P. 194 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19301-19400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mulgnnsubcl 19301*	Closure of the group multiple (exponentiation) operation in a submagma. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆)
Theorem	mulgnn0subcl 19302*	Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 0 ∈ 𝑆)    ⇒   ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆)
Theorem	mulgsubcl 19303*	Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 0 ∈ 𝑆)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐼‘𝑥) ∈ 𝑆)    ⇒   ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆)
Theorem	mulgnncl 19304	Closure of the group multiple (exponentiation) operation for a positive multiplier in a magma. (Contributed by Mario Carneiro, 11-Dec-2014.) (Revised by AV, 29-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mgm ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵)
Theorem	mulgnn0cl 19305	Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. (Contributed by Mario Carneiro, 11-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵)
Theorem	mulgcl 19306	Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵)
Theorem	mulgneg 19307	Group multiple (exponentiation) operation at a negative integer. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 11-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ( -𝑁 · 𝑋) = (𝐼‘(𝑁 · 𝑋)))
Theorem	mulgnegneg 19308	The inverse of a negative group multiple is the positive group multiple. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘( -𝑁 · 𝑋)) = (𝑁 · 𝑋))
Theorem	mulgm1 19309	Group multiple (exponentiation) operation at negative one. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 20-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( -1 · 𝑋) = (𝐼‘𝑋))
Theorem	mulgnn0cld 19310	Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 19305. (Contributed by SN, 1-Feb-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑁 · 𝑋) ∈ 𝐵)
Theorem	mulgcld 19311	Deduction associated with mulgcl 19306. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑁 · 𝑋) ∈ 𝐵)
Theorem	mulgaddcomlem 19312	Lemma for mulgaddcom 19313. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ ((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋))) → (( -𝑦 · 𝑋) + 𝑋) = (𝑋 + ( -𝑦 · 𝑋)))
Theorem	mulgaddcom 19313	The group multiple operator commutes with the group operation. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((𝑁 · 𝑋) + 𝑋) = (𝑋 + (𝑁 · 𝑋)))
Theorem	mulginvcom 19314	The group multiple operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · (𝐼‘𝑋)) = (𝐼‘(𝑁 · 𝑋)))
Theorem	mulginvinv 19315	The group multiple operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘(𝑁 · (𝐼‘𝑋))) = (𝑁 · 𝑋))
Theorem	mulgnn0z 19316	A group multiple of the identity, for nonnegative multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 )
Theorem	mulgz 19317	A group multiple of the identity, for integer multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝑁 · 0 ) = 0 )
Theorem	mulgnndir 19318	Sum of group multiples, for positive multiples. (Contributed by Mario Carneiro, 11-Dec-2014.) (Revised by AV, 29-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Smgrp ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋)))
Theorem	mulgnn0dir 19319	Sum of group multiples, generalized to ℕ0. (Contributed by Mario Carneiro, 11-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋)))
Theorem	mulgdirlem 19320	Lemma for mulgdir 19321. (Contributed by Mario Carneiro, 13-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ (𝑀 + 𝑁) ∈ ℕ0) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋)))
Theorem	mulgdir 19321	Sum of group multiples, generalized to ℤ. (Contributed by Mario Carneiro, 13-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋)))
Theorem	mulgp1 19322	Group multiple (exponentiation) operation at a successor, extended to ℤ. (Contributed by Mario Carneiro, 11-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋))
Theorem	mulgneg2 19323	Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 13-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ( -𝑁 · 𝑋) = (𝑁 · (𝐼‘𝑋)))
Theorem	mulgnnass 19324	Product of group multiples, for positive multiples in a semigroup. (Contributed by Mario Carneiro, 13-Dec-2014.) (Revised by AV, 29-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Smgrp ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))
Theorem	mulgnn0ass 19325	Product of group multiples, generalized to ℕ0. (Contributed by Mario Carneiro, 13-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))
Theorem	mulgass 19326	Product of group multiples, generalized to ℤ. (Contributed by Mario Carneiro, 13-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))
Theorem	mulgassr 19327	Reversed product of group multiples. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((𝑁 · 𝑀) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))
Theorem	mulgmodid 19328	Casting out multiples of the identity element leaves the group multiple unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ (𝑋 ∈ 𝐵 ∧ (𝑀 · 𝑋) = 0 )) → ((𝑁 mod 𝑀) · 𝑋) = (𝑁 · 𝑋))
Theorem	mulgsubdir 19329	Distribution of group multiples over subtraction for group elements, subdir 11753 analog. (Contributed by Mario Carneiro, 13-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 − 𝑁) · 𝑋) = ((𝑀 · 𝑋) − (𝑁 · 𝑋)))
Theorem	mhmmulg 19330	A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ × = (.g‘𝐻)    ⇒   ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹‘𝑋)))
Theorem	mulgpropd 19331*	Two structures with the same group-nature have the same group multiple function. 𝐾 is expected to either be V (when strong equality is available) or 𝐵 (when closure is available). (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ · = (.g‘𝐺)    &   ⊢ × = (.g‘𝐻)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐻))    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐾)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐾)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))    ⇒   ⊢ (𝜑 → · = × )
Theorem	submmulgcl 19332	Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 13-Jan-2015.)
⊢ ∙ = (.g‘𝐺)    ⇒   ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 ∙ 𝑋) ∈ 𝑆)
Theorem	submmulg 19333	A group multiple is the same if evaluated in a submonoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ ∙ = (.g‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾s 𝑆)    &   ⊢ · = (.g‘𝐻)    ⇒   ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 ∙ 𝑋) = (𝑁 · 𝑋))
Theorem	pwsmulg 19334	Value of a group multiple in a structure power. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ ∙ = (.g‘𝑌)    &   ⊢ · = (.g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴)))
10.2.3  Subgroups and Quotient groups
Syntax	csubg 19335	Extend class notation with all subgroups of a group.
class SubGrp
Syntax	cnsg 19336	Extend class notation with all normal subgroups of a group.
class NrmSGrp
Syntax	cqg 19337	Quotient group equivalence class.
class ~QG
Definition	df-subg 19338*	Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19357), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19351), contains the neutral element of the group (see subg0 19347) and contains the inverses for all of its elements (see subginvcl 19350). (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
Definition	df-nsg 19339*	Define the equivalence relation in a quotient ring or quotient group (where 𝑖 is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ NrmSGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑤) ∣ [(Base‘𝑤) / 𝑏][(+g‘𝑤) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})
Definition	df-eqg 19340*	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 19341	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 19342	A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝐵)
Theorem	subgid 19343	A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
Theorem	subggrp 19344	A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
Theorem	subgbas 19345	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 19346	Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
Theorem	subg0 19347	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 19348	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 19349	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 19350	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 19351	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	subgcld 19352	A subgroup is closed under group operation. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑆)
Theorem	subgsubcl 19353	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 19354	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 19355	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 19356	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 19357*	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 19358*	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 19359*	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 19360*	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 19361*	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 19362	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 19363	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 19364	A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubMnd‘𝐺))
Theorem	subsubg 19365	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 19366	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 19367	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 19368	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 19369	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 19370*	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 19371*	Weaken the condition of isnsg 19370 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 19372	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 19373	A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
Theorem	nsgconj 19374	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 19375*	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 19376	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 19377	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 19378*	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 19379*	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 19380*	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 19381*	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 19382*	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 19383*	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 19384	The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → { 0 } ∈ (NrmSGrp‘𝐺))
Theorem	nsgid 19385	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 19386	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 19387	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 19388	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 19389	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 19390	The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑅 = (𝐺 ~QG 𝑆)    ⇒   ⊢ Rel 𝑅
Theorem	ecxpid 19391	The equivalence class of a cartesian product is the whole set. (Contributed by Thierry Arnoux, 15-Jan-2024.)
⊢ (𝑋 ∈ 𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴)
Theorem	qsxpid 19392	The quotient set of a cartesian product is trivial. (Contributed by Thierry Arnoux, 16-Jan-2024.)
⊢ (𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴})
Theorem	eqgfval 19393*	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 19394	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 19395	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 19396*	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 19397	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 19398	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 19399	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	qusxpid 19400	The Group quotient equivalence relation for the whole group is the cartesian product, i.e. all elements are in the same equivalence class. (Contributed by Thierry Arnoux, 16-Jan-2024.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → (𝐺 ~QG 𝐵) = (𝐵 × 𝐵))