P. 142 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14101-14200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ghmgrp1 14101	A group homomorphism is only defined when the domain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
Theorem	ghmgrp2 14102	A group homomorphism is only defined when the codomain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
Theorem	ghmf 14103	A group homomorphism is a function. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ 𝑋 = (Base‘𝑆)    &   ⊢ 𝑌 = (Base‘𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋⟶𝑌)
Theorem	ghmlin 14104	A homomorphism of groups is linear. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ 𝑋 = (Base‘𝑆)    &   ⊢ + = (+g‘𝑆)    &   ⊢ ⨣ = (+g‘𝑇)    ⇒   ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ∈ 𝑋) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉)))
Theorem	ghmid 14105	A homomorphism of groups preserves the identity. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ 𝑌 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) = 0 )
Theorem	ghminv 14106	A homomorphism of groups preserves inverses. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑀 = (invg‘𝑆)    &   ⊢ 𝑁 = (invg‘𝑇)    ⇒   ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑀‘𝑋)) = (𝑁‘(𝐹‘𝑋)))
Theorem	ghmsub 14107	Linearity of subtraction through a group homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ − = (-g‘𝑆)    &   ⊢ 𝑁 = (-g‘𝑇)    ⇒   ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝐹‘(𝑈 − 𝑉)) = ((𝐹‘𝑈)𝑁(𝐹‘𝑉)))
Theorem	isghmd 14108*	Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
⊢ 𝑋 = (Base‘𝑆)    &   ⊢ 𝑌 = (Base‘𝑇)    &   ⊢ + = (+g‘𝑆)    &   ⊢ ⨣ = (+g‘𝑇)    &   ⊢ (𝜑 → 𝑆 ∈ Grp)    &   ⊢ (𝜑 → 𝑇 ∈ Grp)    &   ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇))
Theorem	ghmmhm 14109	A group homomorphism is a monoid homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))
Theorem	ghmmhmb 14110	Group homomorphisms and monoid homomorphisms coincide. (Thus, GrpHom is somewhat redundant, although its stronger reverse closure properties are sometimes useful.) (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
Theorem	ghmex 14111	The set of group homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)
⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) ∈ V)
Theorem	ghmmulg 14112	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 14113	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 14114	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 14115	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 14116	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 14117	One direction of resghm2b 14118. (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 14118	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 14119	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 14120	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 14121	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 14122	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 14123	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 14124	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 14125	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 14126	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 14127	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 14128	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 14129*	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 14130	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 14131	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 14132*	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 14133*	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 14134*	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 14135*	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 14136*	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 14137*	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 14138*	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 14139*	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  Centralizers and centers
Syntax	ccntz 14140	Syntax for the centralizer of a set in a monoid.
class Cntz
Syntax	ccntr 14141	Syntax for the centralizer of a monoid.
class Cntr
Definition	df-cntz 14142*	Define the centralizer of a subset of a magma, which is the set of elements each of which commutes with each element of the given subset. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)}))
Definition	df-cntr 14143	Define the center of a magma, which is the elements that commute with all others. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
Theorem	cntzex 14144	Set existence of the centralizer. (Contributed by Jim Kingdon, 15-Sep-2026.)
⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑀 ∈ 𝑉 → 𝑍 ∈ V)
Theorem	cntrval 14145	Substitute definition of the center. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑍‘𝐵) = (Cntr‘𝑀)
Theorem	cntzfval 14146*	First level substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑀 ∈ 𝑉 → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
Theorem	cntzval 14147*	Definition substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
Theorem	elcntz 14148*	Elementhood in the centralizer. (Contributed by Mario Carneiro, 22-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑆 ⊆ 𝐵 → (𝐴 ∈ (𝑍‘𝑆) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴))))
Theorem	cntzel 14149*	Membership in a centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ (𝑍‘𝑆) ↔ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))
Theorem	cntzsnval 14150*	Special substitution for the centralizer of a singleton. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑌 ∈ 𝐵 → (𝑍‘{𝑌}) = {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})
Theorem	elcntzsn 14151	Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑌 ∈ 𝐵 → (𝑋 ∈ (𝑍‘{𝑌}) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 + 𝑌) = (𝑌 + 𝑋))))
Theorem	sscntz 14152*	A centralizer expression for two sets elementwise commuting. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Theorem	cntzrcl 14153	Reverse closure for elements of the centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵))
Theorem	cntzssv 14154	The centralizer is unconditionally a subset. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑍‘𝑆) ⊆ 𝐵
Theorem	cntzm 14155*	If the centralizer of a subset of a magma has an element, the magma is inhabited. (Contributed by Jim Kingdon, 16-Sep-2026.)
⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑋 ∈ (𝑍‘𝑆) → ∃𝑤 𝑤 ∈ 𝑀)
Theorem	cntzi 14156	Membership in a centralizer (inference). (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Theorem	elcntr 14157*	Elementhood in the center of a magma. (Contributed by SN, 21-Mar-2025.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntr‘𝑀)    ⇒   ⊢ (𝐴 ∈ 𝑍 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
Theorem	cntrss 14158	The center is a subset of the base field. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ (Cntr‘𝑀) ⊆ 𝐵
Theorem	cntri 14159	Defining property of the center of a magma. (Contributed by Mario Carneiro, 22-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntr‘𝑀)    ⇒   ⊢ ((𝑋 ∈ 𝑍 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Theorem	resscntz 14160	Centralizer in a substructure. (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ 𝑌 = (Cntz‘𝐻)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ⊆ 𝐴) → (𝑌‘𝑆) = ((𝑍‘𝑆) ∩ 𝐴))
Theorem	cntzsgrpcl 14161*	Centralizers are closed under the semigroup operation. (Contributed by AV, 17-Feb-2025.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    &   ⊢ 𝐶 = (𝑍‘𝑆)    ⇒   ⊢ ((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) → ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑦(+g‘𝑀)𝑧) ∈ 𝐶)
Theorem	cntz2ss 14162	Centralizers reverse the subset relation. (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝑆) → (𝑍‘𝑆) ⊆ (𝑍‘𝑇))
Theorem	cntzrec 14163	Reciprocity relationship for centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ 𝑇 ⊆ (𝑍‘𝑆)))
Theorem	cntzsubm 14164	Centralizers in a monoid are submonoids. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))
Theorem	cntzsubg 14165	Centralizers in a group are subgroups. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubGrp‘𝑀))
Theorem	cntzidss 14166	If the elements of 𝑆 commute, the elements of a subset 𝑇 also commute. (Contributed by Mario Carneiro, 25-Apr-2016.)
⊢ 𝑍 = (Cntz‘𝐺)    ⇒   ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ (𝑍‘𝑇))
Theorem	cntzmhm 14167	Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ 𝑌 = (Cntz‘𝐻)    ⇒   ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍‘𝑆)) → (𝐹‘𝐴) ∈ (𝑌‘(𝐹 “ 𝑆)))
Theorem	cntzmhm2 14168	Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ 𝑌 = (Cntz‘𝐻)    ⇒   ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → (𝐹 “ 𝑆) ⊆ (𝑌‘(𝐹 “ 𝑇)))
Theorem	cntrsubgnsg 14169	A central subgroup is normal. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝑍 = (Cntr‘𝑀)    ⇒   ⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → 𝑋 ∈ (NrmSGrp‘𝑀))
Theorem	cntrnsg 14170	The center of a group is a normal subgroup. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝑍 = (Cntr‘𝑀)    ⇒   ⊢ (𝑀 ∈ Grp → 𝑍 ∈ (NrmSGrp‘𝑀))
7.2.6  Abelian groups
7.2.6.1  Definition and basic properties
Syntax	ccmn 14171	Extend class notation with class of all commutative monoids.
class CMnd
Syntax	cabl 14172	Extend class notation with class of all Abelian groups.
class Abel
Definition	df-cmn 14173*	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 14174	Define class of all Abelian groups. (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ Abel = (Grp ∩ CMnd)
Theorem	isabl 14175	The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.)
⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
Theorem	ablgrp 14176	An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
Theorem	ablgrpd 14177	An Abelian group is a group, deduction form of ablgrp 14176. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → 𝐺 ∈ Abel)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Grp)
Theorem	ablcmn 14178	An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
Theorem	ablcmnd 14179	An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
⊢ (𝜑 → 𝐺 ∈ Abel)    ⇒   ⊢ (𝜑 → 𝐺 ∈ CMnd)
Theorem	iscmn 14180*	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 14181*	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 14182*	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 14183*	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 14184	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 14185*	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 14186*	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 14187*	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 14188	A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
Theorem	cmncom 14189	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 14190	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 14191	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 14192	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 14193	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 14194	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 14195	A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
⊢ (𝜑 → 𝐺 ∈ CMnd)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Mnd)
Theorem	cmnsubm 14196	A submonoid of a commutative monoid is commutative. (Contributed by Jim Kingdon, 7-Jul-2026.)
⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺))    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ (𝜑 → 𝐻 ∈ CMnd)
Theorem	rinvmod 14197*	Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6283. (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 14198	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 14199	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 14200	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))