P. 202 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20101-20200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	omndmnd 20101	A left-ordered monoid is a monoid. (Contributed by Thierry Arnoux, 13-Mar-2018.)
⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
Theorem	omndtos 20102	A left-ordered monoid is a totally ordered set. (Contributed by Thierry Arnoux, 13-Mar-2018.)
⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
Theorem	omndadd 20103	In an ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))
Theorem	omndaddr 20104	In a right ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ (((oppg‘𝑀) ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑍 + 𝑋) ≤ (𝑍 + 𝑌))
Theorem	omndadd2d 20105	In a commutative left ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ oMnd)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑍)    &   ⊢ (𝜑 → 𝑌 ≤ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ CMnd)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊))
Theorem	omndadd2rd 20106	In a left- and right- ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ oMnd)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑍)    &   ⊢ (𝜑 → 𝑌 ≤ 𝑊)    &   ⊢ (𝜑 → (oppg‘𝑀) ∈ oMnd)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊))
Theorem	submomnd 20107	A submonoid of an ordered monoid is also ordered. (Contributed by Thierry Arnoux, 23-Mar-2018.)
⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (𝑀 ↾s 𝐴) ∈ oMnd)
Theorem	omndmul2 20108	In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ · = (.g‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    ⇒   ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ (𝑁 · 𝑋))
Theorem	omndmul3 20109	In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ · = (.g‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ oMnd)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑃 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ≤ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 0 ≤ 𝑋)    ⇒   ⊢ (𝜑 → (𝑁 · 𝑋) ≤ (𝑃 · 𝑋))
Theorem	omndmul 20110	In a commutative ordered monoid, the ordering is compatible with group power. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ · = (.g‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ oMnd)    &   ⊢ (𝜑 → 𝑀 ∈ CMnd)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    ⇒   ⊢ (𝜑 → (𝑁 · 𝑋) ≤ (𝑁 · 𝑌))
Theorem	ogrpinv0le 20111	In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ≤ = (le‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 ≤ 𝑋 ↔ (𝐼‘𝑋) ≤ 0 ))
Theorem	ogrpsub 20112	In an ordered group, the ordering is compatible with group subtraction. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ≤ = (le‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 − 𝑍) ≤ (𝑌 − 𝑍))
Theorem	ogrpaddlt 20113	In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
Theorem	ogrpaddltbi 20114	In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))
Theorem	ogrpaddltrd 20115	In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → (oppg‘𝐺) ∈ oGrp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 < 𝑌)    ⇒   ⊢ (𝜑 → (𝑍 + 𝑋) < (𝑍 + 𝑌))
Theorem	ogrpaddltrbid 20116	In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 4-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → (oppg‘𝐺) ∈ oGrp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 < 𝑌 ↔ (𝑍 + 𝑋) < (𝑍 + 𝑌)))
Theorem	ogrpsublt 20117	In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 − 𝑍) < (𝑌 − 𝑍))
Theorem	ogrpinv0lt 20118	In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ (𝐼‘𝑋) < 0 ))
Theorem	ogrpinvlt 20119	In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ (((𝐺 ∈ oGrp ∧ (oppg‘𝐺) ∈ oGrp) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝐼‘𝑌) < (𝐼‘𝑋)))
Theorem	gsumle 20120	A finite sum in an ordered monoid is monotonic. This proof would be much easier in an ordered group, where an inverse element would be available. (Contributed by Thierry Arnoux, 13-Mar-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ oMnd)    &   ⊢ (𝜑 → 𝑀 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 ∘r ≤ 𝐺)    ⇒   ⊢ (𝜑 → (𝑀 Σg 𝐹) ≤ (𝑀 Σg 𝐺))
10.3  Rings
10.3.1  Multiplicative Group
Syntax	cmgp 20121	Multiplicative group.
class mulGrp
Definition	df-mgp 20122	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 20363 shows that we get a group if we restrict to the elements that have inverses. This allows to formalize such notions as "the multiplication operation of a ring is a monoid" (ringmgp 20220) or "the multiplicative identity" in terms of the identity of a monoid (df-ur 20163). (Contributed by Mario Carneiro, 21-Dec-2014.)
⊢ mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r‘𝑤)⟩))
Theorem	fnmgp 20123	The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
⊢ mulGrp Fn V
Theorem	mgpval 20124	Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ 𝑀 = (𝑅 sSet ⟨(+g‘ndx), · ⟩)
Theorem	mgpplusg 20125	Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ · = (+g‘𝑀)
Theorem	mgpbas 20126	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)
Theorem	mgpsca 20127	The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 20306. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑆 = (Scalar‘𝑅)    ⇒   ⊢ 𝑆 = (Scalar‘𝑀)
Theorem	mgptset 20128	Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (TopSet‘𝑅) = (TopSet‘𝑀)
Theorem	mgptopn 20129	Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑅)    ⇒   ⊢ 𝐽 = (TopOpen‘𝑀)
Theorem	mgpds 20130	Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝐵 = (dist‘𝑅)    ⇒   ⊢ 𝐵 = (dist‘𝑀)
Theorem	mgpress 20131	Subgroup commutes with the multiplicative group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (mulGrp‘𝑆))
Theorem	prdsmgp 20132	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))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 Fn 𝐼)    ⇒   ⊢ (𝜑 → ((Base‘𝑀) = (Base‘𝑍) ∧ (+g‘𝑀) = (+g‘𝑍)))
10.3.2  Non-unital rings ("rngs") According to Wikipedia, "... in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a [unital] ring, without assuming the existence of a multiplicative identity. The term "rng" (pronounced rung) is meant to suggest that it is a "ring" without "i", i.e. without the requirement for an "identity element"." (see https://en.wikipedia.org/wiki/Rng_(algebra), 28-Mar-2025).
Syntax	crng 20133	Extend class notation with class of all non-unital rings.
class Rng
Definition	df-rng 20134*	Define the class of all non-unital rings. A non-unital ring (or rng, or pseudoring) is a set equipped with two everywhere-defined internal operations, whose first one is an additive abelian group operation and the second one is a multiplicative semigroup operation, and where the addition is left- and right-distributive for the multiplication. Definition of a pseudo-ring in section I.8.1 of [BourbakiAlg1] p. 93 or the definition of a ring in part Preliminaries of [Roman] p. 18. As almost always in mathematics, "non-unital" means "not necessarily unital". Therefore, by talking about a ring (in general) or a non-unital ring the "unital" case is always included. In contrast to a unital ring, the commutativity of addition must be postulated and cannot be proven from the other conditions. (Contributed by AV, 6-Jan-2020.)
⊢ Rng = {𝑓 ∈ Abel ∣ ((mulGrp‘𝑓) ∈ Smgrp ∧ [(Base‘𝑓) / 𝑏][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))}
Theorem	isrng 20135*	The predicate "is a non-unital ring." (Contributed by AV, 6-Jan-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Rng ↔ (𝑅 ∈ Abel ∧ 𝐺 ∈ Smgrp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
Theorem	rngabl 20136	A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
Theorem	rngmgp 20137	A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020.)
⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ Rng → 𝐺 ∈ Smgrp)
Theorem	rngmgpf 20138	Restricted functionality of the multiplicative group on non-unital rings (mgpf 20229 analog). (Contributed by AV, 22-Feb-2025.)
⊢ (mulGrp ↾ Rng):Rng⟶Smgrp
Theorem	rnggrp 20139	A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)
⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
Theorem	rngass 20140	Associative law for the multiplication operation of a non-unital ring. (Contributed by NM, 27-Aug-2011.) (Revised by AV, 13-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍)))
Theorem	rngdi 20141	Distributive law for the multiplication operation of a non-unital ring (left-distributivity). (Contributed by AV, 14-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
Theorem	rngdir 20142	Distributive law for the multiplication operation of a non-unital ring (right-distributivity). (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
Theorem	rngacl 20143	Closure of the addition operation of a non-unital ring. (Contributed by AV, 16-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
Theorem	rng0cl 20144	The zero element of a non-unital ring belongs to its base set. (Contributed by AV, 16-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Rng → 0 ∈ 𝐵)
Theorem	rngcl 20145	Closure of the multiplication operation of a non-unital ring. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵)
Theorem	rnglz 20146	The zero of a non-unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringlz 20274. (Revised by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) = 0 )
Theorem	rngrz 20147	The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 20275. (Revised by AV, 16-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )
Theorem	rngmneg1 20148	Negation of a product in a non-unital ring (mulneg1 11586 analog). In contrast to ringmneg1 20285, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑁‘𝑋) · 𝑌) = (𝑁‘(𝑋 · 𝑌)))
Theorem	rngmneg2 20149	Negation of a product in a non-unital ring (mulneg2 11587 analog). In contrast to ringmneg2 20286, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 · (𝑁‘𝑌)) = (𝑁‘(𝑋 · 𝑌)))
Theorem	rngm2neg 20150	Double negation of a product in a non-unital ring (mul2neg 11589 analog). (Contributed by Mario Carneiro, 4-Dec-2014.) Generalization of ringm2neg 20287. (Revised by AV, 17-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑁‘𝑋) · (𝑁‘𝑌)) = (𝑋 · 𝑌))
Theorem	rngansg 20151	Every additive subgroup of a non-unital ring is normal. (Contributed by AV, 25-Feb-2025.)
⊢ (𝑅 ∈ Rng → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))
Theorem	rngsubdi 20152	Ring multiplication distributes over subtraction. (subdi 11583 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdi 20288. (Revised by AV, 23-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ − = (-g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 · (𝑌 − 𝑍)) = ((𝑋 · 𝑌) − (𝑋 · 𝑍)))
Theorem	rngsubdir 20153	Ring multiplication distributes over subtraction. (subdir 11584 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 20289. (Revised by AV, 23-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ − = (-g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 − 𝑌) · 𝑍) = ((𝑋 · 𝑍) − (𝑌 · 𝑍)))
Theorem	isrngd 20154*	Properties that determine a non-unital ring. (Contributed by AV, 14-Feb-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → + = (+g‘𝑅))    &   ⊢ (𝜑 → · = (.r‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Abel)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))    ⇒   ⊢ (𝜑 → 𝑅 ∈ Rng)
Theorem	rngpropd 20155*	If two structures have the same base set, and the values of their group (addition) and ring (multiplication) operations are equal for all pairs of elements of the base set, one is a non-unital ring iff the other one is. (Contributed by AV, 15-Feb-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ Rng ↔ 𝐿 ∈ Rng))
Theorem	prdsmulrngcl 20156	Closure of the multiplication in a structure product of non-unital rings. (Contributed by Mario Carneiro, 11-Mar-2015.) Generalization of prdsmulrcl 20299. (Revised by AV, 21-Feb-2025.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ · = (.r‘𝑌)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Rng)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵)
Theorem	prdsrngd 20157	A product of non-unital rings is a non-unital ring. (Contributed by AV, 22-Feb-2025.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Rng)    ⇒   ⊢ (𝜑 → 𝑌 ∈ Rng)
Theorem	imasrng 20158*	The image structure of a non-unital ring is a non-unital ring (imasring 20310 analog). (Contributed by AV, 22-Feb-2025.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    ⇒   ⊢ (𝜑 → 𝑈 ∈ Rng)
Theorem	imasrngf1 20159	The image of a non-unital ring under an injection is a non-unital ring (imasmndf1 18744 analog). (Contributed by AV, 22-Feb-2025.)
⊢ 𝑈 = (𝐹 “s 𝑅)    &   ⊢ 𝑉 = (Base‘𝑅)    ⇒   ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Rng) → 𝑈 ∈ Rng)
Theorem	xpsrngd 20160	A product of two non-unital rings is a non-unital ring (xpsmnd 18745 analog). (Contributed by AV, 22-Feb-2025.)
⊢ 𝑌 = (𝑆 ×s 𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ Rng)    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    ⇒   ⊢ (𝜑 → 𝑌 ∈ Rng)
Theorem	qusrng 20161*	The quotient structure of a non-unital ring is a non-unital ring (qusring2 20314 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)
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 20216). 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 20338), and we have "the ring unity is a unit", see 1unit 20354.
Syntax	cur 20162	Extend class notation with ring unity.
class 1r
Definition	df-ur 20163	Define the multiplicative identity, i.e., the monoid identity (df-0g 17404) of the multiplicative monoid (df-mgp 20122) 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 20165, 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 20164	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 20165*	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 20166*	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 20167	Extend class notation with the class of all semirings.
class SRing
Definition	df-srg 20168*	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 20216), 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 20169*	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 20170	A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd)
Theorem	srgmnd 20171	A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
Theorem	srgmgp 20172	A semiring is a monoid under multiplication. (Contributed by Thierry Arnoux, 21-Mar-2018.)
⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd)
Theorem	srgdilem 20173	Lemma for srgdi 20178 and srgdir 20179. (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 20174	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 20175	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 20176*	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 20177	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 20178	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 20179	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 20180	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 20181	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 20182	Lemma for srglidm 20183 and srgridm 20184. (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 20183	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 20184	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 20185*	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 20186	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 20187	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 20188	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 20189	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 20190*	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 20191*	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 20261 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 20192*	Restricted commutativity of the addition in a ring-like structure. This (formerly) part of the proof for ringcom 20261 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 20193	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 20191 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 20194	Lemma for srgcom4 20195. This (formerly) part of the proof for ringcom 20261 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 20195	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 20196	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)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ∗ = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
Theorem	srgen1zr 20197	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 20198	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 20199	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 20200	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)))