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Theorem List for Metamath Proof Explorer - 18501-18600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsrgidmlem 18501 Lemma for srglidm 18502 and srgridm 18503. (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 18502 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 18503 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 18504* 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 18505 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 18506 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 18507 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 18508 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 18509* 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 18510 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 18511 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 18512 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 18513 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 18514 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 18515 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 18516* Left-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism, analogous to ringlghm 18585. (Contributed by AV, 23-Aug-2019.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑥𝐵 ↦ (𝑋 · 𝑥)) ∈ (𝑅 MndHom 𝑅))
 
Theoremsrgrmhm 18517* Right-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism, analogous to ringrghm 18586. (Contributed by AV, 23-Aug-2019.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑥𝐵 ↦ (𝑥 · 𝑋)) ∈ (𝑅 MndHom 𝑅))
 
Theoremsrgsummulcr 18518* A finite semiring sum multiplied by a constant, analogous to gsummulc1 18587. (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 18519* A finite semiring sum multiplied by a constant, analogous to gsummulc2 18588. (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 18520 The exponentiation (by a nonnegative integer) of the unity element of a (semi)ring, analogous to mulgnn0z 17548. (Contributed by AV, 25-Nov-2019.)
𝐺 = (mulGrp‘𝑅)    &    · = (.g𝐺)    &    1 = (1r𝑅)       ((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) → (𝑁 · 1 ) = 1 )
 
10.4.2.2  The binomial theorem for semirings

In this section, we prove the binomial theorem for semirings, srgbinom 18526, which is a generalization of the binomial theorem for complex numbers, binom 14543: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴𝑘) · (𝐵↑(𝑁𝑘)).

Notice that the binomial theorem would also hold in the non-unital case (that is, in a "rg") and actually, the additive unit is not needed in its proof either. Therefore, it could be proven for even more general cases. An example would be the integrable nonnegative (resp. positive) bounded functions on .

Special cases of the binomial theorem are csrgbinom 18527 (binomial theorem for commutative semirings) and crngbinom 18602 (binomial theorem for commutative rings).

 
Theoremsrgbinomlem1 18521 Lemma 1 for srgbinomlem 18525. (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 18522 Lemma 2 for srgbinomlem 18525. (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 18523* Lemma 3 for srgbinomlem 18525. (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 18524* Lemma 4 for srgbinomlem 18525. (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 18525* Lemma for srgbinom 18526. Inductive step, analogous to binomlem 14542. (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 18526* 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 14543). (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 18527* 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))))))
 
10.4.3  Definition and basic properties of unital rings
 
Syntaxcrg 18528 Extend class notation with class of all (unital) rings.
class Ring
 
Syntaxccrg 18529 Extend class notation with class of all (unital) commutative rings.
class CRing
 
Definitiondf-ring 18530* 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 18560), 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 18531 Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd}
 
Theoremisring 18532* 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 18533 A ring is a group. (Contributed by NM, 15-Sep-2011.)
(𝑅 ∈ Ring → 𝑅 ∈ Grp)
 
Theoremringmgp 18534 A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ Ring → 𝐺 ∈ Mnd)
 
Theoremiscrng 18535 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 18536 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ CRing → 𝐺 ∈ CMnd)
 
Theoremringmnd 18537 A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ Ring → 𝑅 ∈ Mnd)
 
Theoremringmgm 18538 A ring is a magma. (Contributed by AV, 31-Jan-2020.)
(𝑅 ∈ Ring → 𝑅 ∈ Mgm)
 
Theoremcrngring 18539 A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ CRing → 𝑅 ∈ Ring)
 
Theoremmgpf 18540 Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
(mulGrp ↾ Ring):Ring⟶Mnd
 
Theoremringi 18541 Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))
 
Theoremringcl 18542 Closure of the multiplication operation of a ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝐵) → (𝑋 · 𝑌) ∈ 𝐵)
 
Theoremcrngcom 18543 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ CRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 · 𝑌) = (𝑌 · 𝑋))
 
Theoremiscrng2 18544* A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 · 𝑦) = (𝑦 · 𝑥)))
 
Theoremringass 18545 Associative law for the multiplication operation of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍)))
 
Theoremringideu 18546* The unit element of a ring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring → ∃!𝑢𝐵𝑥𝐵 ((𝑢 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑢) = 𝑥))
 
Theoremringdi 18547 Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
 
Theoremringdir 18548 Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
 
Theoremringidcl 18549 The unit element of a ring belongs to the base set of the ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → 1𝐵)
 
Theoremring0cl 18550 The zero element of a ring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → 0𝐵)
 
Theoremringidmlem 18551 Lemma for ringlidm 18552 and ringridm 18553. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋))
 
Theoremringlidm 18552 The unit element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → ( 1 · 𝑋) = 𝑋)
 
Theoremringridm 18553 The unit element of a ring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝑋 · 1 ) = 𝑋)
 
Theoremisringid 18554* Properties showing that an element 𝐼 is the unity element of a ring. (Contributed by NM, 7-Aug-2013.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → ((𝐼𝐵 ∧ ∀𝑥𝐵 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥)) ↔ 1 = 𝐼))
 
Theoremringid 18555* The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → ∃𝑢𝐵 ((𝑢 · 𝑋) = 𝑋 ∧ (𝑋 · 𝑢) = 𝑋))
 
Theoremringadd2 18556* A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → ∃𝑥𝐵 (𝑋 + 𝑋) = ((𝑥 + 𝑥) · 𝑋))
 
Theoremrngo2times 18557 A ring element plus itself is two times the element. "Two" in an arbitrary unital ring is the sum of the unit with itself. (Contributed by AV, 24-Aug-2021.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝐴𝐵) → (𝐴 + 𝐴) = (( 1 + 1 ) · 𝐴))
 
Theoremringidss 18558 A subset of the multiplicative group has the multiplicative identity as its identity if the identity is in the subset. (Contributed by Mario Carneiro, 27-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝑀 = ((mulGrp‘𝑅) ↾s 𝐴)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝐴𝐵1𝐴) → 1 = (0g𝑀))
 
Theoremringacl 18559 Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
 
Theoremringcom 18560 Commutativity of the additive group of a ring. (See also lmodcom 18890.) (Contributed by Gérard Lang, 4-Dec-2014.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
 
Theoremringabl 18561 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
(𝑅 ∈ Ring → 𝑅 ∈ Abel)
 
Theoremringcmn 18562 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ Ring → 𝑅 ∈ CMnd)
 
Theoremringpropd 18563* If two structures have the same group components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 6-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
 
Theoremcrngpropd 18564* If two structures have the same group components (properties), one is a commutative ring iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (𝐾 ∈ CRing ↔ 𝐿 ∈ CRing))
 
Theoremringprop 18565 If two structures have the same ring components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)
(Base‘𝐾) = (Base‘𝐿)    &   (+g𝐾) = (+g𝐿)    &   (.r𝐾) = (.r𝐿)       (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)
 
Theoremisringd 18566* Properties that determine a ring. (Contributed by NM, 2-Aug-2013.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑𝑅 ∈ Grp)    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 · 𝑦) ∈ 𝐵)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))    &   (𝜑1𝐵)    &   ((𝜑𝑥𝐵) → ( 1 · 𝑥) = 𝑥)    &   ((𝜑𝑥𝐵) → (𝑥 · 1 ) = 𝑥)       (𝜑𝑅 ∈ Ring)
 
Theoremiscrngd 18567* Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑𝑅 ∈ Grp)    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 · 𝑦) ∈ 𝐵)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))    &   (𝜑1𝐵)    &   ((𝜑𝑥𝐵) → ( 1 · 𝑥) = 𝑥)    &   ((𝜑𝑥𝐵) → (𝑥 · 1 ) = 𝑥)    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 · 𝑦) = (𝑦 · 𝑥))       (𝜑𝑅 ∈ CRing)
 
Theoremringlz 18568 The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → ( 0 · 𝑋) = 0 )
 
Theoremringrz 18569 The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝑋 · 0 ) = 0 )
 
Theoremringsrg 18570 Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑅 ∈ Ring → 𝑅 ∈ SRing)
 
Theoremring1eq0 18571 If one and zero are equal, then any two elements of a ring are equal. Alternatively, every ring has one distinct from zero except the zero ring containing the single element {0}. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝐵) → ( 1 = 0𝑋 = 𝑌))
 
Theoremring1ne0 18572 If a ring has at least two elements, its one and zero are different. (Contributed by AV, 13-Apr-2019.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 1 < (#‘𝐵)) → 10 )
 
Theoremringinvnz1ne0 18573* In a unitary ring, a left invertible element is different from zero iff 10. (Contributed by FL, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑 → ∃𝑎𝐵 (𝑎 · 𝑋) = 1 )       (𝜑 → (𝑋010 ))
 
Theoremringinvnzdiv 18574* In a unitary ring, a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.) (Revised by Jeff Madsen, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑 → ∃𝑎𝐵 (𝑎 · 𝑋) = 1 )    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑋 · 𝑌) = 0𝑌 = 0 ))
 
Theoremringnegl 18575 Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 33711 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   𝑁 = (invg𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → ((𝑁1 ) · 𝑋) = (𝑁𝑋))
 
Theoremrngnegr 18576 Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 33712 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   𝑁 = (invg𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑋 · (𝑁1 )) = (𝑁𝑋))
 
Theoremringmneg1 18577 Negation of a product in a ring. (mulneg1 10451 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑁 = (invg𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑁𝑋) · 𝑌) = (𝑁‘(𝑋 · 𝑌)))
 
Theoremringmneg2 18578 Negation of a product in a ring. (mulneg2 10452 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑁 = (invg𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 · (𝑁𝑌)) = (𝑁‘(𝑋 · 𝑌)))
 
Theoremringm2neg 18579 Double negation of a product in a ring. (mul2neg 10454 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑁 = (invg𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑁𝑋) · (𝑁𝑌)) = (𝑋 · 𝑌))
 
Theoremringsubdi 18580 Ring multiplication distributes over subtraction. (subdi 10448 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    = (-g𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑋 · (𝑌 𝑍)) = ((𝑋 · 𝑌) (𝑋 · 𝑍)))
 
Theoremrngsubdir 18581 Ring multiplication distributes over subtraction. (subdir 10449 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    = (-g𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 𝑌) · 𝑍) = ((𝑋 · 𝑍) (𝑌 · 𝑍)))
 
Theoremmulgass2 18582 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐵 = (Base‘𝑅)    &    · = (.g𝑅)    &    × = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))
 
Theoremring1 18583 The (smallest) structure representing a zero ring. (Contributed by AV, 28-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}       (𝑍𝑉𝑀 ∈ Ring)
 
Theoremringn0 18584 Rings exist. (Contributed by AV, 29-Apr-2019.)
Ring ≠ ∅
 
Theoremringlghm 18585* Left-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝑥𝐵 ↦ (𝑋 · 𝑥)) ∈ (𝑅 GrpHom 𝑅))
 
Theoremringrghm 18586* Right-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝑥𝐵 ↦ (𝑥 · 𝑋)) ∈ (𝑅 GrpHom 𝑅))
 
Theoremgsummulc1 18587* A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐴𝑉)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )       (𝜑 → (𝑅 Σg (𝑘𝐴 ↦ (𝑋 · 𝑌))) = ((𝑅 Σg (𝑘𝐴𝑋)) · 𝑌))
 
Theoremgsummulc2 18588* A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐴𝑉)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )       (𝜑 → (𝑅 Σg (𝑘𝐴 ↦ (𝑌 · 𝑋))) = (𝑌 · (𝑅 Σg (𝑘𝐴𝑋))))
 
Theoremgsummgp0 18589* If one factor in a finite group sum of the multiplicative group of a commutative ring is 0, the whole "sum" (i.e. product) is 0. (Contributed by AV, 3-Jan-2019.)
𝐺 = (mulGrp‘𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   ((𝜑𝑛𝑁) → 𝐴 ∈ (Base‘𝑅))    &   ((𝜑𝑛 = 𝑖) → 𝐴 = 𝐵)    &   (𝜑 → ∃𝑖𝑁 𝐵 = 0 )       (𝜑 → (𝐺 Σg (𝑛𝑁𝐴)) = 0 )
 
Theoremgsumdixp 18590* Distribute a binary product of sums to a sum of binary products in a ring. (Contributed by Mario Carneiro, 8-Mar-2015.) (Revised by AV, 10-Jul-2019.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑𝑥𝐼) → 𝑋𝐵)    &   ((𝜑𝑦𝐽) → 𝑌𝐵)    &   (𝜑 → (𝑥𝐼𝑋) finSupp 0 )    &   (𝜑 → (𝑦𝐽𝑌) finSupp 0 )       (𝜑 → ((𝑅 Σg (𝑥𝐼𝑋)) · (𝑅 Σg (𝑦𝐽𝑌))) = (𝑅 Σg (𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌))))
 
Theoremprdsmgp 18591 The multiplicative monoid of a product is the product of the multiplicative monoids of the factors. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝑀 = (mulGrp‘𝑌)    &   𝑍 = (𝑆Xs(mulGrp ∘ 𝑅))    &   (𝜑𝐼𝑉)    &   (𝜑𝑆𝑊)    &   (𝜑𝑅 Fn 𝐼)       (𝜑 → ((Base‘𝑀) = (Base‘𝑍) ∧ (+g𝑀) = (+g𝑍)))
 
Theoremprdsmulrcl 18592 A structure product of rings has closed binary operation. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    · = (.r𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐹 · 𝐺) ∈ 𝐵)
 
Theoremprdsringd 18593 A product of rings is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Ring)       (𝜑𝑌 ∈ Ring)
 
Theoremprdscrngd 18594 A product of commutative rings is a commutative ring. Since the resulting ring will have zero divisors in all nontrivial cases, this cannot be strengthened much further. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶CRing)       (𝜑𝑌 ∈ CRing)
 
Theoremprds1 18595 Value of the ring unit in a structure family product. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Ring)       (𝜑 → (1r𝑅) = (1r𝑌))
 
Theorempwsring 18596 A structure power of a ring is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ Ring ∧ 𝐼𝑉) → 𝑌 ∈ Ring)
 
Theorempws1 18597 Value of the ring unit in a structure power. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑅s 𝐼)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑉) → (𝐼 × { 1 }) = (1r𝑌))
 
Theorempwscrng 18598 A structure power of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ CRing ∧ 𝐼𝑉) → 𝑌 ∈ CRing)
 
Theorempwsmgp 18599 The multiplicative group of the power structure resembles the power of the multiplicative group. (Contributed by Mario Carneiro, 12-Mar-2015.)
𝑌 = (𝑅s 𝐼)    &   𝑀 = (mulGrp‘𝑅)    &   𝑍 = (𝑀s 𝐼)    &   𝑁 = (mulGrp‘𝑌)    &   𝐵 = (Base‘𝑁)    &   𝐶 = (Base‘𝑍)    &    + = (+g𝑁)    &    = (+g𝑍)       ((𝑅𝑉𝐼𝑊) → (𝐵 = 𝐶+ = ))
 
Theoremimasring 18600* The image structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &    + = (+g𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑𝑅 ∈ Ring)       (𝜑 → (𝑈 ∈ Ring ∧ (𝐹1 ) = (1r𝑈)))
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