P. 139 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13801-13900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iscmnd 13801*	Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    ⇒   ⊢ (𝜑 → 𝐺 ∈ CMnd)
Theorem	isabld 13802*	Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.)
⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    ⇒   ⊢ (𝜑 → 𝐺 ∈ Abel)
Theorem	isabli 13803*	Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.)
⊢ 𝐺 ∈ Grp    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    ⇒   ⊢ 𝐺 ∈ Abel
Theorem	cmnmnd 13804	A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
Theorem	cmncom 13805	A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Theorem	ablcom 13806	An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Theorem	cmn32 13807	Commutative/associative law for commutative monoids. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))
Theorem	cmn4 13808	Commutative/associative law for commutative monoids. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))
Theorem	cmn12 13809	Commutative/associative law for commutative monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
Theorem	abl32 13810	Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))
Theorem	cmnmndd 13811	A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
⊢ (𝜑 → 𝐺 ∈ CMnd)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Mnd)
Theorem	rinvmod 13812*	Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6170. (Contributed by AV, 31-Dec-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 (𝐴 + 𝑤) = 0 )
Theorem	ablinvadd 13813	The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁‘𝑋) + (𝑁‘𝑌)))
Theorem	ablsub2inv 13814	Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑁‘𝑋) − (𝑁‘𝑌)) = (𝑌 − 𝑋))
Theorem	ablsubadd 13815	Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑌) = 𝑍 ↔ (𝑌 + 𝑍) = 𝑋))
Theorem	ablsub4 13816	Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 + 𝑌) − (𝑍 + 𝑊)) = ((𝑋 − 𝑍) + (𝑌 − 𝑊)))
Theorem	abladdsub4 13817	Abelian group addition/subtraction law. (Contributed by NM, 31-Mar-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 + 𝑌) = (𝑍 + 𝑊) ↔ (𝑋 − 𝑍) = (𝑊 − 𝑌)))
Theorem	abladdsub 13818	Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) − 𝑍) = ((𝑋 − 𝑍) + 𝑌))
Theorem	ablpncan2 13819	Cancellation law for subtraction in an Abelian group. (Contributed by NM, 2-Oct-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑌) − 𝑋) = 𝑌)
Theorem	ablpncan3 13820	A cancellation law for Abelian groups. (Contributed by NM, 23-Mar-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 + (𝑌 − 𝑋)) = 𝑌)
Theorem	ablsubsub 13821	Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 − (𝑌 − 𝑍)) = ((𝑋 − 𝑌) + 𝑍))
Theorem	ablsubsub4 13822	Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = (𝑋 − (𝑌 + 𝑍)))
Theorem	ablpnpcan 13823	Cancellation law for mixed addition and subtraction. (pnpcan 8353 analog.) (Contributed by NM, 29-May-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = (𝑌 − 𝑍))
Theorem	ablnncan 13824	Cancellation law for group subtraction. (nncan 8343 analog.) (Contributed by NM, 7-Apr-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = 𝑌)
Theorem	ablsub32 13825	Swap the second and third terms in a double group subtraction. (Contributed by NM, 7-Apr-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = ((𝑋 − 𝑍) − 𝑌))
Theorem	ablnnncan 13826	Cancellation law for group subtraction. (nnncan 8349 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 − (𝑌 − 𝑍)) − 𝑍) = (𝑋 − 𝑌))
Theorem	ablnnncan1 13827	Cancellation law for group subtraction. (nnncan1 8350 analog.) (Contributed by NM, 7-Apr-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = (𝑍 − 𝑌))
Theorem	ablsubsub23 13828	Swap subtrahend and result of group subtraction. (Contributed by NM, 14-Dec-2007.) (Revised by AV, 7-Oct-2021.)
⊢ 𝑉 = (Base‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐴 − 𝐶) = 𝐵))
Theorem	ghmfghm 13829*	The function fulfilling the conditions of ghmgrp 13621 is a group homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑌 = (Base‘𝐻)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+g‘𝐻)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
Theorem	ghmcmn 13830*	The image of a commutative monoid 𝐺 under a group homomorphism 𝐹 is a commutative monoid. (Contributed by Thierry Arnoux, 26-Jan-2020.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑌 = (Base‘𝐻)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+g‘𝐻)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    ⇒   ⊢ (𝜑 → 𝐻 ∈ CMnd)
Theorem	ghmabl 13831*	The image of an abelian group 𝐺 under a group homomorphism 𝐹 is an abelian group. (Contributed by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑌 = (Base‘𝐻)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+g‘𝐻)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    ⇒   ⊢ (𝜑 → 𝐻 ∈ Abel)
Theorem	invghm 13832	The inversion map is a group automorphism if and only if the group is abelian. (In general it is only a group homomorphism into the opposite group, but in an abelian group the opposite group coincides with the group itself.) (Contributed by Mario Carneiro, 4-May-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺))
Theorem	eqgabl 13833	Value of the subgroup coset equivalence relation on an abelian group. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝑆)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐵 − 𝐴) ∈ 𝑆)))
Theorem	qusecsub 13834	Two subgroup cosets are equal if and only if the difference of their representatives is a member of the subgroup. (Contributed by AV, 7-Mar-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝑆)    ⇒   ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ([𝑋] ∼ = [𝑌] ∼ ↔ (𝑌 − 𝑋) ∈ 𝑆))
Theorem	subgabl 13835	A subgroup of an abelian group is also abelian. (Contributed by Mario Carneiro, 3-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)
Theorem	subcmnd 13836	A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ (𝜑 → 𝐻 = (𝐺 ↾s 𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐻 ∈ Mnd)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐻 ∈ CMnd)
Theorem	ablnsg 13837	Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))
Theorem	ablressid 13838	A commutative group restricted to its base set is a commutative group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 13070. (Contributed by Jim Kingdon, 5-May-2025.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Abel → (𝐺 ↾s 𝐵) ∈ Abel)
Theorem	imasabl 13839*	The image structure of an abelian group is an abelian group (imasgrp 13614 analog). (Contributed by AV, 22-Feb-2025.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → + = (+g‘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 → 𝑅 ∈ Abel)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘ 0 ) = (0g‘𝑈)))
7.2.5.2  Group sum operation
Theorem	gsumfzreidx 13840	Re-index a finite group sum using a bijection. Corresponds to the first equation in [Lang] p. 5 with 𝑀 = 1. (Contributed by AV, 26-Dec-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    &   ⊢ (𝜑 → 𝐻:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))
Theorem	gsumfzsubmcl 13841	Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 30-Aug-2025.)
⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺))    &   ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝑆)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆)
Theorem	gsumfzmptfidmadd 13842*	The sum of two group sums expressed as mappings with finite domain. (Contributed by AV, 23-Jul-2019.) (Revised by Jim Kingdon, 31-Aug-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐷 ∈ 𝐵)    &   ⊢ 𝐹 = (𝑥 ∈ (𝑀...𝑁) ↦ 𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ (𝑀...𝑁) ↦ 𝐷)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝑀...𝑁) ↦ (𝐶 + 𝐷))) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
Theorem	gsumfzmptfidmadd2 13843*	The sum of two group sums expressed as mappings with finite domain, using a function operation. (Contributed by AV, 23-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐷 ∈ 𝐵)    &   ⊢ 𝐹 = (𝑥 ∈ (𝑀...𝑁) ↦ 𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ (𝑀...𝑁) ↦ 𝐷)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘𝑓 + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
Theorem	gsumfzconst 13844*	Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Jim Kingdon, 6-Sep-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) = (((𝑁 − 𝑀) + 1) · 𝑋))
Theorem	gsumfzconstf 13845*	Sum of a constant series. (Contributed by Thierry Arnoux, 5-Jul-2017.)
⊢ Ⅎ𝑘𝑋    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) = (((𝑁 − 𝑀) + 1) · 𝑋))
Theorem	gsumfzmhm 13846	Apply a monoid homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 8-Sep-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐻 ∈ Mnd)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻))    &   ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
Theorem	gsumfzmhm2 13847*	Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 9-Sep-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐻 ∈ Mnd)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ (𝐺 MndHom 𝐻))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑋 ∈ 𝐵)    &   ⊢ (𝑥 = 𝑋 → 𝐶 = 𝐷)    &   ⊢ (𝑥 = (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑋)) → 𝐶 = 𝐸)    ⇒   ⊢ (𝜑 → (𝐻 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐷)) = 𝐸)
Theorem	gsumfzsnfd 13848*	Group sum of a singleton, deduction form, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Thierry Arnoux, 28-Mar-2018.) (Revised by AV, 11-Dec-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐶)    &   ⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐶    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)
7.3  Rings
7.3.1  Multiplicative Group
Syntax	cmgp 13849	Multiplicative group.
class mulGrp
Definition	df-mgp 13850	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 we get a group if we restrict to the elements that have inverses. This allows us to formalize such notions as "the multiplication operation of a ring is a monoid" or "the multiplicative identity" in terms of the identity of a monoid (df-ur 13889). (Contributed by Mario Carneiro, 21-Dec-2014.)
⊢ mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r‘𝑤)⟩))
Theorem	fnmgp 13851	The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
⊢ mulGrp Fn V
Theorem	mgpvalg 13852	Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝑀 = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
Theorem	mgpplusgg 13853	Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → · = (+g‘𝑀))
Theorem	mgpex 13854	Existence of the multiplication group. If 𝑅 is known to be a semiring, see srgmgp 13897. (Contributed by Jim Kingdon, 10-Jan-2025.)
⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝑀 ∈ V)
Theorem	mgpbasg 13855	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)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝐵 = (Base‘𝑀))
Theorem	mgpscag 13856	The multiplication monoid has the same (if any) scalars as the original ring. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑆 = (Scalar‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (Scalar‘𝑀))
Theorem	mgptsetg 13857	Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → (TopSet‘𝑅) = (TopSet‘𝑀))
Theorem	mgptopng 13858	Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝐽 = (TopOpen‘𝑀))
Theorem	mgpdsg 13859	Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝐵 = (dist‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝐵 = (dist‘𝑀))
Theorem	mgpress 13860	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))
7.3.2  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 13861	Extend class notation with class of all non-unital rings.
class Rng
Definition	df-rng 13862*	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 13863*	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 13864	A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
Theorem	rngmgp 13865	A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020.)
⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ Rng → 𝐺 ∈ Smgrp)
Theorem	rngmgpf 13866	Restricted functionality of the multiplicative group on non-unital rings (mgpf 13940 analog). (Contributed by AV, 22-Feb-2025.)
⊢ (mulGrp ↾ Rng):Rng⟶Smgrp
Theorem	rnggrp 13867	A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)
⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
Theorem	rngass 13868	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 13869	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 13870	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 13871	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 13872	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 13873	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 13874	The zero of a non-unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringlz 13972. (Revised by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) = 0 )
Theorem	rngrz 13875	The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 13973. (Revised by AV, 16-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )
Theorem	rngmneg1 13876	Negation of a product in a non-unital ring (mulneg1 8509 analog). In contrast to ringmneg1 13982, 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 13877	Negation of a product in a non-unital ring (mulneg2 8510 analog). In contrast to ringmneg2 13983, 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 13878	Double negation of a product in a non-unital ring (mul2neg 8512 analog). (Contributed by Mario Carneiro, 4-Dec-2014.) Generalization of ringm2neg 13984. (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 13879	Every additive subgroup of a non-unital ring is normal. (Contributed by AV, 25-Feb-2025.)
⊢ (𝑅 ∈ Rng → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))
Theorem	rngsubdi 13880	Ring multiplication distributes over subtraction. (subdi 8499 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdi 13985. (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 13881	Ring multiplication distributes over subtraction. (subdir 8500 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 13986. (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 13882*	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	rngressid 13883	A non-unital ring restricted to its base set is a non-unital ring. It will usually be the original non-unital ring exactly, of course, but to show that needs additional conditions such as those in strressid 13070. (Contributed by Jim Kingdon, 5-May-2025.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Rng → (𝐺 ↾s 𝐵) ∈ Rng)
Theorem	rngpropd 13884*	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	imasrng 13885*	The image structure of a non-unital ring is a non-unital ring (imasring 13993 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 13886	The image of a non-unital ring under an injection is a non-unital ring. (Contributed by AV, 22-Feb-2025.)
⊢ 𝑈 = (𝐹 “s 𝑅)    &   ⊢ 𝑉 = (Base‘𝑅)    ⇒   ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Rng) → 𝑈 ∈ Rng)
Theorem	qusrng 13887*	The quotient structure of a non-unital ring is a non-unital ring (qusring2 13995 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)
7.3.3  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 13927). 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 https://us.metamath.org/mpeuni/df-unit.html 13927 in set.mm), and we have "the ring unity is a unit", see https://us.metamath.org/mpeuni/1unit.html 13927.
Syntax	cur 13888	Extend class notation with ring unity.
class 1r
Definition	df-ur 13889	Define the multiplicative identity, i.e., the monoid identity (df-0g 13257) of the multiplicative monoid (df-mgp 13850) 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 dfur2g 13891, 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	ringidvalg 13890	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)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 1 = (0g‘𝐺))
Theorem	dfur2g 13891*	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)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 1 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥))))
7.3.4  Semirings
Syntax	csrg 13892	Extend class notation with the class of all semirings.
class SRing
Definition	df-srg 13893*	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, 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 13894*	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 13895	A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd)
Theorem	srgmnd 13896	A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.)
⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
Theorem	srgmgp 13897	A semiring is a monoid under multiplication. (Contributed by Thierry Arnoux, 21-Mar-2018.)
⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd)
Theorem	srgdilem 13898	Lemma for srgdi 13903 and srgdir 13904. (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 13899	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 13900	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)))