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Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcycpmco2lem6 30701 Lemma for cycpmco2 30703. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    &   (𝜑𝐾 ∈ ran 𝑊)    &   (𝜑𝐾𝐼)    &   (𝜑 → (𝑈𝐾) ∈ (𝐸..^((♯‘𝑈) − 1)))       (𝜑 → ((𝑀𝑈)‘𝐾) = ((𝑀𝑊)‘𝐾))
 
Theoremcycpmco2lem7 30702 Lemma for cycpmco2 30703. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    &   (𝜑𝐾 ∈ ran 𝑊)    &   (𝜑𝐾𝐽)    &   (𝜑 → (𝑈𝐾) ∈ (0..^𝐸))       (𝜑 → ((𝑀𝑈)‘𝐾) = ((𝑀𝑊)‘𝐾))
 
Theoremcycpmco2 30703 The composition of a cyclic permutation and a transposition of one element in the cycle and one outside the cycle results in a cyclic permutation with one more element in its orbit. (Contributed by Thierry Arnoux, 2-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → ((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) = (𝑀𝑈))
 
Theoremcyc2fvx 30704 Function value of a 2-cycle outside of its orbit. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾)
 
Theoremcycpm3cl 30705 Closure of the 3-cycles in the permutations. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆))
 
Theoremcycpm3cl2 30706 Closure of the 3-cycles in the class of 3-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (𝐶 “ (♯ “ {3})))
 
Theoremcyc3fv1 30707 Function value of a 3-cycle at the first point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽)
 
Theoremcyc3fv2 30708 Function value of a 3-cycle at the second point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾)
 
Theoremcyc3fv3 30709 Function value of a 3-cycle at the third point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼)
 
Theoremcyc3co2 30710 Represent a 3-cycle as a composition of two 2-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)    &    · = (+g𝑆)       (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) = ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)))
 
Theoremcycpmconjvlem 30711 Lemma for cycpmconjv 30712 (Contributed by Thierry Arnoux, 9-Oct-2023.)
(𝜑𝐹:𝐷1-1-onto𝐷)    &   (𝜑𝐵𝐷)       (𝜑 → ((𝐹 ↾ (𝐷𝐵)) ∘ 𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹𝐵))))
 
Theoremcycpmconjv 30712 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 = (SymGrp‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &    + = (+g𝑆)    &    = (-g𝑆)    &   𝐵 = (Base‘𝑆)       ((𝐷𝑉𝐺𝐵𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀𝑊)) 𝐺) = (𝑀‘(𝐺𝑊)))
 
Theoremcycpmrn 30713 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 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑 → 1 < (♯‘𝑊))       (𝜑 → ran 𝑊 = dom ((𝑀𝑊) ∖ I ))
 
Theoremtocyccntz 30714* All elements of a (finite) set of cycles commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 27-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑍 = (Cntz‘𝑆)    &   𝑀 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑Disj 𝑥𝐴 ran 𝑥)    &   (𝜑𝐴 ⊆ dom 𝑀)       (𝜑 → (𝑀𝐴) ⊆ (𝑍‘(𝑀𝐴)))
 
20.3.9.9  The Alternating Group
 
Theoremevpmval 30715 Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝐴 = (pmEven‘𝐷)       (𝐷𝑉𝐴 = ((pmSgn‘𝐷) “ {1}))
 
Theoremcnmsgn0g 30716 The neutral element of the sign subgroup of the complex numbers. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1})       1 = (0g𝑈)
 
Theoremevpmsubg 30717 The alternating group is a subgroup of the symmetric group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝐴 = (pmEven‘𝐷)       (𝐷 ∈ Fin → 𝐴 ∈ (SubGrp‘𝑆))
 
Theoremevpmid 30718 The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝑆 = (SymGrp‘𝐷)       (𝐷 ∈ Fin → ( I ↾ 𝐷) ∈ (pmEven‘𝐷))
 
Theoremaltgnsg 30719 The alternating group (pmEven‘𝐷) is a normal subgroup of the symmetric group. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝑆 = (SymGrp‘𝐷)       (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆))
 
Theoremcyc3evpm 30720 3-Cycles are even permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = ((toCyc‘𝐷) “ (♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)       (𝐷 ∈ Fin → 𝐶𝐴)
 
Theoremcyc3genpmlem 30721* Lemma for cyc3genpm 30722. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (𝑀 “ (♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &    · = (+g𝑆)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐿𝐷)    &   (𝜑𝐸 = (𝑀‘⟨“𝐼𝐽”⟩))    &   (𝜑𝐹 = (𝑀‘⟨“𝐾𝐿”⟩))    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐽)    &   (𝜑𝐾𝐿)       (𝜑 → ∃𝑐 ∈ Word 𝐶(𝐸 · 𝐹) = (𝑆 Σg 𝑐))
 
Theoremcyc3genpm 30722* The alternating group 𝐴 is generated by 3-cycles. Property (a) of [Lang] p. 32 . (Contributed by Thierry Arnoux, 27-Sep-2023.)
𝐶 = (𝑀 “ (♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)       (𝐷 ∈ Fin → (𝑄𝐴 ↔ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)))
 
Theoremcycpmgcl 30723 Cyclic permutations are permutations, similar to cycpmcl 30686, but where the set of cyclic permutations of length 𝑃 is expressed in terms of a preimage. (Contributed by Thierry Arnoux, 13-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   𝐵 = (Base‘𝑆)       ((𝐷𝑉𝑃 ∈ (0...𝑁)) → 𝐶𝐵)
 
Theoremcycpmconjslem1 30724 Lemma for cycpmconjs 30726 (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑 → (♯‘𝑊) = 𝑃)       (𝜑 → ((𝑊 ∘ (𝑀𝑊)) ∘ 𝑊) = (( I ↾ (0..^𝑃)) cyclShift 1))
 
Theoremcycpmconjslem2 30725* Lemma for cycpmconjs 30726 (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    = (-g𝑆)    &   (𝜑𝑃 ∈ (0...𝑁))    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑄𝐶)       (𝜑 → ∃𝑞(𝑞:(0..^𝑁)–1-1-onto𝐷 ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))))
 
Theoremcycpmconjs 30726* All cycles of the same length are conjugate in the symmetric group. (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    = (-g𝑆)    &   (𝜑𝑃 ∈ (0...𝑁))    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑄𝐶)    &   (𝜑𝑇𝐶)       (𝜑 → ∃𝑝𝐵 𝑄 = ((𝑝 + 𝑇) 𝑝))
 
Theoremcyc3conja 30727* 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.)
𝐶 = (𝑀 “ (♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &    + = (+g𝑆)    &    = (-g𝑆)    &   (𝜑 → 5 ≤ 𝑁)    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑄𝐶)    &   (𝜑𝑇𝐶)       (𝜑 → ∃𝑝𝐴 𝑄 = ((𝑝 + 𝑇) 𝑝))
 
20.3.9.10  Signum in an ordered monoid
 
Syntaxcsgns 30728 Extend class notation to include the Signum function.
class sgns
 
Definitiondf-sgns 30729* Signum function for a structure. See also df-sgn 14436 for the version for extended reals. (Contributed by Thierry Arnoux, 10-Sep-2018.)
sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g𝑟), 0, if((0g𝑟)(lt‘𝑟)𝑥, 1, -1))))
 
Theoremsgnsv 30730* The sign mapping. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   𝑆 = (sgns𝑅)       (𝑅𝑉𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
 
Theoremsgnsval 30731 The sign value. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   𝑆 = (sgns𝑅)       ((𝑅𝑉𝑋𝐵) → (𝑆𝑋) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
 
Theoremsgnsf 30732 The sign function. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   𝑆 = (sgns𝑅)       (𝑅𝑉𝑆:𝐵⟶{-1, 0, 1})
 
20.3.9.11  The Archimedean property for generic ordered algebraic structures
 
Syntaxcinftm 30733 Class notation for the infinitesimal relation.
class
 
Syntaxcarchi 30734 Class notation for the Archimedean property.
class Archi
 
Definitiondf-inftm 30735* Define the relation "𝑥 is infinitesimal with respect to 𝑦 " for a structure 𝑤. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⋘ = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦))})
 
Definitiondf-archi 30736 A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
 
Theoreminftmrel 30737 The infinitesimal relation for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)       (𝑊𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
 
Theoremisinftm 30738* Express 𝑥 is infinitesimal with respect to 𝑦 for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    · = (.g𝑊)    &    < = (lt‘𝑊)       ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋(⋘‘𝑊)𝑌 ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))
 
Theoremisarchi 30739* Express the predicate "𝑊 is Archimedean ". (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (⋘‘𝑊)       (𝑊𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦))
 
Theorempnfinf 30740 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+𝐴(⋘‘ℝ*𝑠)+∞)
 
Theoremxrnarchi 30741 The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.)
¬ ℝ*𝑠 ∈ Archi
 
Theoremisarchi2 30742* Alternative way to express the predicate "𝑊 is Archimedean ", for Tosets. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    · = (.g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)       ((𝑊 ∈ Toset ∧ 𝑊 ∈ Mnd) → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑦𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 (𝑛 · 𝑥))))
 
Theoremsubmarchi 30743 A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
(((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Archi)
 
Theoremisarchi3 30744* This is the usual definition of the Archimedean property for an ordered group. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)       (𝑊 ∈ oGrp → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑦𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))))
 
Theoremarchirng 30745* Property of Archimedean ordered groups, framing positive 𝑌 between multiples of 𝑋. (Contributed by Thierry Arnoux, 12-Apr-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (lt‘𝑊)    &    = (le‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑0 < 𝑋)    &   (𝜑0 < 𝑌)       (𝜑 → ∃𝑛 ∈ ℕ0 ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
 
Theoremarchirngz 30746* Property of Archimedean left and right ordered groups. (Contributed by Thierry Arnoux, 6-May-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (lt‘𝑊)    &    = (le‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑0 < 𝑋)    &   (𝜑 → (oppg𝑊) ∈ oGrp)       (𝜑 → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
 
Theoremarchiexdiv 30747* In an Archimedean group, given two positive elements, there exists a "divisor" 𝑛. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)       (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋𝐵𝑌𝐵) ∧ 0 < 𝑋) → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋))
 
Theoremarchiabllem1a 30748* Lemma for archiabl 30755: In case an archimedean group 𝑊 admits a smallest positive element 𝑈, then any positive element 𝑋 of 𝑊 can be written as (𝑛 · 𝑈) with 𝑛 ∈ ℕ. Since the reciprocal holds for negative elements, 𝑊 is then isomorphic to . (Contributed by Thierry Arnoux, 12-Apr-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &   (𝜑𝑈𝐵)    &   (𝜑0 < 𝑈)    &   ((𝜑𝑥𝐵0 < 𝑥) → 𝑈 𝑥)    &   (𝜑𝑋𝐵)    &   (𝜑0 < 𝑋)       (𝜑 → ∃𝑛 ∈ ℕ 𝑋 = (𝑛 · 𝑈))
 
Theoremarchiabllem1b 30749* Lemma for archiabl 30755. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &   (𝜑𝑈𝐵)    &   (𝜑0 < 𝑈)    &   ((𝜑𝑥𝐵0 < 𝑥) → 𝑈 𝑥)       ((𝜑𝑦𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
 
Theoremarchiabllem1 30750* Archimedean ordered groups with a minimal positive value are abelian. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &   (𝜑𝑈𝐵)    &   (𝜑0 < 𝑈)    &   ((𝜑𝑥𝐵0 < 𝑥) → 𝑈 𝑥)       (𝜑𝑊 ∈ Abel)
 
Theoremarchiabllem2a 30751* Lemma for archiabl 30755, which requires the group to be both left- and right-ordered. (Contributed by Thierry Arnoux, 13-Apr-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)    &   (𝜑0 < 𝑋)       (𝜑 → ∃𝑐𝐵 ( 0 < 𝑐 ∧ (𝑐 + 𝑐) 𝑋))
 
Theoremarchiabllem2c 30752* Lemma for archiabl 30755. (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))
 
Theoremarchiabllem2b 30753* Lemma for archiabl 30755. (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 30754* 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 30755 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 30756 Extend class notation with class of all semimodules.
class SLMod
 
Definitiondf-slmd 30757* 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 30758* 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 30759 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 30760 Left semimodules generalize the notion of left modules. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑊 ∈ LMod → 𝑊 ∈ SLMod)
 
Theoremslmdcmn 30761 A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑊 ∈ SLMod → 𝑊 ∈ CMnd)
 
Theoremslmdmnd 30762 A semimodule is a monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑊 ∈ SLMod → 𝑊 ∈ Mnd)
 
Theoremslmdsrg 30763 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 30764 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 30765 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 30766 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 30767 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 30768 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 30769 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 30770 Closure of scalar product for a semiring left module. (hvmulcl 28718 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 30771 Distributive law for scalar product. (ax-hvdistr1 28713 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 30772 Distributive law for scalar product. (ax-hvdistr1 28713 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 30773 Associative law for scalar product. (ax-hvmulass 28712 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 30774 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 30775 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 30776 Scalar product with ring unit. (ax-hvmulid 28711 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 30777 The zero vector is a vector. (ax-hv0cl 28708 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 30778 Left identity law for the zero vector. (hvaddid2 28728 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 30779 Right identity law for the zero vector. (ax-hvaddid 28709 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 30780 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 28715 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 30781 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 28729 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 30782* 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 30783* 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 30784 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 30785* 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 30786 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 30787 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)))
 
Theoremress1r 30788 1r is unaffected by restriction. This is a bit more generic than subrg1 19476. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑆 = (𝑅s 𝐴)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 1𝐴𝐴𝐵) → 1 = (1r𝑆))
 
Theoremdvrdir 30789 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 30790 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 30791* 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 30792 Cancellation law for common factor in ratio. (divcan5 11331 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 30793 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 30794* 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 30795 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 30796 Extend class notation with the class of all ordered rings.
class oRing
 
Syntaxcofld 30797 Extend class notation with the class of all ordered fields.
class oField
 
Definitiondf-orng 30798* 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 30799 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 30800* 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))))
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