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Theorem List for Metamath Proof Explorer - 19201-19300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremablfaclem3 19201* Lemma for ablfac 19202. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   𝑂 = (od‘𝐺)    &   𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})    &   𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})       (𝜑 → (𝑊𝐵) ≠ ∅)
 
Theoremablfac 19202* The Fundamental Theorem of (finite) Abelian Groups. Any finite abelian group is a direct product of cyclic p-groups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))
 
Theoremablfac2 19203* Choose generators for each cyclic group in ablfac 19202. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &    · = (.g𝐺)    &   𝑆 = (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))))       (𝜑 → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤𝐶𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))
 
10.2.15  Simple groups
 
10.2.15.1  Definition and basic properties
 
Syntaxcsimpg 19204 Extend class notation with the class of simple groups.
class SimpGrp
 
Definitiondf-simpg 19205 Define class of all simple groups. A simple group is a group (df-grp 18098) with exactly two normal subgroups. These are always the subgroup of all elements and the subgroup containing only the identity (simpgnsgbid 19217). (Contributed by Rohan Ridenour, 3-Aug-2023.)
SimpGrp = {𝑔 ∈ Grp ∣ (NrmSGrp‘𝑔) ≈ 2o}
 
Theoremissimpg 19206 The predicate "is a simple group". (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
 
Theoremissimpgd 19207 Deduce a simple group from its properties. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑𝐺 ∈ Grp)    &   (𝜑 → (NrmSGrp‘𝐺) ≈ 2o)       (𝜑𝐺 ∈ SimpGrp)
 
Theoremsimpggrp 19208 A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)
 
Theoremsimpggrpd 19209 A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑𝐺 ∈ SimpGrp)       (𝜑𝐺 ∈ Grp)
 
Theoremsimpg2nsg 19210 A simple group has two normal subgroups. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝐺 ∈ SimpGrp → (NrmSGrp‘𝐺) ≈ 2o)
 
Theoremtrivnsimpgd 19211 Trivial groups are not simple. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐵 = { 0 })       (𝜑 → ¬ 𝐺 ∈ SimpGrp)
 
Theoremsimpgntrivd 19212 Simple groups are nontrivial. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ SimpGrp)       (𝜑 → ¬ 𝐵 = { 0 })
 
Theoremsimpgnideld 19213* A simple group contains a nonidentity element. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ SimpGrp)       (𝜑 → ∃𝑥𝐵 ¬ 𝑥 = 0 )
 
Theoremsimpgnsgd 19214 The only normal subgroups of a simple group are the group itself and the trivial group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ SimpGrp)       (𝜑 → (NrmSGrp‘𝐺) = {{ 0 }, 𝐵})
 
Theoremsimpgnsgeqd 19215 A normal subgroup of a simple group is either the whole group or the trivial subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ SimpGrp)    &   (𝜑𝐴 ∈ (NrmSGrp‘𝐺))       (𝜑 → (𝐴 = { 0 } ∨ 𝐴 = 𝐵))
 
Theorem2nsgsimpgd 19216* If any normal subgroup of a nontrivial group is either the trivial subgroup or the whole group, the group is simple. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑 → ¬ { 0 } = 𝐵)    &   ((𝜑𝑥 ∈ (NrmSGrp‘𝐺)) → (𝑥 = { 0 } ∨ 𝑥 = 𝐵))       (𝜑𝐺 ∈ SimpGrp)
 
Theoremsimpgnsgbid 19217 A nontrivial group is simple if and only if its normal subgroups are exactly the group itself and the trivial subgroup. (Contributed by Rohan Ridenour, 4-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑 → ¬ { 0 } = 𝐵)       (𝜑 → (𝐺 ∈ SimpGrp ↔ (NrmSGrp‘𝐺) = {{ 0 }, 𝐵}))
 
10.2.15.2  Classification of abelian simple groups
 
Theoremablsimpnosubgd 19218 A subgroup of an abelian simple group containing a nonidentity element is the whole group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝐴𝑆)    &   (𝜑 → ¬ 𝐴 = 0 )       (𝜑𝑆 = 𝐵)
 
Theoremablsimpg1gend 19219* An abelian simple group is generated by any non-identity element. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)    &   (𝜑𝐴𝐵)    &   (𝜑 → ¬ 𝐴 = 0 )    &   (𝜑𝐶𝐵)       (𝜑 → ∃𝑛 ∈ ℤ 𝐶 = (𝑛 · 𝐴))
 
Theoremablsimpgcygd 19220 An abelian simple group is cyclic. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Proof shortened by Rohan Ridenour, 31-Oct-2023.)
(𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)       (𝜑𝐺 ∈ CycGrp)
 
Theoremablsimpgfindlem1 19221* Lemma for ablsimpgfind 19224. An element of an abelian finite simple group which doesn't square to the identity has finite order. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Proof shortened by Rohan Ridenour, 31-Oct-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)    &   𝑂 = (od‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)       (((𝜑𝑥𝐵) ∧ (2 · 𝑥) ≠ 0 ) → (𝑂𝑥) ≠ 0)
 
Theoremablsimpgfindlem2 19222* Lemma for ablsimpgfind 19224. An element of an abelian finite simple group which squares to the identity has finite order. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)    &   𝑂 = (od‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)       (((𝜑𝑥𝐵) ∧ (2 · 𝑥) = 0 ) → (𝑂𝑥) ≠ 0)
 
Theoremcycsubggenodd 19223* Relationship between the order of a subgroup and the order of a generator of the subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝑂 = (od‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶 = ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝐴)))       (𝜑 → (𝑂𝐴) = if(𝐶 ∈ Fin, (♯‘𝐶), 0))
 
Theoremablsimpgfind 19224 An abelian simple group is finite. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)       (𝜑𝐵 ∈ Fin)
 
Theoremfincygsubgd 19225* The subgroup referenced in fincygsubgodd 19226 is a subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐻 = (𝑛 ∈ ℤ ↦ (𝑛 · (𝐶 · 𝐴)))    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶 ∈ ℕ)       (𝜑 → ran 𝐻 ∈ (SubGrp‘𝐺))
 
Theoremfincygsubgodd 19226* Calculate the order of a subgroup of a finite cyclic group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐷 = ((♯‘𝐵) / 𝐶)    &   𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 𝐴))    &   𝐻 = (𝑛 ∈ ℤ ↦ (𝑛 · (𝐶 · 𝐴)))    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐴𝐵)    &   (𝜑 → ran 𝐹 = 𝐵)    &   (𝜑𝐶 ∥ (♯‘𝐵))    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐶 ∈ ℕ)       (𝜑 → (♯‘ran 𝐻) = 𝐷)
 
Theoremfincygsubgodexd 19227* A finite cyclic group has subgroups of every possible order. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CycGrp)    &   (𝜑𝐶 ∥ (♯‘𝐵))    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐶 ∈ ℕ)       (𝜑 → ∃𝑥 ∈ (SubGrp‘𝐺)(♯‘𝑥) = 𝐶)
 
Theoremprmgrpsimpgd 19228 A group of prime order is simple. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑 → (♯‘𝐵) ∈ ℙ)       (𝜑𝐺 ∈ SimpGrp)
 
Theoremablsimpgprmd 19229 An abelian simple group has prime order. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)       (𝜑 → (♯‘𝐵) ∈ ℙ)
 
Theoremablsimpgd 19230 An abelian group is simple if and only if its order is prime. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)       (𝜑 → (𝐺 ∈ SimpGrp ↔ (♯‘𝐵) ∈ ℙ))
 
10.3  Rings
 
10.3.1  Multiplicative Group
 
Syntaxcmgp 19231 Multiplicative group.
class mulGrp
 
Definitiondf-mgp 19232 Define a structure that puts the multiplication operation of a ring in the addition slot. Note that this will not actually be a group for the average ring, or even for a field, but it will be a monoid, and unitgrp 19409 shows that we get a group if we restrict to the elements that have inverses. This allows us to formalize such notions as "the multiplication operation of a ring is a monoid" (ringmgp 19295) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 10475). (Contributed by Mario Carneiro, 21-Dec-2014.)
mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r𝑤)⟩))
 
Theoremfnmgp 19233 The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
mulGrp Fn V
 
Theoremmgpval 19234 Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
𝑀 = (mulGrp‘𝑅)    &    · = (.r𝑅)       𝑀 = (𝑅 sSet ⟨(+g‘ndx), · ⟩)
 
Theoremmgpplusg 19235 Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
𝑀 = (mulGrp‘𝑅)    &    · = (.r𝑅)        · = (+g𝑀)
 
Theoremmgplem 19236 Lemma for mgpbas 19237. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 ≠ 2       (𝐸𝑅) = (𝐸𝑀)
 
Theoremmgpbas 19237 Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝐵 = (Base‘𝑅)       𝐵 = (Base‘𝑀)
 
Theoremmgpsca 19238 The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 19360. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝑆 = (Scalar‘𝑅)       𝑆 = (Scalar‘𝑀)
 
Theoremmgptset 19239 Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)       (TopSet‘𝑅) = (TopSet‘𝑀)
 
Theoremmgptopn 19240 Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝐽 = (TopOpen‘𝑅)       𝐽 = (TopOpen‘𝑀)
 
Theoremmgpds 19241 Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝐵 = (dist‘𝑅)       𝐵 = (dist‘𝑀)
 
Theoremmgpress 19242 Subgroup commutes with the multiplication group operator. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑆 = (𝑅s 𝐴)    &   𝑀 = (mulGrp‘𝑅)       ((𝑅𝑉𝐴𝑊) → (𝑀s 𝐴) = (mulGrp‘𝑆))
 
10.3.2  Ring unit
 
Syntaxcur 19243 Extend class notation with ring unit.
class 1r
 
Definitiondf-ur 19244 Define the multiplicative neutral element of a ring. This definition works by extracting the 0g element, i.e. the neutral element in a group or monoid, and transferring it to the multiplicative monoid via the mulGrp function (df-mgp 19232). See also dfur2 19246, which derives the "traditional" definition as the unique element of a ring which is left- and right-neutral under multiplication. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
1r = (0g ∘ mulGrp)
 
Theoremringidval 19245 The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐺 = (mulGrp‘𝑅)    &    1 = (1r𝑅)        1 = (0g𝐺)
 
Theoremdfur2 19246* The multiplicative identity is the unique element of the ring that is left- and right-neutral on all elements under multiplication. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)        1 = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥)))
 
10.3.2.1  Semirings
 
Syntaxcsrg 19247 Extend class notation with the class of all semirings.
class SRing
 
Definitiondf-srg 19248* Define class of all semirings. A semiring is a set equipped with two everywhere-defined internal operations, whose first one is an additive commutative monoid structure and the second one is a multiplicative monoid structure, and where multiplication is (left- and right-) distributive over addition. Compared to the definition of a ring, this definition also adds that the additive identity is an absorbing element of the multiplicative law, as this cannot be deduced from distributivity alone. Definition of [Golan] p. 1. Note that our semirings are unital. Such semirings are sometimes called "rigs", being "rings without negatives". (Contributed by Thierry Arnoux, 21-Mar-2018.)
SRing = {𝑓 ∈ CMnd ∣ ((mulGrp‘𝑓) ∈ Mnd ∧ [(Base‘𝑓) / 𝑟][(+g𝑓) / 𝑝][(.r𝑓) / 𝑡][(0g𝑓) / 𝑛]𝑥𝑟 (∀𝑦𝑟𝑧𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))}
 
Theoremissrg 19249* The predicate "is a semiring". (Contributed by Thierry Arnoux, 21-Mar-2018.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
 
Theoremsrgcmn 19250 A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
(𝑅 ∈ SRing → 𝑅 ∈ CMnd)
 
Theoremsrgmnd 19251 A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
(𝑅 ∈ SRing → 𝑅 ∈ Mnd)
 
Theoremsrgmgp 19252 A semiring is a monoid under multiplication. (Contributed by Thierry Arnoux, 21-Mar-2018.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ SRing → 𝐺 ∈ Mnd)
 
Theoremsrgi 19253 Properties of a semiring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))
 
Theoremsrgcl 19254 Closure of the multiplication operation of a semiring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 · 𝑌) ∈ 𝐵)
 
Theoremsrgass 19255 Associative law for the multiplication operation of a semiring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍)))
 
Theoremsrgideu 19256* The unit element of a semiring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ SRing → ∃!𝑢𝐵𝑥𝐵 ((𝑢 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑢) = 𝑥))
 
Theoremsrgfcl 19257 Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by AV, 24-Aug-2021.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → · :(𝐵 × 𝐵)⟶𝐵)
 
Theoremsrgdi 19258 Distributive law for the multiplication operation of a semiring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
 
Theoremsrgdir 19259 Distributive law for the multiplication operation of a semiring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
 
Theoremsrgidcl 19260 The unit element of a semiring belongs to the base set of the semiring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ SRing → 1𝐵)
 
Theoremsrg0cl 19261 The zero element of a semiring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ SRing → 0𝐵)
 
Theoremsrgidmlem 19262 Lemma for srglidm 19263 and srgridm 19264. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋))
 
Theoremsrglidm 19263 The unit element of a semiring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ( 1 · 𝑋) = 𝑋)
 
Theoremsrgridm 19264 The unit element of a semiring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑋 · 1 ) = 𝑋)
 
Theoremissrgid 19265* Properties showing that an element 𝐼 is the unity element of a semiring. (Contributed by NM, 7-Aug-2013.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ SRing → ((𝐼𝐵 ∧ ∀𝑥𝐵 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥)) ↔ 1 = 𝐼))
 
Theoremsrgacl 19266 Closure of the addition operation of a semiring. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
 
Theoremsrgcom 19267 Commutativity of the additive group of a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
 
Theoremsrgrz 19268 The zero of a semiring is a right-absorbing element. (Contributed by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑋 · 0 ) = 0 )
 
Theoremsrglz 19269 The zero of a semiring is a left-absorbing element. (Contributed by AV, 23-Aug-2019.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ( 0 · 𝑋) = 0 )
 
Theoremsrgisid 19270* In a semiring, the only left-absorbing element is the additive identity. Remark in [Golan] p. 1. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝑍𝐵)    &   ((𝜑𝑥𝐵) → (𝑍 · 𝑥) = 𝑍)       (𝜑𝑍 = 0 )
 
Theoremsrg1zr 19271 The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)       (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
Theoremsrgen1zr 19272 The only semiring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)    &   𝑍 = (0g𝑅)       ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
Theoremsrgmulgass 19273 An associative property between group multiple and ring multiplication for semirings. (Contributed by AV, 23-Aug-2019.)
𝐵 = (Base‘𝑅)    &    · = (.g𝑅)    &    × = (.r𝑅)       ((𝑅 ∈ SRing ∧ (𝑁 ∈ ℕ0𝑋𝐵𝑌𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))
 
Theoremsrgpcomp 19274 If two elements of a semiring commute, they also commute if one of the elements is raised to a higher power. (Contributed by AV, 23-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))       (𝜑 → ((𝐾 𝐵) × 𝐴) = (𝐴 × (𝐾 𝐵)))
 
Theoremsrgpcompp 19275 If two elements of a semiring commute, they also commute if the elements are raised to a higher power. (Contributed by AV, 23-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (((𝑁 𝐴) × (𝐾 𝐵)) × 𝐴) = (((𝑁 + 1) 𝐴) × (𝐾 𝐵)))
 
Theoremsrgpcomppsc 19276 If two elements of a semiring commute, they also commute if the elements are raised to a higher power and a scalar multiplication is involved. (Contributed by AV, 23-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   (𝜑𝑁 ∈ ℕ0)    &    · = (.g𝑅)    &   (𝜑𝐶 ∈ ℕ0)       (𝜑 → ((𝐶 · ((𝑁 𝐴) × (𝐾 𝐵))) × 𝐴) = (𝐶 · (((𝑁 + 1) 𝐴) × (𝐾 𝐵))))
 
Theoremsrglmhm 19277* Left-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism, analogous to ringlghm 19346. (Contributed by AV, 23-Aug-2019.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑥𝐵 ↦ (𝑋 · 𝑥)) ∈ (𝑅 MndHom 𝑅))
 
Theoremsrgrmhm 19278* Right-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism, analogous to ringrghm 19347. (Contributed by AV, 23-Aug-2019.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑥𝐵 ↦ (𝑥 · 𝑋)) ∈ (𝑅 MndHom 𝑅))
 
Theoremsrgsummulcr 19279* A finite semiring sum multiplied by a constant, analogous to gsummulc1 19348. (Contributed by AV, 23-Aug-2019.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑉)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )       (𝜑 → (𝑅 Σg (𝑘𝐴 ↦ (𝑋 · 𝑌))) = ((𝑅 Σg (𝑘𝐴𝑋)) · 𝑌))
 
Theoremsgsummulcl 19280* A finite semiring sum multiplied by a constant, analogous to gsummulc2 19349. (Contributed by AV, 23-Aug-2019.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑉)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )       (𝜑 → (𝑅 Σg (𝑘𝐴 ↦ (𝑌 · 𝑋))) = (𝑌 · (𝑅 Σg (𝑘𝐴𝑋))))
 
Theoremsrg1expzeq1 19281 The exponentiation (by a nonnegative integer) of the multiplicative identity of a semiring, analogous to mulgnn0z 18246. (Contributed by AV, 25-Nov-2019.)
𝐺 = (mulGrp‘𝑅)    &    · = (.g𝐺)    &    1 = (1r𝑅)       ((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) → (𝑁 · 1 ) = 1 )
 
10.3.2.2  The binomial theorem for semirings

In this section, we prove the binomial theorem for semirings, srgbinom 19287, which is a generalization of the binomial theorem for complex numbers, binom 15177: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴𝑘) · (𝐵↑(𝑁𝑘)).

Note that the binomial theorem also holds in the non-unital case (that is, in a "rg") and actually, the additive unit is not needed in its proof either. Therefore, it can be proven in even more general cases. An example is the "rg" (resp. "rg without a zero") of integrable nonnegative (resp. positive) functions on .

Special cases of the binomial theorem are csrgbinom 19288 (binomial theorem for commutative semirings) and crngbinom 19363 (binomial theorem for commutative rings).

 
Theoremsrgbinomlem1 19282 Lemma 1 for srgbinomlem 19286. (Contributed by AV, 23-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &    · = (.g𝑅)    &    + = (+g𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   (𝜑𝑁 ∈ ℕ0)       ((𝜑 ∧ (𝐷 ∈ ℕ0𝐸 ∈ ℕ0)) → ((𝐷 𝐴) × (𝐸 𝐵)) ∈ 𝑆)
 
Theoremsrgbinomlem2 19283 Lemma 2 for srgbinomlem 19286. (Contributed by AV, 23-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &    · = (.g𝑅)    &    + = (+g𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   (𝜑𝑁 ∈ ℕ0)       ((𝜑 ∧ (𝐶 ∈ ℕ0𝐷 ∈ ℕ0𝐸 ∈ ℕ0)) → (𝐶 · ((𝐷 𝐴) × (𝐸 𝐵))) ∈ 𝑆)
 
Theoremsrgbinomlem3 19284* Lemma 3 for srgbinomlem 19286. (Contributed by AV, 23-Aug-2019.) (Proof shortened by AV, 27-Oct-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &    · = (.g𝑅)    &    + = (+g𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜓 → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))       ((𝜑𝜓) → ((𝑁 (𝐴 + 𝐵)) × 𝐴) = (𝑅 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ ((𝑁C𝑘) · ((((𝑁 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))))
 
Theoremsrgbinomlem4 19285* Lemma 4 for srgbinomlem 19286. (Contributed by AV, 24-Aug-2019.) (Proof shortened by AV, 19-Nov-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &    · = (.g𝑅)    &    + = (+g𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜓 → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))       ((𝜑𝜓) → ((𝑁 (𝐴 + 𝐵)) × 𝐵) = (𝑅 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ ((𝑁C(𝑘 − 1)) · ((((𝑁 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))))
 
Theoremsrgbinomlem 19286* Lemma for srgbinom 19287. Inductive step, analogous to binomlem 15176. (Contributed by AV, 24-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &    · = (.g𝑅)    &    + = (+g𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ SRing)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜓 → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))       ((𝜑𝜓) → ((𝑁 + 1) (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ (((𝑁 + 1)C𝑘) · ((((𝑁 + 1) − 𝑘) 𝐴) × (𝑘 𝐵))))))
 
Theoremsrgbinom 19287* The binomial theorem for commuting elements of a semiring: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴𝑘) · (𝐵↑(𝑁𝑘)) (generalization of binom 15177). (Contributed by AV, 24-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &    · = (.g𝑅)    &    + = (+g𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)       (((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) ∧ (𝐴𝑆𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))
 
Theoremcsrgbinom 19288* The binomial theorem for commutative semirings. (Contributed by AV, 24-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &    · = (.g𝑅)    &    + = (+g𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)       (((𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ0) ∧ (𝐴𝑆𝐵𝑆)) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))
 
10.3.3  Definition and basic properties of unital rings
 
Syntaxcrg 19289 Extend class notation with class of all (unital) rings.
class Ring
 
Syntaxccrg 19290 Extend class notation with class of all (unital) commutative rings.
class CRing
 
Definitiondf-ring 19291* Define class of all (unital) rings. A unital ring is a set equipped with two everywhere-defined internal operations, whose first one is an additive group structure and the second one is a multiplicative monoid structure, and where the addition is left- and right-distributive for the multiplication. Definition 1 in [BourbakiAlg1] p. 92 or definition of a ring with identity in part Preliminaries of [Roman] p. 19. So that the additive structure must be abelian (see ringcom 19321), care must be taken that in the case of a non-unital ring, the commutativity of addition must be postulated and cannot be proved from the other conditions. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 27-Dec-2014.)
Ring = {𝑓 ∈ Grp ∣ ((mulGrp‘𝑓) ∈ Mnd ∧ [(Base‘𝑓) / 𝑟][(+g𝑓) / 𝑝][(.r𝑓) / 𝑡]𝑥𝑟𝑦𝑟𝑧𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))}
 
Definitiondf-cring 19292 Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd}
 
Theoremisring 19293* The predicate "is a (unital) ring." Definition of ring with unit in [Schechter] p. 187. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ 𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
 
Theoremringgrp 19294 A ring is a group. (Contributed by NM, 15-Sep-2011.)
(𝑅 ∈ Ring → 𝑅 ∈ Grp)
 
Theoremringmgp 19295 A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ Ring → 𝐺 ∈ Mnd)
 
Theoremiscrng 19296 A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))
 
Theoremcrngmgp 19297 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ CRing → 𝐺 ∈ CMnd)
 
Theoremringmnd 19298 A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ Ring → 𝑅 ∈ Mnd)
 
Theoremringmgm 19299 A ring is a magma. (Contributed by AV, 31-Jan-2020.)
(𝑅 ∈ Ring → 𝑅 ∈ Mgm)
 
Theoremcrngring 19300 A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ CRing → 𝑅 ∈ Ring)
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