P. 192 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-eqg 19101*	Define the equivalence relation in a group generated by a subgroup. More precisely, if 𝐺 is a group and 𝐻 is a subgroup, then 𝐺 ~QG 𝐻 is the equivalence relation on 𝐺 associated with the left cosets of 𝐻. A typical application of this definition is the construction of the quotient group (resp. ring) of a group (resp. ring) by a normal subgroup (resp. two-sided ideal). (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖)})
Theorem	issubg 19102	The subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))
Theorem	subgss 19103	A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝐵)
Theorem	subgid 19104	A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
Theorem	subggrp 19105	A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
Theorem	subgbas 19106	The base of the restricted group in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
Theorem	subgrcl 19107	Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
Theorem	subg0 19108	A subgroup of a group must have the same identity as the group. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0g‘𝐻))
Theorem	subginv 19109	The inverse of an element in a subgroup is the same as the inverse in the larger group. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ 𝐽 = (invg‘𝐻)    ⇒   ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐼‘𝑋) = (𝐽‘𝑋))
Theorem	subg0cl 19110	The group identity is an element of any subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆)
Theorem	subginvcl 19111	The inverse of an element is closed in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆) → (𝐼‘𝑋) ∈ 𝑆)
Theorem	subgcl 19112	A subgroup is closed under group operation. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
Theorem	subgsubcl 19113	A subgroup is closed under group subtraction. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 − 𝑌) ∈ 𝑆)
Theorem	subgsub 19114	The subtraction of elements in a subgroup is the same as subtraction in the group. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ − = (-g‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾s 𝑆)    &   ⊢ 𝑁 = (-g‘𝐻)    ⇒   ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 − 𝑌) = (𝑋𝑁𝑌))
Theorem	subgmulgcl 19115	Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
⊢ · = (.g‘𝐺)    ⇒   ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆)
Theorem	subgmulg 19116	A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ · = (.g‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾s 𝑆)    &   ⊢ ∙ = (.g‘𝐻)    ⇒   ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) = (𝑁 ∙ 𝑋))
Theorem	issubg2 19117*	Characterize the subgroups of a group by closure properties. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼‘𝑥) ∈ 𝑆))))
Theorem	issubgrpd2 19118*	Prove a subgroup by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷))    &   ⊢ (𝜑 → 0 = (0g‘𝐼))    &   ⊢ (𝜑 → + = (+g‘𝐼))    &   ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼))    &   ⊢ (𝜑 → 0 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷)    &   ⊢ (𝜑 → 𝐼 ∈ Grp)    ⇒   ⊢ (𝜑 → 𝐷 ∈ (SubGrp‘𝐼))
Theorem	issubgrpd 19119*	Prove a subgroup by closure. (Contributed by Stefan O'Rear, 7-Dec-2014.)
⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷))    &   ⊢ (𝜑 → 0 = (0g‘𝐼))    &   ⊢ (𝜑 → + = (+g‘𝐼))    &   ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼))    &   ⊢ (𝜑 → 0 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷)    &   ⊢ (𝜑 → 𝐼 ∈ Grp)    ⇒   ⊢ (𝜑 → 𝑆 ∈ Grp)
Theorem	issubg3 19120*	A subgroup is a symmetric submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ∈ (SubMnd‘𝐺) ∧ ∀𝑥 ∈ 𝑆 (𝐼‘𝑥) ∈ 𝑆)))
Theorem	issubg4 19121*	A subgroup is a nonempty subset of the group closed under subtraction. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 − 𝑦) ∈ 𝑆)))
Theorem	grpissubg 19122	If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the (base set of the) group is subgroup of the other group. (Contributed by AV, 14-Mar-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑆 = (Base‘𝐻)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) → ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubGrp‘𝐺)))
Theorem	resgrpisgrp 19123	If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the other group restricted to the base set of the group is a group. (Contributed by AV, 14-Mar-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑆 = (Base‘𝐻)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) → ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → (𝐺 ↾s 𝑆) ∈ Grp))
Theorem	subgsubm 19124	A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubMnd‘𝐺))
Theorem	subsubg 19125	A subgroup of a subgroup is a subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)))
Theorem	subgint 19126	The intersection of a nonempty collection of subgroups is a subgroup. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ ((𝑆 ⊆ (SubGrp‘𝐺) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubGrp‘𝐺))
Theorem	0subg 19127	The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.) (Proof shortened by SN, 31-Jan-2025.)
⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))
Theorem	trivsubgd 19128	The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 = { 0 })    &   ⊢ (𝜑 → 𝐴 ∈ (SubGrp‘𝐺))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	trivsubgsnd 19129	The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 = { 0 })    ⇒   ⊢ (𝜑 → (SubGrp‘𝐺) = {𝐵})
Theorem	isnsg 19130*	Property of being a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
Theorem	isnsg2 19131*	Weaken the condition of isnsg 19130 to only one side of the implication. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))
Theorem	nsgbi 19132	Defining property of a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆))
Theorem	nsgsubg 19133	A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
Theorem	nsgconj 19134	The conjugation of an element of a normal subgroup is in the subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → ((𝐴 + 𝐵) − 𝐴) ∈ 𝑆)
Theorem	isnsg3 19135*	A subgroup is normal iff the conjugation of all the elements of the subgroup is in the subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆))
Theorem	subgacs 19136	Subgroups are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
Theorem	nsgacs 19137	Normal subgroups form an algebraic closure system. (Contributed by Stefan O'Rear, 4-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → (NrmSGrp‘𝐺) ∈ (ACS‘𝐵))
Theorem	elnmz 19138*	Elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}    ⇒   ⊢ (𝐴 ∈ 𝑁 ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝐴 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆)))
Theorem	nmzbi 19139*	Defining property of the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}    ⇒   ⊢ ((𝐴 ∈ 𝑁 ∧ 𝐵 ∈ 𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆))
Theorem	nmzsubg 19140*	The normalizer NG(S) of a subset 𝑆 of the group is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))
Theorem	ssnmz 19141*	A subgroup is a subset of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑁)
Theorem	isnsg4 19142*	A subgroup is normal iff its normalizer is the entire group. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 = 𝑋))
Theorem	nmznsg 19143*	Any subgroup is a normal subgroup of its normalizer. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾s 𝑁)    ⇒   ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (NrmSGrp‘𝐻))
Theorem	0nsg 19144	The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → { 0 } ∈ (NrmSGrp‘𝐺))
Theorem	nsgid 19145	The whole group is a normal subgroup of itself. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (NrmSGrp‘𝐺))
Theorem	0idnsgd 19146	The whole group and the zero subgroup are normal subgroups of a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    ⇒   ⊢ (𝜑 → {{ 0 }, 𝐵} ⊆ (NrmSGrp‘𝐺))
Theorem	trivnsgd 19147	The only normal subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 = { 0 })    ⇒   ⊢ (𝜑 → (NrmSGrp‘𝐺) = {𝐵})
Theorem	triv1nsgd 19148	A trivial group has exactly one normal subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 = { 0 })    ⇒   ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 1o)
Theorem	1nsgtrivd 19149	A group with exactly one normal subgroup is trivial. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 1o)    ⇒   ⊢ (𝜑 → 𝐵 = { 0 })
Theorem	releqg 19150	The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑅 = (𝐺 ~QG 𝑆)    ⇒   ⊢ Rel 𝑅
Theorem	eqgfval 19151*	Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑅 = (𝐺 ~QG 𝑆)    ⇒   ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁‘𝑥) + 𝑦) ∈ 𝑆)})
Theorem	eqgval 19152	Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑅 = (𝐺 ~QG 𝑆)    ⇒   ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ⊆ 𝑋) → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((𝑁‘𝐴) + 𝐵) ∈ 𝑆)))
Theorem	eqger 19153	The subgroup coset equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 13-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝑌)    ⇒   ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
Theorem	eqglact 19154*	A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝑌)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))
Theorem	eqgid 19155	The left coset containing the identity is the original subgroup. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝑌)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝑌 ∈ (SubGrp‘𝐺) → [ 0 ] ∼ = 𝑌)
Theorem	eqgen 19156	Each coset is equipotent to the subgroup itself (which is also the coset containing the identity). (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝑌)    ⇒   ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (𝑋 / ∼ )) → 𝑌 ≈ 𝐴)
Theorem	eqgcpbl 19157	The subgroup coset equivalence relation is compatible with addition when the subgroup is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝑌)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷)))
Theorem	eqg0el 19158	Equivalence class of a quotient group for a subgroup. (Contributed by Thierry Arnoux, 15-Jan-2024.)
⊢ ∼ = (𝐺 ~QG 𝐻)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([𝑋] ∼ = 𝐻 ↔ 𝑋 ∈ 𝐻))
Theorem	quselbas 19159*	Membership in the base set of a quotient group. (Contributed by AV, 1-Mar-2025.)
⊢ ∼ = (𝐺 ~QG 𝑆)    &   ⊢ 𝑈 = (𝐺 /s ∼ )    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → (𝑋 ∈ (Base‘𝑈) ↔ ∃𝑥 ∈ 𝐵 𝑋 = [𝑥] ∼ ))
Theorem	quseccl0 19160	Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) Generalization of quseccl 19162 for arbitrary sets 𝐺. (Revised by AV, 24-Feb-2025.)
⊢ ∼ = (𝐺 ~QG 𝑆)    &   ⊢ 𝐻 = (𝐺 /s ∼ )    &   ⊢ 𝐶 = (Base‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐻)    ⇒   ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝐶) → [𝑋] ∼ ∈ 𝐵)
Theorem	qusgrp 19161	If 𝑌 is a normal subgroup of 𝐺, then 𝐻 = 𝐺 / 𝑌 is a group, called the quotient of 𝐺 by 𝑌. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    ⇒   ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp)
Theorem	quseccl 19162	Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) (Proof shortened by AV, 9-Mar-2025.)
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    &   ⊢ 𝑉 = (Base‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐻)    ⇒   ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [𝑋](𝐺 ~QG 𝑆) ∈ 𝐵)
Theorem	qusadd 19163	Value of the group operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    &   ⊢ 𝑉 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ✚ = (+g‘𝐻)    ⇒   ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋](𝐺 ~QG 𝑆) ✚ [𝑌](𝐺 ~QG 𝑆)) = [(𝑋 + 𝑌)](𝐺 ~QG 𝑆))
Theorem	qus0 19164	Value of the group identity operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → [ 0 ](𝐺 ~QG 𝑆) = (0g‘𝐻))
Theorem	qusinv 19165	Value of the group inverse operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    &   ⊢ 𝑉 = (Base‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐻)    ⇒   ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → (𝑁‘[𝑋](𝐺 ~QG 𝑆)) = [(𝐼‘𝑋)](𝐺 ~QG 𝑆))
Theorem	qussub 19166	Value of the group subtraction operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    &   ⊢ 𝑉 = (Base‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ 𝑁 = (-g‘𝐻)    ⇒   ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋](𝐺 ~QG 𝑆)𝑁[𝑌](𝐺 ~QG 𝑆)) = [(𝑋 − 𝑌)](𝐺 ~QG 𝑆))
Theorem	ecqusaddd 19167	Addition of equivalence classes in a quotient group. (Contributed by AV, 25-Feb-2025.)
⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → [(𝐴(+g‘𝑅)𝐶)] ∼ = ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ ))
Theorem	ecqusaddcl 19168	Closure of the addition in a quotient group. (Contributed by AV, 24-Feb-2025.)
⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ ) ∈ (Base‘𝑄))
Theorem	lagsubg2 19169	Lagrange's theorem for finite groups. Call the "order" of a group the cardinal number of the basic set of the group, and "index of a subgroup" the cardinal number of the set of left (or right, this is the same) cosets of this subgroup. Then the order of the group is the (cardinal) product of the order of any of its subgroups by the index of this subgroup. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝑌)    &   ⊢ (𝜑 → 𝑌 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    ⇒   ⊢ (𝜑 → (♯‘𝑋) = ((♯‘(𝑋 / ∼ )) · (♯‘𝑌)))
Theorem	lagsubg 19170	Lagrange's theorem for Groups: the order of any subgroup of a finite group is a divisor of the order of the group. This is Metamath 100 proof #71. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘𝑌) ∥ (♯‘𝑋))
Theorem	eqg0subg 19171	The coset equivalence relation for the trivial (zero) subgroup of a group is the identity relation restricted to the base set of the group. (Contributed by AV, 25-Feb-2025.)
⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑆 = { 0 }    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑅 = (𝐺 ~QG 𝑆)    ⇒   ⊢ (𝐺 ∈ Grp → 𝑅 = ( I ↾ 𝐵))
Theorem	eqg0subgecsn 19172	The equivalence classes modulo the coset equivalence relation for the trivial (zero) subgroup of a group are singletons. (Contributed by AV, 26-Feb-2025.)
⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑆 = { 0 }    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑅 = (𝐺 ~QG 𝑆)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → [𝑋]𝑅 = {𝑋})
Theorem	qus0subgbas 19173*	The base set of a quotient of a group by the trivial (zero) subgroup. (Contributed by AV, 26-Feb-2025.)
⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑆 = { 0 }    &   ⊢ ∼ = (𝐺 ~QG 𝑆)    &   ⊢ 𝑈 = (𝐺 /s ∼ )    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → (Base‘𝑈) = {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
Theorem	qus0subgadd 19174*	The addition in a quotient of a group by the trivial (zero) subgroup. (Contributed by AV, 26-Feb-2025.)
⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑆 = { 0 }    &   ⊢ ∼ = (𝐺 ~QG 𝑆)    &   ⊢ 𝑈 = (𝐺 /s ∼ )    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ({𝑎} (+g‘𝑈){𝑏}) = {(𝑎(+g‘𝐺)𝑏)})
10.2.4  Cyclic monoids and groups This section contains some preliminary results about cyclic monoids and groups before the class CycGrp of cyclic groups (see df-cyg 19853) is defined in the context of Abelian groups.
Theorem	cycsubmel 19175*	Characterization of an element of the set of nonnegative integer powers of an element 𝐴. Although this theorem holds for any class 𝐺, the definition of 𝐹 is only meaningful if 𝐺 is a monoid or at least a unital magma. (Contributed by AV, 28-Dec-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   ⊢ 𝐶 = ran 𝐹    ⇒   ⊢ (𝑋 ∈ 𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴))
Theorem	cycsubmcl 19176*	The set of nonnegative integer powers of an element 𝐴 contains 𝐴. Although this theorem holds for any class 𝐺, the definition of 𝐹 is only meaningful if 𝐺 is a monoid or at least a unital magma. (Contributed by AV, 28-Dec-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   ⊢ 𝐶 = ran 𝐹    ⇒   ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐶)
Theorem	cycsubm 19177*	The set of nonnegative integer powers of an element 𝐴 of a monoid forms a submonoid containing 𝐴 (see cycsubmcl 19176), called the cyclic monoid generated by the element 𝐴. This corresponds to the statement in [Lang] p. 6. (Contributed by AV, 28-Dec-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   ⊢ 𝐶 = ran 𝐹    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → 𝐶 ∈ (SubMnd‘𝐺))
Theorem	cyccom 19178*	Condition for an operation to be commutative. Lemma for cycsubmcom 19179 and cygabl 19866. Formerly part of proof for cygabl 19866. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 20-Jan-2024.)
⊢ (𝜑 → ∀𝑐 ∈ 𝐶 ∃𝑥 ∈ 𝑍 𝑐 = (𝑥 · 𝐴))    &   ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 ∀𝑛 ∈ 𝑍 ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴) + (𝑛 · 𝐴)))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐶)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐶)    &   ⊢ (𝜑 → 𝑍 ⊆ ℂ)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Theorem	cycsubmcom 19179*	The operation of a monoid is commutative over the set of nonnegative integer powers of an element 𝐴 of the monoid. (Contributed by AV, 20-Jan-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   ⊢ 𝐶 = ran 𝐹    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Theorem	cycsubggend 19180*	The cyclic subgroup generated by 𝐴 includes its generator. Although this theorem holds for any class 𝐺, the definition of 𝐹 is only meaningful if 𝐺 is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 𝐴))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ran 𝐹)
Theorem	cycsubgcl 19181*	The set of integer powers of an element 𝐴 of a group forms a subgroup containing 𝐴, called the cyclic group generated by the element 𝐴. (Contributed by Mario Carneiro, 13-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (ran 𝐹 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ran 𝐹))
Theorem	cycsubgss 19182*	The cyclic subgroup generated by an element 𝐴 is a subset of any subgroup containing 𝐴. (Contributed by Mario Carneiro, 13-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))    ⇒   ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑆) → ran 𝐹 ⊆ 𝑆)
Theorem	cycsubg 19183*	The cyclic group generated by 𝐴 is the smallest subgroup containing 𝐴. (Contributed by Mario Carneiro, 13-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ran 𝐹 = ∩ {𝑠 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑠})
Theorem	cycsubgcld 19184*	The cyclic subgroup generated by 𝐴 is a subgroup. Deduction related to cycsubgcl 19181. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 𝐴))    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ran 𝐹 ∈ (SubGrp‘𝐺))
Theorem	cycsubg2 19185*	The subgroup generated by an element is exhausted by its multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))    &   ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐾‘{𝐴}) = ran 𝐹)
Theorem	cycsubg2cl 19186	Any multiple of an element is contained in the generated cyclic subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝑁 · 𝐴) ∈ (𝐾‘{𝐴}))
10.2.5  Elementary theory of group homomorphisms
Syntax	cghm 19187	Extend class notation with the generator of group hom-sets.
class GrpHom
Definition	df-ghm 19188*	A homomorphism of groups is a map between two structures which preserves the group operation. Requiring both sides to be groups simplifies most theorems at the cost of complicating the theorem which pushes forward a group structure. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ GrpHom = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∣ [(Base‘𝑠) / 𝑤](𝑔:𝑤⟶(Base‘𝑡) ∧ ∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑔‘(𝑥(+g‘𝑠)𝑦)) = ((𝑔‘𝑥)(+g‘𝑡)(𝑔‘𝑦)))})
Theorem	reldmghm 19189	Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ Rel dom GrpHom
Theorem	isghm 19190*	Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.) (Proof shortened by SN, 5-Jun-2025.)
⊢ 𝑋 = (Base‘𝑆)    &   ⊢ 𝑌 = (Base‘𝑇)    &   ⊢ + = (+g‘𝑆)    &   ⊢ ⨣ = (+g‘𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))
Theorem	isghmOLD 19191*	Obsolete version of isghm 19190 as of 5-Jun-2025. (Contributed by Stefan O'Rear, 31-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑋 = (Base‘𝑆)    &   ⊢ 𝑌 = (Base‘𝑇)    &   ⊢ + = (+g‘𝑆)    &   ⊢ ⨣ = (+g‘𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))
Theorem	isghm3 19192*	Property of a group homomorphism, similar to ismhm 18753. (Contributed by Mario Carneiro, 7-Mar-2015.)
⊢ 𝑋 = (Base‘𝑆)    &   ⊢ 𝑌 = (Base‘𝑇)    &   ⊢ + = (+g‘𝑆)    &   ⊢ ⨣ = (+g‘𝑇)    ⇒   ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))
Theorem	ghmgrp1 19193	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 19194	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 19195	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 19196	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 19197	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 19198	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 19199	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 19200*	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))