P. 189 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18801-18900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	releqg 18801	The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑅 = (𝐺 ~QG 𝑆)    ⇒   ⊢ Rel 𝑅
Theorem	eqgfval 18802*	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 18803	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 18804	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 18805*	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 18806	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 18807	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 18808	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	qusgrp 18809	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 18810	Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    &   ⊢ 𝑉 = (Base‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐻)    ⇒   ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [𝑋](𝐺 ~QG 𝑆) ∈ 𝐵)
Theorem	qusadd 18811	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 18812	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 18813	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 18814	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	lagsubg2 18815	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 18816	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))
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 19476) is defined in the context of Abelian groups.
Theorem	cycsubmel 18817*	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 18818*	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 18819*	The set of nonnegative integer powers of an element 𝐴 of a monoid forms a submonoid containing 𝐴 (see cycsubmcl 18818), 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 18820*	Condition for an operation to be commutative. Lemma for cycsubmcom 18821 and cygabl 19489. Formerly part of proof for cygabl 19489. (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 18821*	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 18822*	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 18823*	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 18824*	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 18825*	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 18826*	The cyclic subgroup generated by 𝐴 is a subgroup. Deduction related to cycsubgcl 18823. (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 18827*	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 18828	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 18829	Extend class notation with the generator of group hom-sets.
class GrpHom
Definition	df-ghm 18830*	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 18831	Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ Rel dom GrpHom
Theorem	isghm 18832*	Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.)
⊢ 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 18833*	Property of a group homomorphism, similar to ismhm 18430. (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 18834	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 18835	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 18836	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 18837	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 18838	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 18839	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 18840	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 18841*	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 18842	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 18843	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	ghmmulg 18844	A homomorphism of monoids 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 18845	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 18846	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 18847	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 18848	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 18849	One direction of resghm2b 18850. (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 18850	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 18851	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 18852	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 18853	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 18854	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 18855	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 18856	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 18857	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 18858	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 18859	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	pwsdiagghm 18860*	Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥}))    ⇒   ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 GrpHom 𝑌))
Theorem	ghmf1 18861*	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.)
⊢ 𝑋 = (Base‘𝑆)    &   ⊢ 𝑌 = (Base‘𝑇)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝑈 = (0g‘𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1→𝑌 ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )))
Theorem	ghmf1o 18862	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 18863*	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 18864*	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 18865*	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 18866*	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 18867*	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 18868*	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 18869*	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 18870*	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))
10.2.6  Isomorphisms of groups
Syntax	cgim 18871	The class of group isomorphism sets.
class GrpIso
Syntax	cgic 18872	The class of the group isomorphism relation.
class ≃𝑔
Definition	df-gim 18873*	An isomorphism of groups is a homomorphism which is also a bijection, i.e. it preserves equality as well as the group operation. (Contributed by Stefan O'Rear, 21-Jan-2015.)
⊢ GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
Definition	df-gic 18874	Two groups are said to be isomorphic iff they are connected by at least one isomorphism. Isomorphic groups share all global group properties, but to relate local properties requires knowledge of a specific isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1o))
Theorem	gimfn 18875	The group isomorphism function is a well-defined function. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ GrpIso Fn (Grp × Grp)
Theorem	isgim 18876	An isomorphism of groups is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))
Theorem	gimf1o 18877	An isomorphism of groups is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶)
Theorem	gimghm 18878	An isomorphism of groups is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
Theorem	isgim2 18879	A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 22908. (Contributed by Mario Carneiro, 6-May-2015.)
⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 GrpHom 𝑅)))
Theorem	subggim 18880	Behavior of subgroups under isomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝐹 “ 𝐴) ∈ (SubGrp‘𝑆)))
Theorem	gimcnv 18881	The converse of a bijective group homomorphism is a bijective group homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ (𝐹 ∈ (𝑆 GrpIso 𝑇) → ◡𝐹 ∈ (𝑇 GrpIso 𝑆))
Theorem	gimco 18882	The composition of group isomorphisms is a group isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ((𝐹 ∈ (𝑇 GrpIso 𝑈) ∧ 𝐺 ∈ (𝑆 GrpIso 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpIso 𝑈))
Theorem	brgic 18883	The relation "is isomorphic to" for groups. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
Theorem	brgici 18884	Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝑅 ≃𝑔 𝑆)
Theorem	gicref 18885	Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ (𝑅 ∈ Grp → 𝑅 ≃𝑔 𝑅)
Theorem	giclcl 18886	Isomorphism implies the left side is a group. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝑅 ≃𝑔 𝑆 → 𝑅 ∈ Grp)
Theorem	gicrcl 18887	Isomorphism implies the right side is a group. (Contributed by Mario Carneiro, 6-May-2015.)
⊢ (𝑅 ≃𝑔 𝑆 → 𝑆 ∈ Grp)
Theorem	gicsym 18888	Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ (𝑅 ≃𝑔 𝑆 → 𝑆 ≃𝑔 𝑅)
Theorem	gictr 18889	Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ((𝑅 ≃𝑔 𝑆 ∧ 𝑆 ≃𝑔 𝑇) → 𝑅 ≃𝑔 𝑇)
Theorem	gicer 18890	Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 1-May-2021.)
⊢ ≃𝑔 Er Grp
Theorem	gicen 18891	Isomorphic groups have equinumerous base sets. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝑅 ≃𝑔 𝑆 → 𝐵 ≈ 𝐶)
Theorem	gicsubgen 18892	A less trivial example of a group invariant: cardinality of the subgroup lattice. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝑅 ≃𝑔 𝑆 → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
10.2.7  Group actions
Syntax	cga 18893	Extend class definition to include the class of group actions.
class GrpAct
Definition	df-ga 18894*	Define the class of all group actions. A group 𝐺 acts on a set 𝑆 if a permutation on 𝑆 is associated with every element of 𝐺 in such a way that the identity permutation on 𝑆 is associated with the neutral element of 𝐺, and the composition of the permutations associated with two elements of 𝐺 is identical with the permutation associated with the composition of these two elements (in the same order) in the group 𝐺. (Contributed by Jeff Hankins, 10-Aug-2009.)
⊢ GrpAct = (𝑔 ∈ Grp, 𝑠 ∈ V ↦ ⦋(Base‘𝑔) / 𝑏⦌{𝑚 ∈ (𝑠 ↑m (𝑏 × 𝑠)) ∣ ∀𝑥 ∈ 𝑠 (((0g‘𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
Theorem	isga 18895*	The predicate "is a (left) group action". The group 𝐺 is said to act on the base set 𝑌 of the action, which is not assumed to have any special properties. There is a related notion of right group action, but as the Wikipedia article explains, it is not mathematically interesting. The way actions are usually thought of is that each element 𝑔 of 𝐺 is a permutation of the elements of 𝑌 (see gapm 18910). Since group theory was classically about symmetry groups, it is therefore likely that the notion of group action was useful even in early group theory. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))))
Theorem	gagrp 18896	The left argument of a group action is a group. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
Theorem	gaset 18897	The right argument of a group action is a set. (Contributed by Mario Carneiro, 30-Apr-2015.)
⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
Theorem	gagrpid 18898	The identity of the group does not alter the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ( 0 ⊕ 𝐴) = 𝐴)
Theorem	gaf 18899	The mapping of the group action operation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
Theorem	gafo 18900	A group action is onto its base set. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)–onto→𝑌)