 32946: In case an archimedean group π‘Š admits a smallest positive element π‘ˆ, then any positive element 𝑋 of π‘Š can be written as (𝑛 Β· π‘ˆ) with 𝑛 ∈ β„•. Since the reciprocal holds for negative elements, π‘Š is then isomorphic to β„€. (Contributed by Thierry Arnoux, 12-Apr-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &   (πœ‘ β†’ π‘ˆ ∈ 𝐡)    &   (πœ‘ β†’ 0 < π‘ˆ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 0 < π‘₯) β†’ π‘ˆ ≀ π‘₯)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ 0 < 𝑋)    β‡’   (πœ‘ β†’ βˆƒπ‘› ∈ β„• 𝑋 = (𝑛 Β· π‘ˆ))
 
Theoremarchiabllem1b 32940* Lemma for archiabl 32946. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &   (πœ‘ β†’ π‘ˆ ∈ 𝐡)    &   (πœ‘ β†’ 0 < π‘ˆ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 0 < π‘₯) β†’ π‘ˆ ≀ π‘₯)    β‡’   ((πœ‘ ∧ 𝑦 ∈ 𝐡) β†’ βˆƒπ‘› ∈ β„€ 𝑦 = (𝑛 Β· π‘ˆ))
 
Theoremarchiabllem1 32941* Archimedean ordered groups with a minimal positive value are abelian. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &   (πœ‘ β†’ π‘ˆ ∈ 𝐡)    &   (πœ‘ β†’ 0 < π‘ˆ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 0 < π‘₯) β†’ π‘ˆ ≀ π‘₯)    β‡’   (πœ‘ β†’ π‘Š ∈ Abel)
 
Theoremarchiabllem2a 32942* Lemma for archiabl 32946, which requires the group to be both left- and right-ordered. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &    + = (+gβ€˜π‘Š)    &   (πœ‘ β†’ (oppgβ€˜π‘Š) ∈ oGrp)    &   ((πœ‘ ∧ π‘Ž ∈ 𝐡 ∧ 0 < π‘Ž) β†’ βˆƒπ‘ ∈ 𝐡 ( 0 < 𝑏 ∧ 𝑏 < π‘Ž))    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ 0 < 𝑋)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ 𝐡 ( 0 < 𝑐 ∧ (𝑐 + 𝑐) ≀ 𝑋))
 
Theoremarchiabllem2c 32943* Lemma for archiabl 32946. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &    + = (+gβ€˜π‘Š)    &   (πœ‘ β†’ (oppgβ€˜π‘Š) ∈ oGrp)    &   ((πœ‘ ∧ π‘Ž ∈ 𝐡 ∧ 0 < π‘Ž) β†’ βˆƒπ‘ ∈ 𝐡 ( 0 < 𝑏 ∧ 𝑏 < π‘Ž))    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ Β¬ (𝑋 + π‘Œ) < (π‘Œ + 𝑋))
 
Theoremarchiabllem2b 32944* Lemma for archiabl 32946. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &    + = (+gβ€˜π‘Š)    &   (πœ‘ β†’ (oppgβ€˜π‘Š) ∈ oGrp)    &   ((πœ‘ ∧ π‘Ž ∈ 𝐡 ∧ 0 < π‘Ž) β†’ βˆƒπ‘ ∈ 𝐡 ( 0 < 𝑏 ∧ 𝑏 < π‘Ž))    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) = (π‘Œ + 𝑋))
 
Theoremarchiabllem2 32945* Archimedean ordered groups with no minimal positive value are abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &    + = (+gβ€˜π‘Š)    &   (πœ‘ β†’ (oppgβ€˜π‘Š) ∈ oGrp)    &   ((πœ‘ ∧ π‘Ž ∈ 𝐡 ∧ 0 < π‘Ž) β†’ βˆƒπ‘ ∈ 𝐡 ( 0 < 𝑏 ∧ 𝑏 < π‘Ž))    β‡’   (πœ‘ β†’ π‘Š ∈ Abel)
 
Theoremarchiabl 32946 Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
((π‘Š ∈ oGrp ∧ (oppgβ€˜π‘Š) ∈ oGrp ∧ π‘Š ∈ Archi) β†’ π‘Š ∈ Abel)
 
21.3.9.13  Semiring left modules
 
Syntaxcslmd 32947 Extend class notation with class of all semimodules.
class SLMod
 
Definitiondf-slmd 32948* 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 ∣ [(Baseβ€˜π‘”) / 𝑣][(+gβ€˜π‘”) / π‘Ž][( ·𝑠 β€˜π‘”) / 𝑠][(Scalarβ€˜π‘”) / 𝑓][(Baseβ€˜π‘“) / π‘˜][(+gβ€˜π‘“) / 𝑝][(.rβ€˜π‘“) / 𝑑](𝑓 ∈ SRing ∧ βˆ€π‘ž ∈ π‘˜ βˆ€π‘Ÿ ∈ π‘˜ βˆ€π‘₯ ∈ 𝑣 βˆ€π‘€ ∈ 𝑣 (((π‘Ÿπ‘ π‘€) ∈ 𝑣 ∧ (π‘Ÿπ‘ (π‘€π‘Žπ‘₯)) = ((π‘Ÿπ‘ π‘€)π‘Ž(π‘Ÿπ‘ π‘₯)) ∧ ((π‘žπ‘π‘Ÿ)𝑠𝑀) = ((π‘žπ‘ π‘€)π‘Ž(π‘Ÿπ‘ π‘€))) ∧ (((π‘žπ‘‘π‘Ÿ)𝑠𝑀) = (π‘žπ‘ (π‘Ÿπ‘ π‘€)) ∧ ((1rβ€˜π‘“)𝑠𝑀) = 𝑀 ∧ ((0gβ€˜π‘“)𝑠𝑀) = (0gβ€˜π‘”))))}
 
Theoremisslmd 32949* 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 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    ⨣ = (+gβ€˜πΉ)    &    Γ— = (.rβ€˜πΉ)    &    1 = (1rβ€˜πΉ)    &   π‘‚ = (0gβ€˜πΉ)    β‡’   (π‘Š ∈ SLMod ↔ (π‘Š ∈ CMnd ∧ 𝐹 ∈ SRing ∧ βˆ€π‘ž ∈ 𝐾 βˆ€π‘Ÿ ∈ 𝐾 βˆ€π‘₯ ∈ 𝑉 βˆ€π‘€ ∈ 𝑉 (((π‘Ÿ Β· 𝑀) ∈ 𝑉 ∧ (π‘Ÿ Β· (𝑀 + π‘₯)) = ((π‘Ÿ Β· 𝑀) + (π‘Ÿ Β· π‘₯)) ∧ ((π‘ž ⨣ π‘Ÿ) Β· 𝑀) = ((π‘ž Β· 𝑀) + (π‘Ÿ Β· 𝑀))) ∧ (((π‘ž Γ— π‘Ÿ) Β· 𝑀) = (π‘ž Β· (π‘Ÿ Β· 𝑀)) ∧ ( 1 Β· 𝑀) = 𝑀 ∧ (𝑂 Β· 𝑀) = 0 ))))
 
Theoremslmdlema 32950 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 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    ⨣ = (+gβ€˜πΉ)    &    Γ— = (.rβ€˜πΉ)    &    1 = (1rβ€˜πΉ)    &   π‘‚ = (0gβ€˜πΉ)    β‡’   ((π‘Š ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (((𝑅 Β· π‘Œ) ∈ 𝑉 ∧ (𝑅 Β· (π‘Œ + 𝑋)) = ((𝑅 Β· π‘Œ) + (𝑅 Β· 𝑋)) ∧ ((𝑄 ⨣ 𝑅) Β· π‘Œ) = ((𝑄 Β· π‘Œ) + (𝑅 Β· π‘Œ))) ∧ (((𝑄 Γ— 𝑅) Β· π‘Œ) = (𝑄 Β· (𝑅 Β· π‘Œ)) ∧ ( 1 Β· π‘Œ) = π‘Œ ∧ (𝑂 Β· π‘Œ) = 0 )))
 
Theoremlmodslmd 32951 Left semimodules generalize the notion of left modules. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(π‘Š ∈ LMod β†’ π‘Š ∈ SLMod)
 
Theoremslmdcmn 32952 A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(π‘Š ∈ SLMod β†’ π‘Š ∈ CMnd)
 
Theoremslmdmnd 32953 A semimodule is a monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(π‘Š ∈ SLMod β†’ π‘Š ∈ Mnd)
 
Theoremslmdsrg 32954 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 = (Scalarβ€˜π‘Š)    β‡’   (π‘Š ∈ SLMod β†’ 𝐹 ∈ SRing)
 
Theoremslmdbn0 32955 The base set of a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.) (Proof shortened by AV, 10-Jan-2023.)
𝐡 = (Baseβ€˜π‘Š)    β‡’   (π‘Š ∈ SLMod β†’ 𝐡 β‰  βˆ…)
 
Theoremslmdacl 32956 Closure of ring addition for a semimodule. (Contributed by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    + = (+gβ€˜πΉ)    β‡’   ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ π‘Œ ∈ 𝐾) β†’ (𝑋 + π‘Œ) ∈ 𝐾)
 
Theoremslmdmcl 32957 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 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    Β· = (.rβ€˜πΉ)    β‡’   ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ π‘Œ ∈ 𝐾) β†’ (𝑋 Β· π‘Œ) ∈ 𝐾)
 
Theoremslmdsn0 32958 The set of scalars in a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.) (Proof shortened by AV, 10-Jan-2023.)
𝐹 = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜πΉ)    β‡’   (π‘Š ∈ SLMod β†’ 𝐡 β‰  βˆ…)
 
Theoremslmdvacl 32959 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 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    β‡’   ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 + π‘Œ) ∈ 𝑉)
 
Theoremslmdass 32960 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 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    β‡’   ((π‘Š ∈ SLMod ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) β†’ ((𝑋 + π‘Œ) + 𝑍) = (𝑋 + (π‘Œ + 𝑍)))
 
Theoremslmdvscl 32961 Closure of scalar product for a semiring left module. (hvmulcl 30862 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ SLMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (𝑅 Β· 𝑋) ∈ 𝑉)
 
Theoremslmdvsdi 32962 Distributive law for scalar product. (ax-hvdistr1 30857 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ)))
 
Theoremslmdvsdir 32963 Distributive law for scalar product. (ax-hvdistr1 30857 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    ⨣ = (+gβ€˜πΉ)    β‡’   ((π‘Š ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑄 ⨣ 𝑅) Β· 𝑋) = ((𝑄 Β· 𝑋) + (𝑅 Β· 𝑋)))
 
Theoremslmdvsass 32964 Associative law for scalar product. (ax-hvmulass 30856 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    Γ— = (.rβ€˜πΉ)    β‡’   ((π‘Š ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑄 Γ— 𝑅) Β· 𝑋) = (𝑄 Β· (𝑅 Β· 𝑋)))
 
Theoremslmd0cl 32965 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 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    0 = (0gβ€˜πΉ)    β‡’   (π‘Š ∈ SLMod β†’ 0 ∈ 𝐾)
 
Theoremslmd1cl 32966 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 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    1 = (1rβ€˜πΉ)    β‡’   (π‘Š ∈ SLMod β†’ 1 ∈ 𝐾)
 
Theoremslmdvs1 32967 Scalar product with ring unity. (ax-hvmulid 30855 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &    1 = (1rβ€˜πΉ)    β‡’   ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝑉) β†’ ( 1 Β· 𝑋) = 𝑋)
 
Theoremslmd0vcl 32968 The zero vector is a vector. (ax-hv0cl 30852 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    β‡’   (π‘Š ∈ SLMod β†’ 0 ∈ 𝑉)
 
Theoremslmd0vlid 32969 Left identity law for the zero vector. (hvaddlid 30872 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    β‡’   ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝑉) β†’ ( 0 + 𝑋) = 𝑋)
 
Theoremslmd0vrid 32970 Right identity law for the zero vector. (ax-hvaddid 30853 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    β‡’   ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝑉) β†’ (𝑋 + 0 ) = 𝑋)
 
Theoremslmd0vs 32971 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 30859 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘‚ = (0gβ€˜πΉ)    &    0 = (0gβ€˜π‘Š)    β‡’   ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝑉) β†’ (𝑂 Β· 𝑋) = 0 )
 
Theoremslmdvs0 32972 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 30873 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    0 = (0gβ€˜π‘Š)    β‡’   ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾) β†’ (𝑋 Β· 0 ) = 0 )
 
Theoremgsumvsca1 32973* Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.)
𝐡 = (Baseβ€˜π‘Š)    &   πΊ = (Scalarβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &   (πœ‘ β†’ 𝐾 βŠ† (Baseβ€˜πΊ))    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ π‘Š ∈ SLMod)    &   (πœ‘ β†’ 𝑃 ∈ 𝐾)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝑄 ∈ 𝐡)    β‡’   (πœ‘ β†’ (π‘Š Ξ£g (π‘˜ ∈ 𝐴 ↦ (𝑃 Β· 𝑄))) = (𝑃 Β· (π‘Š Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑄))))
 
Theoremgsumvsca2 32974* 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.)
𝐡 = (Baseβ€˜π‘Š)    &   πΊ = (Scalarβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &   (πœ‘ β†’ 𝐾 βŠ† (Baseβ€˜πΊ))    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ π‘Š ∈ SLMod)    &   (πœ‘ β†’ 𝑄 ∈ 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝑃 ∈ 𝐾)    β‡’   (πœ‘ β†’ (π‘Š Ξ£g (π‘˜ ∈ 𝐴 ↦ (𝑃 Β· 𝑄))) = ((𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑃)) Β· 𝑄))
 
21.3.9.14  Simple groups
 
Theoremprmsimpcyc 32975 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.)
𝐡 = (Baseβ€˜πΊ)    β‡’   ((β™―β€˜π΅) ∈ β„™ β†’ (𝐺 ∈ SimpGrp ↔ 𝐺 ∈ CycGrp))
 
21.3.9.15  Rings - misc additions
 
Theoremcringmul32d 32976 Commutative/associative law that swaps the last two factors in a triple product in a commutative ring. See also mul32 11405. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    β‡’   (πœ‘ β†’ ((𝑋 Β· π‘Œ) Β· 𝑍) = ((𝑋 Β· 𝑍) Β· π‘Œ))
 
Theoremringdid 32977 Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Thierry Arnoux, 4-May-2025.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 Β· (π‘Œ + 𝑍)) = ((𝑋 Β· π‘Œ) + (𝑋 Β· 𝑍)))
 
Theoremringdird 32978 Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Thierry Arnoux, 4-May-2025.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    β‡’   (πœ‘ β†’ ((𝑋 + π‘Œ) Β· 𝑍) = ((𝑋 Β· 𝑍) + (π‘Œ Β· 𝑍)))
 
Theoremurpropd 32979* Sufficient condition for ring unities to be equal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝐡 = (Baseβ€˜π‘†)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝑇 ∈ π‘Š)    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘‡))    &   (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯(.rβ€˜π‘†)𝑦) = (π‘₯(.rβ€˜π‘‡)𝑦))    β‡’   (πœ‘ β†’ (1rβ€˜π‘†) = (1rβ€˜π‘‡))
 
Theoremfrobrhm 32980* 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.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ƒ = (chrβ€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘…))    &   πΉ = (π‘₯ ∈ 𝐡 ↦ (𝑃 ↑ π‘₯))    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑃 ∈ β„™)    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝑅 RingHom 𝑅))
 
Theoremress1r 32981 1r is unaffected by restriction. This is a bit more generic than subrg1 20520. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑆 = (𝑅 β†Ύs 𝐴)    &   π΅ = (Baseβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 βŠ† 𝐡) β†’ 1 = (1rβ€˜π‘†))
 
Theoremringinvval 32982* The ring inverse expressed in terms of multiplication. (Contributed by Thierry Arnoux, 23-Oct-2017.)
𝐡 = (Baseβ€˜π‘…)    &    βˆ— = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   π‘ = (invrβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ π‘ˆ) β†’ (π‘β€˜π‘‹) = (℩𝑦 ∈ π‘ˆ (𝑦 βˆ— 𝑋) = 1 ))
 
Theoremdvrcan5 32983 Cancellation law for common factor in ratio. (divcan5 11941 analog.) (Contributed by Thierry Arnoux, 26-Oct-2016.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &    / = (/rβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ π‘ˆ ∧ 𝑍 ∈ π‘ˆ)) β†’ ((𝑋 Β· 𝑍) / (π‘Œ Β· 𝑍)) = (𝑋 / π‘Œ))
 
Theoremsubrgchr 32984 If 𝐴 is a subring of 𝑅, then they have the same characteristic. (Contributed by Thierry Arnoux, 24-Feb-2018.)
(𝐴 ∈ (SubRingβ€˜π‘…) β†’ (chrβ€˜(𝑅 β†Ύs 𝐴)) = (chrβ€˜π‘…))
 
Theoremrmfsupp2 32985* 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 = (Baseβ€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ Ring)    &   (πœ‘ β†’ 𝑉 ∈ 𝑋)    &   ((πœ‘ ∧ 𝑣 ∈ 𝑉) β†’ 𝐢 ∈ 𝑅)    &   (πœ‘ β†’ 𝐴:π‘‰βŸΆπ‘…)    &   (πœ‘ β†’ 𝐴 finSupp (0gβ€˜π‘€))    β‡’   (πœ‘ β†’ (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)(.rβ€˜π‘€)𝐢)) finSupp (0gβ€˜π‘€))
 
Theoremunitnz 32986 In a nonzero ring, a unit cannot be zero. (Contributed by Thierry Arnoux, 25-Apr-2025.)
π‘ˆ = (Unitβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ NzRing)    &   (πœ‘ β†’ 𝑋 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ 𝑋 β‰  0 )
 
21.3.9.16  The zero ring
 
Theoremirrednzr 32987 A ring with an irreducible element cannot be the zero ring. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐼 = (Irredβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐼)    β‡’   (πœ‘ β†’ 𝑅 ∈ NzRing)
 
Theorem0ringsubrg 32988 A subring of a zero ring is a zero ring. (Contributed by Thierry Arnoux, 5-Feb-2025.)
𝐡 = (Baseβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ (β™―β€˜π΅) = 1)    &   (πœ‘ β†’ 𝑆 ∈ (SubRingβ€˜π‘…))    β‡’   (πœ‘ β†’ (β™―β€˜π‘†) = 1)
 
Theorem0ringcring 32989 The zero ring is commutative. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐡 = (Baseβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ (β™―β€˜π΅) = 1)    β‡’   (πœ‘ β†’ 𝑅 ∈ CRing)
 
21.3.9.17  Localization of rings
 
Syntaxcerl 32990 Syntax for ring localization equivalence class operation.
class ~RL
 
Syntaxcrloc 32991 Syntax for ring localization operation.
class RLocal
 
Definitiondf-erl 32992* Define the operation giving the equivalence relation used in the localization of a ring π‘Ÿ by a set 𝑠. Two pairs π‘Ž = ⟨π‘₯, π‘¦βŸ© and 𝑏 = βŸ¨π‘§, π‘€βŸ© are equivalent if there exists 𝑑 ∈ 𝑠 such that 𝑑 Β· (π‘₯ Β· 𝑀 βˆ’ 𝑧 Β· 𝑦) = 0. This corresponds to the usual comparison of fractions π‘₯ / 𝑦 and 𝑧 / 𝑀. (Contributed by Thierry Arnoux, 28-Apr-2025.)
~RL = (π‘Ÿ ∈ V, 𝑠 ∈ V ↦ ⦋(.rβ€˜π‘Ÿ) / π‘₯β¦Œβ¦‹((Baseβ€˜π‘Ÿ) Γ— 𝑠) / π‘€β¦Œ{βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ 𝑀 ∧ 𝑏 ∈ 𝑀) ∧ βˆƒπ‘‘ ∈ 𝑠 (𝑑π‘₯(((1st β€˜π‘Ž)π‘₯(2nd β€˜π‘))(-gβ€˜π‘Ÿ)((1st β€˜π‘)π‘₯(2nd β€˜π‘Ž)))) = (0gβ€˜π‘Ÿ))})
 
Definitiondf-rloc 32993* Define the operation giving the localization of a ring π‘Ÿ by a given set 𝑠. The localized ring π‘Ÿ RLocal 𝑠 is the set of equivalence classes of pairs of elements in π‘Ÿ over the relation π‘Ÿ ~RL 𝑠 with addition and multiplication defined naturally. (Contributed by Thierry Arnoux, 27-Apr-2025.)
RLocal = (π‘Ÿ ∈ V, 𝑠 ∈ V ↦ ⦋(.rβ€˜π‘Ÿ) / π‘₯β¦Œβ¦‹((Baseβ€˜π‘Ÿ) Γ— 𝑠) / π‘€β¦Œ((({⟨(Baseβ€˜ndx), π‘€βŸ©, ⟨(+gβ€˜ndx), (π‘Ž ∈ 𝑀, 𝑏 ∈ 𝑀 ↦ ⟨(((1st β€˜π‘Ž)π‘₯(2nd β€˜π‘))(+gβ€˜π‘Ÿ)((1st β€˜π‘)π‘₯(2nd β€˜π‘Ž))), ((2nd β€˜π‘Ž)π‘₯(2nd β€˜π‘))⟩)⟩, ⟨(.rβ€˜ndx), (π‘Ž ∈ 𝑀, 𝑏 ∈ 𝑀 ↦ ⟨((1st β€˜π‘Ž)π‘₯(1st β€˜π‘)), ((2nd β€˜π‘Ž)π‘₯(2nd β€˜π‘))⟩)⟩} βˆͺ {⟨(Scalarβ€˜ndx), (Scalarβ€˜π‘Ÿ)⟩, ⟨( ·𝑠 β€˜ndx), (π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Ÿ)), π‘Ž ∈ 𝑀 ↦ ⟨(π‘˜( ·𝑠 β€˜π‘Ÿ)(1st β€˜π‘Ž)), (2nd β€˜π‘Ž)⟩)⟩, ⟨(Β·π‘–β€˜ndx), βˆ…βŸ©}) βˆͺ {⟨(TopSetβ€˜ndx), ((TopSetβ€˜π‘Ÿ) Γ—t ((TopSetβ€˜π‘Ÿ) β†Ύt 𝑠))⟩, ⟨(leβ€˜ndx), {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ 𝑀 ∧ 𝑏 ∈ 𝑀) ∧ ((1st β€˜π‘Ž)π‘₯(2nd β€˜π‘))(leβ€˜π‘Ÿ)((1st β€˜π‘)π‘₯(2nd β€˜π‘Ž)))}⟩, ⟨(distβ€˜ndx), (π‘Ž ∈ 𝑀, 𝑏 ∈ 𝑀 ↦ (((1st β€˜π‘Ž)π‘₯(2nd β€˜π‘))(distβ€˜π‘Ÿ)((1st β€˜π‘)π‘₯(2nd β€˜π‘Ž))))⟩}) /s (π‘Ÿ ~RL 𝑠)))
 
Theoremreldmrloc 32994 Ring localization is a proper operator, so it can be used with ovprc1 7452. (Contributed by Thierry Arnoux, 10-May-2025.)
Rel dom RLocal
 
Theoremerlval 32995* Value of the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    βˆ’ = (-gβ€˜π‘…)    &   π‘Š = (𝐡 Γ— 𝑆)    &    ∼ = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ π‘Š ∧ 𝑏 ∈ π‘Š) ∧ βˆƒπ‘‘ ∈ 𝑆 (𝑑 Β· (((1st β€˜π‘Ž) Β· (2nd β€˜π‘)) βˆ’ ((1st β€˜π‘) Β· (2nd β€˜π‘Ž)))) = 0 )}    &   (πœ‘ β†’ 𝑆 βŠ† 𝐡)    β‡’   (πœ‘ β†’ (𝑅 ~RL 𝑆) = ∼ )
 
Theoremrlocval 32996* Expand the value of the ring localization operation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    βˆ’ = (-gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    ≀ = (leβ€˜π‘…)    &   πΉ = (Scalarβ€˜π‘…)    &   πΎ = (Baseβ€˜πΉ)    &   πΆ = ( ·𝑠 β€˜π‘…)    &   π‘Š = (𝐡 Γ— 𝑆)    &    ∼ = (𝑅 ~RL 𝑆)    &   π½ = (TopSetβ€˜π‘…)    &   π· = (distβ€˜π‘…)    &    βŠ• = (π‘Ž ∈ π‘Š, 𝑏 ∈ π‘Š ↦ ⟨(((1st β€˜π‘Ž) Β· (2nd β€˜π‘)) + ((1st β€˜π‘) Β· (2nd β€˜π‘Ž))), ((2nd β€˜π‘Ž) Β· (2nd β€˜π‘))⟩)    &    βŠ— = (π‘Ž ∈ π‘Š, 𝑏 ∈ π‘Š ↦ ⟨((1st β€˜π‘Ž) Β· (1st β€˜π‘)), ((2nd β€˜π‘Ž) Β· (2nd β€˜π‘))⟩)    &    Γ— = (π‘˜ ∈ 𝐾, π‘Ž ∈ π‘Š ↦ ⟨(π‘˜πΆ(1st β€˜π‘Ž)), (2nd β€˜π‘Ž)⟩)    &    ≲ = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ π‘Š ∧ 𝑏 ∈ π‘Š) ∧ ((1st β€˜π‘Ž) Β· (2nd β€˜π‘)) ≀ ((1st β€˜π‘) Β· (2nd β€˜π‘Ž)))}    &   πΈ = (π‘Ž ∈ π‘Š, 𝑏 ∈ π‘Š ↦ (((1st β€˜π‘Ž) Β· (2nd β€˜π‘))𝐷((1st β€˜π‘) Β· (2nd β€˜π‘Ž))))    &   (πœ‘ β†’ 𝑅 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 βŠ† 𝐡)    β‡’   (πœ‘ β†’ (𝑅 RLocal 𝑆) = ((({⟨(Baseβ€˜ndx), π‘ŠβŸ©, ⟨(+gβ€˜ndx), βŠ• ⟩, ⟨(.rβ€˜ndx), βŠ— ⟩} βˆͺ {⟨(Scalarβ€˜ndx), 𝐹⟩, ⟨( ·𝑠 β€˜ndx), Γ— ⟩, ⟨(Β·π‘–β€˜ndx), βˆ…βŸ©}) βˆͺ {⟨(TopSetβ€˜ndx), (𝐽 Γ—t (𝐽 β†Ύt 𝑆))⟩, ⟨(leβ€˜ndx), ≲ ⟩, ⟨(distβ€˜ndx), 𝐸⟩}) /s ∼ ))
 
Theoremerlcl1 32997 Closure for the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐡 = (Baseβ€˜π‘…)    &    ∼ = (𝑅 ~RL 𝑆)    &   (πœ‘ β†’ 𝑆 βŠ† 𝐡)    &   (πœ‘ β†’ π‘ˆ ∼ 𝑉)    β‡’   (πœ‘ β†’ π‘ˆ ∈ (𝐡 Γ— 𝑆))
 
Theoremerlcl2 32998 Closure for the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐡 = (Baseβ€˜π‘…)    &    ∼ = (𝑅 ~RL 𝑆)    &   (πœ‘ β†’ 𝑆 βŠ† 𝐡)    &   (πœ‘ β†’ π‘ˆ ∼ 𝑉)    β‡’   (πœ‘ β†’ 𝑉 ∈ (𝐡 Γ— 𝑆))
 
Theoremerldi 32999* Main property of the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐡 = (Baseβ€˜π‘…)    &    ∼ = (𝑅 ~RL 𝑆)    &   (πœ‘ β†’ 𝑆 βŠ† 𝐡)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    βˆ’ = (-gβ€˜π‘…)    &   (πœ‘ β†’ π‘ˆ ∼ 𝑉)    β‡’   (πœ‘ β†’ βˆƒπ‘‘ ∈ 𝑆 (𝑑 Β· (((1st β€˜π‘ˆ) Β· (2nd β€˜π‘‰)) βˆ’ ((1st β€˜π‘‰) Β· (2nd β€˜π‘ˆ)))) = 0 )
 
Theoremerlbrd 33000 Deduce the ring localization equivalence relation. If for some 𝑇 ∈ 𝑆 we have 𝑇 Β· (𝐸 Β· 𝐻 βˆ’ 𝐹 Β· 𝐺) = 0, then pairs ⟨𝐸, 𝐺⟩ and ⟨𝐹, 𝐻⟩ are equivalent under the localization relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐡 = (Baseβ€˜π‘…)    &    ∼ = (𝑅 ~RL 𝑆)    &   (πœ‘ β†’ 𝑆 βŠ† 𝐡)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    βˆ’ = (-gβ€˜π‘…)    &   (πœ‘ β†’ π‘ˆ = ⟨𝐸, 𝐺⟩)    &   (πœ‘ β†’ 𝑉 = ⟨𝐹, 𝐻⟩)    &   (πœ‘ β†’ 𝐸 ∈ 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐺 ∈ 𝑆)    &   (πœ‘ β†’ 𝐻 ∈ 𝑆)    &   (πœ‘ β†’ 𝑇 ∈ 𝑆)    &   (πœ‘ β†’ (𝑇 Β· ((𝐸 Β· 𝐻) βˆ’ (𝐹 Β· 𝐺))) = 0 )    β‡’   (πœ‘ β†’ π‘ˆ ∼ 𝑉)
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