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	ghmmulg 13801	A group homomorphism preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ × = (.g‘𝐻)    ⇒   ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋)))
Theorem	ghmrn 13802	The range of a homomorphism is a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ∈ (SubGrp‘𝑇))
Theorem	0ghm 13803	The constant zero linear function between two groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 0 = (0g‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁))
Theorem	idghm 13804	The identity homomorphism on a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺))
Theorem	resghm 13805	Restriction of a homomorphism to a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ 𝑈 = (𝑆 ↾s 𝑋)    ⇒   ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 GrpHom 𝑇))
Theorem	resghm2 13806	One direction of resghm2b 13807. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
⊢ 𝑈 = (𝑇 ↾s 𝑋)    ⇒   ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝑋 ∈ (SubGrp‘𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
Theorem	resghm2b 13807	Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
⊢ 𝑈 = (𝑇 ↾s 𝑋)    ⇒   ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
Theorem	ghmghmrn 13808	A group homomorphism from 𝐺 to 𝐻 is also a group homomorphism from 𝐺 to its image in 𝐻. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by AV, 26-Aug-2021.)
⊢ 𝑈 = (𝑇 ↾s ran 𝐹)    ⇒   ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑈))
Theorem	ghmco 13809	The composition of group homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈))
Theorem	ghmima 13810	The image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑆)) → (𝐹 “ 𝑈) ∈ (SubGrp‘𝑇))
Theorem	ghmpreima 13811	The inverse image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆))
Theorem	ghmeql 13812	The equalizer of two group homomorphisms is a subgroup. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆))
Theorem	ghmnsgima 13813	The image of a normal subgroup under a surjective homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 𝑌 = (Base‘𝑇)    ⇒   ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹 “ 𝑈) ∈ (NrmSGrp‘𝑇))
Theorem	ghmnsgpreima 13814	The inverse image of a normal subgroup under a homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (NrmSGrp‘𝑆))
Theorem	ghmker 13815	The kernel of a homomorphism is a normal subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 0 = (0g‘𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝑆))
Theorem	ghmeqker 13816	Two source points map to the same destination point under a group homomorphism iff their difference belongs to the kernel. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ − = (-g‘𝑆)    ⇒   ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → ((𝐹‘𝑈) = (𝐹‘𝑉) ↔ (𝑈 − 𝑉) ∈ 𝐾))
Theorem	f1ghm0to0 13817	If a group homomorphism 𝐹 is injective, it maps the zero of one group (and only the zero) to the zero of the other group. (Contributed by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 13-May-2023.)
⊢ 𝐴 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑁 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 0 ↔ 𝑋 = 𝑁))
Theorem	ghmf1 13818*	Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 4-Apr-2025.)
⊢ 𝐴 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑁 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)))
Theorem	kerf1ghm 13819	A group homomorphism 𝐹 is injective if and only if its kernel is the singleton {𝑁}. (Contributed by Thierry Arnoux, 27-Oct-2017.) (Proof shortened by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 13-May-2023.)
⊢ 𝐴 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑁 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ (◡𝐹 “ { 0 }) = {𝑁}))
Theorem	ghmf1o 13820	A bijective group homomorphism is an isomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.)
⊢ 𝑋 = (Base‘𝑆)    &   ⊢ 𝑌 = (Base‘𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)))
Theorem	conjghm 13821*	Conjugation is an automorphism of the group. (Contributed by Mario Carneiro, 13-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ (𝐺 GrpHom 𝐺) ∧ 𝐹:𝑋–1-1-onto→𝑋))
Theorem	conjsubg 13822*	A conjugated subgroup is also a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))    ⇒   ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ran 𝐹 ∈ (SubGrp‘𝐺))
Theorem	conjsubgen 13823*	A conjugated subgroup is equinumerous to the original subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))    ⇒   ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 ≈ ran 𝐹)
Theorem	conjnmz 13824*	A subgroup is unchanged under conjugation by an element of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))    &   ⊢ 𝑁 = {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}    ⇒   ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 = ran 𝐹)
Theorem	conjnmzb 13825*	Alternative condition for elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))    &   ⊢ 𝑁 = {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}    ⇒   ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ 𝑁 ↔ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)))
Theorem	conjnsg 13826*	A normal subgroup is unchanged under conjugation. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))    ⇒   ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 = ran 𝐹)
Theorem	qusghm 13827*	If 𝑌 is a normal subgroup of 𝐺, then the "natural map" from elements to their cosets is a group homomorphism from 𝐺 to 𝐺 / 𝑌. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 18-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌))    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌))    ⇒   ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
Theorem	ghmpropd 13828*	Group homomorphism depends only on the group attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐽))    &   ⊢ (𝜑 → 𝐶 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → 𝐶 = (Base‘𝑀))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))    ⇒   ⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
7.2.5  Abelian groups
7.2.5.1  Definition and basic properties
Syntax	ccmn 13829	Extend class notation with class of all commutative monoids.
class CMnd
Syntax	cabl 13830	Extend class notation with class of all Abelian groups.
class Abel
Definition	df-cmn 13831*	Define class of all commutative monoids. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ CMnd = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)(𝑎(+g‘𝑔)𝑏) = (𝑏(+g‘𝑔)𝑎)}
Definition	df-abl 13832	Define class of all Abelian groups. (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ Abel = (Grp ∩ CMnd)
Theorem	isabl 13833	The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.)
⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
Theorem	ablgrp 13834	An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
Theorem	ablgrpd 13835	An Abelian group is a group, deduction form of ablgrp 13834. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → 𝐺 ∈ Abel)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Grp)
Theorem	ablcmn 13836	An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
Theorem	ablcmnd 13837	An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
⊢ (𝜑 → 𝐺 ∈ Abel)    ⇒   ⊢ (𝜑 → 𝐺 ∈ CMnd)
Theorem	iscmn 13838*	The predicate "is a commutative monoid". (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Theorem	isabl2 13839*	The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Theorem	cmnpropd 13840*	If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))
Theorem	ablpropd 13841*	If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 6-Dec-2014.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
Theorem	ablprop 13842	If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 11-Oct-2013.)
⊢ (Base‘𝐾) = (Base‘𝐿)    &   ⊢ (+g‘𝐾) = (+g‘𝐿)    ⇒   ⊢ (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)
Theorem	iscmnd 13843*	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 13844*	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 13845*	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 13846	A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
Theorem	cmncom 13847	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 13848	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 13849	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 13850	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 13851	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 13852	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 13853	A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
⊢ (𝜑 → 𝐺 ∈ CMnd)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Mnd)
Theorem	rinvmod 13854*	Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6205. (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 13855	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 13856	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 13857	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 13858	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 13859	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 13860	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 13861	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 13862	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 13863	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 13864	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 13865	Cancellation law for mixed addition and subtraction. (pnpcan 8393 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 13866	Cancellation law for group subtraction. (nncan 8383 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 13867	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 13868	Cancellation law for group subtraction. (nnncan 8389 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 13869	Cancellation law for group subtraction. (nnncan1 8390 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 13870	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 13871*	The function fulfilling the conditions of ghmgrp 13663 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 13872*	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 13873*	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 13874	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 13875	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 13876	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 13877	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 13878	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 13879	Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))
Theorem	ablressid 13880	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 13112. (Contributed by Jim Kingdon, 5-May-2025.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Abel → (𝐺 ↾s 𝐵) ∈ Abel)
Theorem	imasabl 13881*	The image structure of an abelian group is an abelian group (imasgrp 13656 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 13882	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 13883	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 13884*	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 13885*	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 13886*	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 13887*	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 13888	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 13889*	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 13890*	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 13891	Multiplicative group.
class mulGrp
Definition	df-mgp 13892	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 13931). (Contributed by Mario Carneiro, 21-Dec-2014.)
⊢ mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r‘𝑤)⟩))
Theorem	fnmgp 13893	The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
⊢ mulGrp Fn V
Theorem	mgpvalg 13894	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 13895	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 13896	Existence of the multiplication group. If 𝑅 is known to be a semiring, see srgmgp 13939. (Contributed by Jim Kingdon, 10-Jan-2025.)
⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝑀 ∈ V)
Theorem	mgpbasg 13897	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 13898	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 13899	Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → (TopSet‘𝑅) = (TopSet‘𝑀))
Theorem	mgptopng 13900	Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝐽 = (TopOpen‘𝑀))