P. 191 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19001-19100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mhmlem 19001*	Lemma for mhmmnd 19003 and ghmgrp 19005. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐹‘(𝐴 + 𝐵)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵)))
Theorem	mhmid 19002*	A surjective monoid morphism preserves identity element. (Contributed by Thierry Arnoux, 25-Jan-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑌 = (Base‘𝐻)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+g‘𝐻)    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝐻))
Theorem	mhmmnd 19003*	The image of a monoid 𝐺 under a monoid homomorphism 𝐹 is a monoid. (Contributed by Thierry Arnoux, 25-Jan-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑌 = (Base‘𝐻)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+g‘𝐻)    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    ⇒   ⊢ (𝜑 → 𝐻 ∈ Mnd)
Theorem	mhmfmhm 19004*	The function fulfilling the conditions of mhmmnd 19003 is a monoid homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑌 = (Base‘𝐻)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+g‘𝐻)    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐺 MndHom 𝐻))
Theorem	ghmgrp 19005*	The image of a group 𝐺 under a group homomorphism 𝐹 is a group. This is a stronger result than that usually found in the literature, since the target of the homomorphism (operator 𝑂 in our model) need not have any of the properties of a group as a prerequisite. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑌 = (Base‘𝐻)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+g‘𝐻)    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    ⇒   ⊢ (𝜑 → 𝐻 ∈ Grp)
10.2.2  Group multiple operation The "group multiple" operation (if the group is multiplicative, also called "group power" or "group exponentiation" operation), can be defined for arbitrary magmas, if the multiplier/exponent is a nonnegative integer. See also the definition in [Lang] p. 6, where an element 𝑥(of a monoid) to the power of a nonnegative integer 𝑛 is defined and denoted by 𝑥↑𝑛. Definition df-mulg 19007, however, defines the group multiple for arbitrary (i.e. also negative) integers. This is meaningful for groups only, and requires Definition df-minusg 18876 of the inverse operation invg.
Syntax	cmg 19006	Extend class notation with a function mapping a group operation to the multiple/power operation for the magma/group.
class .g
Definition	df-mulg 19007*	Define the group multiple function, also known as group exponentiation when viewed multiplicatively. (Contributed by Mario Carneiro, 11-Dec-2014.)
⊢ .g = (𝑔 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑔) ↦ if(𝑛 = 0, (0g‘𝑔), ⦋seq1((+g‘𝑔), (ℕ × {𝑥})) / 𝑠⦌if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑔)‘(𝑠‘-𝑛))))))
Theorem	mulgfval 19008*	Group multiple (exponentiation) operation. For a shorter proof using ax-rep 5237, see mulgfvalALT 19009. (Contributed by Mario Carneiro, 11-Dec-2014.) Remove dependency on ax-rep 5237. (Revised by Rohan Ridenour, 17-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
Theorem	mulgfvalALT 19009*	Shorter proof of mulgfval 19008 using ax-rep 5237. (Contributed by Mario Carneiro, 11-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, 0 , if(0 < 𝑛, (seq1( + , (ℕ × {𝑥}))‘𝑛), (𝐼‘(seq1( + , (ℕ × {𝑥}))‘-𝑛)))))
Theorem	mulgval 19010	Value of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝑆 = seq1( + , (ℕ × {𝑋}))    ⇒   ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, 0 , if(0 < 𝑁, (𝑆‘𝑁), (𝐼‘(𝑆‘-𝑁)))))
Theorem	mulgfn 19011	Functionality of the group multiple operation. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ · Fn (ℤ × 𝐵)
Theorem	mulgfvi 19012	The group multiple operation is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ · = (.g‘𝐺)    ⇒   ⊢ · = (.g‘( I ‘𝐺))
Theorem	mulg0 19013	Group multiple (exponentiation) operation at zero. (Contributed by Mario Carneiro, 11-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 )
Theorem	mulgnn 19014	Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝑆 = seq1( + , (ℕ × {𝑋}))    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝑆‘𝑁))
Theorem	ressmulgnn 19015	Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 12-Jun-2017.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝐴 ⊆ (Base‘𝐺)    &   ⊢ ∗ = (.g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋))
Theorem	ressmulgnn0 19016	Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝐴 ⊆ (Base‘𝐺)    &   ⊢ ∗ = (.g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ (0g‘𝐺) = (0g‘𝐻)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋))
Theorem	ressmulgnnd 19017	Values for the group multiple function in a restricted structure, a deduction version. (Contributed by metakunt, 14-May-2025.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ (𝜑 → 𝐴 ⊆ (Base‘𝐺))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝑁(.g‘𝐻)𝑋) = (𝑁(.g‘𝐺)𝑋))
Theorem	mulgnngsum 19018*	Group multiple (exponentiation) operation at a positive integer expressed by a group sum. (Contributed by AV, 28-Dec-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))
Theorem	mulgnn0gsum 19019*	Group multiple (exponentiation) operation at a nonnegative integer expressed by a group sum. This corresponds to the definition in [Lang] p. 6, second formula. (Contributed by AV, 28-Dec-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))
Theorem	mulg1 19020	Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋)
Theorem	mulgnnp1 19021	Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋))
Theorem	mulg2 19022	Group multiple (exponentiation) operation at two. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝐵 → (2 · 𝑋) = (𝑋 + 𝑋))
Theorem	mulgnegnn 19023	Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (-𝑁 · 𝑋) = (𝐼‘(𝑁 · 𝑋)))
Theorem	mulgnn0p1 19024	Group multiple (exponentiation) operation at a successor, extended to ℕ0. (Contributed by Mario Carneiro, 11-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋))
Theorem	mulgnnsubcl 19025*	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 19026*	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 19027*	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 19028	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 19029	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 19030	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 19031	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 19032	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 19033	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 19034	Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 19029. (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 19035	Deduction associated with mulgcl 19030. (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 19036	Lemma for mulgaddcom 19037. (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 19037	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 19038	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 19039	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 19040	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 19041	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 19042	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 19043	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 19044	Lemma for mulgdir 19045. (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 19045	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 19046	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 19047	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 19048	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 19049	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 19050	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 19051	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 19052	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 19053	Distribution of group multiples over subtraction for group elements, subdir 11619 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 19054	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 19055*	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 19056	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 19057	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 19058	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 19059	Extend class notation with all subgroups of a group.
class SubGrp
Syntax	cnsg 19060	Extend class notation with all normal subgroups of a group.
class NrmSGrp
Syntax	cqg 19061	Quotient group equivalence class.
class ~QG
Definition	df-subg 19062*	Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19080), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19075), contains the neutral element of the group (see subg0 19071) and contains the inverses for all of its elements (see subginvcl 19074). (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
Definition	df-nsg 19063*	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 19064*	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 19065	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 19066	A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝐵)
Theorem	subgid 19067	A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))
Theorem	subggrp 19068	A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
Theorem	subgbas 19069	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 19070	Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
Theorem	subg0 19071	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 19072	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 19073	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 19074	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 19075	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 19076	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 19077	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 19078	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 19079	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 19080*	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 19081*	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 19082*	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 19083*	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 19084*	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 19085	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 19086	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 19087	A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubMnd‘𝐺))
Theorem	subsubg 19088	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 19089	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 19090	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	0subgOLD 19091	Obsolete version of 0subg 19090 as of 31-Jan-2025. (Contributed by Stefan O'Rear, 10-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))
Theorem	trivsubgd 19092	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 19093	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 19094*	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 19095*	Weaken the condition of isnsg 19094 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 19096	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 19097	A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
Theorem	nsgconj 19098	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 19099*	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 19100	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))