P. 205 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20401-20500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xpsrngd 20401	A product of two non-unital rings is a non-unital ring (xpsmnd 18971 analog). (Contributed by AV, 22-Feb-2025.)
⊢ 𝑌 = (𝑆 ×s 𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ Rng)    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    ⇒   ⊢ (𝜑 → 𝑌 ∈ Rng)
Theorem	qusrng 20402*	The quotient structure of a non-unital ring is a non-unital ring (qusring2 20564 analog). (Contributed by AV, 23-Feb-2025.)
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → ∼ Er 𝑉)    &   ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞)))    &   ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    ⇒   ⊢ (𝜑 → 𝑈 ∈ Rng)
Theorem	rng1zrlem 20403	Lemma for rng1zr 20404 and srg1zr 20441. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 18-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ∗ = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
Theorem	rng1zr 20404	The only ring 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, 18-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ∗ = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Rng ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
Theorem	rngen1zr 20405	The only ring 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, 18-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ∗ = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Rng ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
Theorem	rngen1zr0 20406	The only ring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 15-Feb-2010.) (Revised by AV, 18-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ∗ = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨ 0 , 0 ⟩, 0 ⟩} ∧ ∗ = {⟨⟨ 0 , 0 ⟩, 0 ⟩})))
10.3.3  Ring unity (multiplicative identity) In Wikipedia "Identity element", see https://en.wikipedia.org/wiki/Identity_element (18-Jan-2025): "... an identity with respect to multiplication is called a multiplicative identity (often denoted as 1). ... The distinction between additive and multiplicative identity is used most often for sets that support both binary operations, such as rings, integral domains, and fields. The multiplicative identity is often called unity in the latter context (a ring with unity). This should not be confused with a unit in ring theory, which is any element having a multiplicative inverse. By its own definition, unity itself is necessarily a unit." Calling the multiplicative identity of a ring a unity is taken from the definition of a ring with unity in section 17.3 of [BeauregardFraleigh] p. 135, "A ring ( R , + , . ) is a ring with unity if R is not the zero ring and ( R , . ) is a monoid. In this case, the identity element of ( R , . ) is denoted by 1 and is called the unity of R." This definition of a "ring with unity" corresponds to our definition of a unital ring (see df-ring 20461). Some authors call the multiplicative identity "unit" or "unit element" (for example in section I, 2.2 of [BourbakiAlg1] p. 14, definition in section 1.3 of [Hall] p. 4, or in section I, 1 of [Lang] p. 3), whereas other authors use the term "unit" for an element having a multiplicative inverse (for example in section 17.3 of [BeauregardFraleigh] p. 135, in definition in [Roman] p. 26, or even in section II, 1 of [Lang] p. 84). Sometimes, the multiplicative identity is simply called "one" (see, for example, chapter 8 in [Schechter] p. 180). To avoid this ambiguity of the term "unit", also mentioned in Wikipedia, we call the multiplicative identity of a structure with a multiplication (usually a ring) a "ring unity", or straightly "multiplicative identity". The term "unit" will be used for an element having a multiplicative inverse (see df-unit 20588), and we have "the ring unity is a unit", see 1unit 20604.
Syntax	cur 20407	Extend class notation with ring unity.
class 1r
Definition	df-ur 20408	Define the multiplicative identity, i.e., the monoid identity (df-0g 17612) of the multiplicative monoid (df-mgp 20361) of a ring-like structure. This multiplicative identity is also called "ring unity" or "unity element". This definition works by transferring the multiplicative operation from the .r slot to the +g slot and then looking at the element which is then the 0g element, that is an identity with respect to the operation which started out in the .r slot. See also dfur2 20410, 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)
Theorem	ringidval 20409	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 = (1r‘𝑅)    ⇒   ⊢ 1 = (0g‘𝐺)
Theorem	dfur2 20410*	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)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ 1 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥)))
Theorem	ringurd 20411*	Deduce the unity element of a ring from its properties. (Contributed by Thierry Arnoux, 6-Sep-2016.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → · = (.r‘𝑅))    &   ⊢ (𝜑 → 1 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 · 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 · 1 ) = 𝑥)    ⇒   ⊢ (𝜑 → 1 = (1r‘𝑅))
10.3.4  Semirings
Syntax	csrg 20412	Extend class notation with the class of all semirings.
class SRing
Definition	df-srg 20413*	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. Like with rings (df-ring 20461), the additive identity is an absorbing element of the multiplicative law, but in the case of semirings, this has to be part of the definition, as it 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][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡][(0g‘𝑓) / 𝑛]∀𝑥 ∈ 𝑟 (∀𝑦 ∈ 𝑟 ∀𝑧 ∈ 𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))}
Theorem	issrg 20414*	The predicate "is a semiring". (Contributed by Thierry Arnoux, 21-Mar-2018.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
Theorem	srgcmn 20415	A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd)
Theorem	srgmnd 20416	A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
Theorem	srgmgp 20417	A semiring is a monoid under multiplication. (Contributed by Thierry Arnoux, 21-Mar-2018.)
⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd)
Theorem	srgdilem 20418	Lemma for srgdi 20423 and srgdir 20424. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))
Theorem	srgcl 20419	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)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵)
Theorem	srgass 20420	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)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍)))
Theorem	srgideu 20421*	The unity 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)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ SRing → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑢) = 𝑥))
Theorem	srgfcl 20422	Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by AV, 24-Aug-2021.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → · :(𝐵 × 𝐵)⟶𝐵)
Theorem	srgdi 20423	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)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
Theorem	srgdir 20424	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)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
Theorem	srgidcl 20425	The unity 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 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ SRing → 1 ∈ 𝐵)
Theorem	srg0cl 20426	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 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ SRing → 0 ∈ 𝐵)
Theorem	srgidmlem 20427	Lemma for srglidm 20428 and srgridm 20429. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋))
Theorem	srglidm 20428	The unity 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)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → ( 1 · 𝑋) = 𝑋)
Theorem	srgridm 20429	The unity 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)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋)
Theorem	issrgid 20430*	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)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ SRing → ((𝐼 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥)) ↔ 1 = 𝐼))
Theorem	srgacl 20431	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)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
Theorem	srgcom 20432	Commutativity of the additive group of a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Theorem	srgrz 20433	The zero of a semiring is a right-absorbing element. (Contributed by Thierry Arnoux, 1-Apr-2018.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )
Theorem	srglz 20434	The zero of a semiring is a left-absorbing element. (Contributed by AV, 23-Aug-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) = 0 )
Theorem	srgisid 20435*	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)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍 · 𝑥) = 𝑍)    ⇒   ⊢ (𝜑 → 𝑍 = 0 )
Theorem	o2timesd 20436*	An element of a ring-like structure plus itself is two times the element. "Two" in such a structure is the sum of the unity element with itself. This (formerly) part of the proof for ringcom 20509 depends on the (right) distributivity and the existence of a (left) multiplicative identity only. (Contributed by Gérard Lang, 4-Dec-2014.) (Revised by AV, 1-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))    &   ⊢ (𝜑 → 1 ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 1 · 𝑥) = 𝑥)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑋) = (( 1 + 1 ) · 𝑋))
Theorem	rglcom4d 20437*	Restricted commutativity of the addition in a ring-like structure. This (formerly) part of the proof for ringcom 20509 depends on the closure of the addition, the (left and right) distributivity and the existence of a (left) multiplicative identity only. (Contributed by Gérard Lang, 4-Dec-2014.) (Revised by AV, 1-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))    &   ⊢ (𝜑 → 1 ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 1 · 𝑥) = 𝑥)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 + 𝑋) + (𝑌 + 𝑌)) = ((𝑋 + 𝑌) + (𝑋 + 𝑌)))
Theorem	srgo2times 20438	A semiring element plus itself is two times the element. "Two" in an arbitrary (unital) semiring is the sum of the unity element with itself. (Contributed by AV, 24-Aug-2021.) Variant of o2timesd 20436 for semirings. (Revised by AV, 1-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ 𝐴 ∈ 𝐵) → (𝐴 + 𝐴) = (( 1 + 1 ) · 𝐴))
Theorem	srgcom4lem 20439	Lemma for srgcom4 20440. This (formerly) part of the proof for ringcom 20509 is applicable for semirings (without using the commutativity of the addition given per definition of a semiring). (Contributed by Gérard Lang, 4-Dec-2014.) (Revised by AV, 1-Feb-2025.) (Proof modification is discouraged.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑋) + (𝑌 + 𝑌)) = ((𝑋 + 𝑌) + (𝑋 + 𝑌)))
Theorem	srgcom4 20440	Restricted commutativity of the addition in semirings (without using the commutativity of the addition given per definition of a semiring). (Contributed by AV, 1-Feb-2025.) (Proof modification is discouraged.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑋 + 𝑌)) + 𝑌) = ((𝑋 + (𝑌 + 𝑋)) + 𝑌))
Theorem	srg1zr 20441	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.) (Proof shortened by AV, 19-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ∗ = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
Theorem	srgen1zr0 20442	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)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ∗ = (.r‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
Theorem	srgmulgass 20443	An associative property between group multiple and ring multiplication for semirings. (Contributed by AV, 23-Aug-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.g‘𝑅)    &   ⊢ × = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))
Theorem	srgpcomp 20444	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)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    ⇒   ⊢ (𝜑 → ((𝐾 ↑ 𝐵) × 𝐴) = (𝐴 × (𝐾 ↑ 𝐵)))
Theorem	srgpcompp 20445	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)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × (𝐾 ↑ 𝐵)) × 𝐴) = (((𝑁 + 1) ↑ 𝐴) × (𝐾 ↑ 𝐵)))
Theorem	srgpcomppsc 20446	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)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ · = (.g‘𝑅)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝐶 · ((𝑁 ↑ 𝐴) × (𝐾 ↑ 𝐵))) × 𝐴) = (𝐶 · (((𝑁 + 1) ↑ 𝐴) × (𝐾 ↑ 𝐵))))
Theorem	srglmhm 20447*	Left-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism, analogous to ringlghm 20543. (Contributed by AV, 23-Aug-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥)) ∈ (𝑅 MndHom 𝑅))
Theorem	srgrmhm 20448*	Right-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism, analogous to ringrghm 20544. (Contributed by AV, 23-Aug-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋)) ∈ (𝑅 MndHom 𝑅))
Theorem	srgsummulcr 20449*	A finite semiring sum multiplied by a constant, analogous to gsummulc1 20545. (Contributed by AV, 23-Aug-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 )    ⇒   ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑋 · 𝑌))) = ((𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) · 𝑌))
Theorem	sgsummulcl 20450*	A finite semiring sum multiplied by a constant, analogous to gsummulc2 20546. (Contributed by AV, 23-Aug-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 )    ⇒   ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑌 · 𝑋))) = (𝑌 · (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋))))
Theorem	srg1expzeq1 20451	The exponentiation (by a nonnegative integer) of the multiplicative identity of a semiring, analogous to mulgnn0z 19311. (Contributed by AV, 25-Nov-2019.)
⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) → (𝑁 · 1 ) = 1 )
10.3.4.1  The binomial theorem for semirings In this section, we prove the binomial theorem for semirings, srgbinom 20457, which is a generalization of the binomial theorem for complex numbers, binom 15999: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑𝑘) · (𝐵↑(𝑁 − 𝑘)). Note that the binomial theorem also holds in the non-unital case (that is, in a "rg") and actually, the additive identity 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 ℝ. Special cases of the binomial theorem are csrgbinom 20458 (binomial theorem for commutative semirings) and crngbinom 20565 (binomial theorem for commutative rings).
Theorem	srgbinomlem1 20452	Lemma 1 for srgbinomlem 20456. (Contributed by AV, 23-Aug-2019.)
⊢ 𝑆 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ · = (.g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆)
Theorem	srgbinomlem2 20453	Lemma 2 for srgbinomlem 20456. (Contributed by AV, 23-Aug-2019.)
⊢ 𝑆 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ · = (.g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ ((𝜑 ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → (𝐶 · ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵))) ∈ 𝑆)
Theorem	srgbinomlem3 20454*	Lemma 3 for srgbinomlem 20456. (Contributed by AV, 23-Aug-2019.) (Proof shortened by AV, 27-Oct-2019.)
⊢ 𝑆 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ · = (.g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜓 → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ((𝑁 ↑ (𝐴 + 𝐵)) × 𝐴) = (𝑅 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ ((𝑁C𝑘) · ((((𝑁 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))
Theorem	srgbinomlem4 20455*	Lemma 4 for srgbinomlem 20456. (Contributed by AV, 24-Aug-2019.) (Proof shortened by AV, 19-Nov-2019.)
⊢ 𝑆 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ · = (.g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜓 → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ((𝑁 ↑ (𝐴 + 𝐵)) × 𝐵) = (𝑅 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ ((𝑁C(𝑘 − 1)) · ((((𝑁 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))
Theorem	srgbinomlem 20456*	Lemma for srgbinom 20457. Inductive step, analogous to binomlem 15998. (Contributed by AV, 24-Aug-2019.)
⊢ 𝑆 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ · = (.g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜓 → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ((𝑁 + 1) ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ (((𝑁 + 1)C𝑘) · ((((𝑁 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))
Theorem	srgbinom 20457*	The binomial theorem for commuting elements of a semiring: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑𝑘) · (𝐵↑(𝑁 − 𝑘)) (generalization of binom 15999). (Contributed by AV, 24-Aug-2019.)
⊢ 𝑆 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ · = (.g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    ⇒   ⊢ (((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))
Theorem	csrgbinom 20458*	The binomial theorem for commutative semirings. (Contributed by AV, 24-Aug-2019.)
⊢ 𝑆 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ · = (.g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ ↑ = (.g‘𝐺)    ⇒   ⊢ (((𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵))))))
10.3.5  Unital rings
Syntax	crg 20459	Extend class notation with class of all (unital) rings.
class Ring
Syntax	ccrg 20460	Extend class notation with class of all (unital) commutative rings.
class CRing
Definition	df-ring 20461*	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 20509), 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. Therefore, it can be shown that a unital ring is a non-unital ring (ringrng 20514) only after ringabl 20510 was proven. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 27-Dec-2014.)
⊢ Ring = {𝑓 ∈ Grp ∣ ((mulGrp‘𝑓) ∈ Mnd ∧ [(Base‘𝑓) / 𝑟][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡]∀𝑥 ∈ 𝑟 ∀𝑦 ∈ 𝑟 ∀𝑧 ∈ 𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))}
Definition	df-cring 20462	Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd}
Theorem	isring 20463*	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)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
Theorem	ringgrp 20464	A ring is a group. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by NM, 15-Sep-2011.)
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
Theorem	ringmgp 20465	A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Mnd)
Theorem	iscrng 20466	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))
Theorem	crngmgp 20467	A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd)
Theorem	ringbn0 20468	The base set of a ring is not empty. (Contributed by FL, 24-Jan-2010.) (Revised by AV, 25-Aug-2011.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Ring → 𝐵 ≠ ∅)
Theorem	ringgrpd 20469	A ring is a group. (Contributed by SN, 16-May-2024.)
⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Grp)
Theorem	ringmnd 20470	A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
Theorem	ringmgm 20471	A ring is a magma. (Contributed by AV, 31-Jan-2020.)
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mgm)
Theorem	crngring 20472	A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
Theorem	crngringd 20473	A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by SN, 16-May-2024.)
⊢ (𝜑 → 𝑅 ∈ CRing)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Ring)
Theorem	crnggrpd 20474	A commutative ring is a group. (Contributed by SN, 16-May-2024.)
⊢ (𝜑 → 𝑅 ∈ CRing)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Grp)
Theorem	mgpf 20475	Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
⊢ (mulGrp ↾ Ring):Ring⟶Mnd
Theorem	ringdilem 20476	Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))
Theorem	ringcl 20477	Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵)
Theorem	crngcom 20478	A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = (𝑌 · 𝑋))
Theorem	iscrng2 20479*	A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 · 𝑦) = (𝑦 · 𝑥)))
Theorem	ringass 20480	Associative law for multiplication in a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍)))
Theorem	ringideu 20481*	The unity element of a ring is unique. (Contributed by NM, 4-Apr-2009.) (Revised by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑢) = 𝑥))
Theorem	crngcomd 20482	Multiplication is commutative in a commutative ring. (Contributed by SN, 8-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑌 · 𝑋))
Theorem	crngbascntr 20483	The base set of a commutative ring is its center. (Contributed by SN, 21-Mar-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑍 = (Cntr‘(mulGrp‘𝐺))    ⇒   ⊢ (𝐺 ∈ CRing → 𝐵 = 𝑍)
Theorem	ringcld 20484	Closure of the multiplication operation of a ring. (Contributed by SN, 29-Jul-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)
Theorem	ringassd 20485	Associative law for multiplication in a ring. (Contributed by SN, 14-Aug-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍)))
Theorem	crng12d 20486	Commutative/associative law that swaps the first two factors in a triple product in a commutative ring. See also mul12d 11519. (Contributed by SN, 8-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 · (𝑌 · 𝑍)) = (𝑌 · (𝑋 · 𝑍)))
Theorem	crng32d 20487	Commutative/associative law that swaps the last two factors in a triple product in a commutative ring. See also mul32d 11520. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 · 𝑌) · 𝑍) = ((𝑋 · 𝑍) · 𝑌))
Theorem	crng4 20488	Commutative/associative law for commutative rings. See also mul4d 11522. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by AV, 20-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 · 𝑌) · (𝑍 · 𝑈)) = ((𝑋 · 𝑍) · (𝑌 · 𝑈)))
Theorem	ringdi 20489	Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
Theorem	ringdir 20490	Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
Theorem	ringdid 20491	Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
Theorem	ringdird 20492	Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
Theorem	ringdi22 20493	Expand the product of two sums in a ring. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))
Theorem	ringidcl 20494	The unity element of a ring belongs to the base set of the ring. (Contributed by FL, 12-Feb-2010.) (Revised by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵)
Theorem	ringidcld 20495	The unity element of a ring belongs to the base set of the ring, deduction version. (Contributed by SN, 16-Oct-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 1 ∈ 𝐵)
Theorem	ring0cl 20496	The zero element of a ring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵)
Theorem	ringidmlem 20497	Lemma for ringlidm 20498 and ringridm 20499. (Contributed by FL, 18-Feb-2010.) (Revised by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋))
Theorem	ringlidm 20498	The unity element of a ring is a left multiplicative identity. (Contributed by FL, 18-Apr-2010.) (Revised by NM, 15-Sep-2011.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 1 · 𝑋) = 𝑋)
Theorem	ringridm 20499	The unity element of a ring is a right multiplicative identity. (Contributed by FL, 18-Apr-2010.) (Revised by NM, 15-Sep-2011.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋)
Theorem	isringid 20500*	Properties showing that an element 𝐼 is the unity element of a ring. (Contributed by NM, 7-Aug-2013.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → ((𝐼 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥)) ↔ 1 = 𝐼))