P. 198 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19701-19800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eqgabl 19701	Value of the subgroup coset equivalence relation on an abelian group. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝑆)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐵 − 𝐴) ∈ 𝑆)))
Theorem	qusecsub 19702	Two subgroup cosets are equal if and only if the difference of their representatives is a member of the subgroup. (Contributed by AV, 7-Mar-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝑆)    ⇒   ⊢ (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ([𝑋] ∼ = [𝑌] ∼ ↔ (𝑌 − 𝑋) ∈ 𝑆))
Theorem	subgabl 19703	A subgroup of an abelian group is also abelian. (Contributed by Mario Carneiro, 3-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)
Theorem	subcmn 19704	A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐻 ∈ CMnd)
Theorem	submcmn 19705	A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ CMnd)
Theorem	submcmn2 19706	A submonoid is commutative iff it is a subset of its own centralizer. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    &   ⊢ 𝑍 = (Cntz‘𝐺)    ⇒   ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (𝐻 ∈ CMnd ↔ 𝑆 ⊆ (𝑍‘𝑆)))
Theorem	cntzcmn 19707	The centralizer of any subset in a commutative monoid is the whole monoid. (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    ⇒   ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) = 𝐵)
Theorem	cntzcmnss 19708	Any subset in a commutative monoid is a subset of its centralizer. (Contributed by AV, 12-Jan-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    ⇒   ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ (𝑍‘𝑆))
Theorem	cntrcmnd 19709	The center of a monoid is a commutative submonoid. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝑍 = (𝑀 ↾s (Cntr‘𝑀))    ⇒   ⊢ (𝑀 ∈ Mnd → 𝑍 ∈ CMnd)
Theorem	cntrabl 19710	The center of a group is an abelian group. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝑍 = (𝑀 ↾s (Cntr‘𝑀))    ⇒   ⊢ (𝑀 ∈ Grp → 𝑍 ∈ Abel)
Theorem	cntzspan 19711	If the generators commute, the generated monoid is commutative. (Contributed by Mario Carneiro, 25-Apr-2016.)
⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ 𝐾 = (mrCls‘(SubMnd‘𝐺))    &   ⊢ 𝐻 = (𝐺 ↾s (𝐾‘𝑆))    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → 𝐻 ∈ CMnd)
Theorem	cntzcmnf 19712	Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
Theorem	ghmplusg 19713	The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ + = (+g‘𝑁)    ⇒   ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))
Theorem	ablnsg 19714	Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))
Theorem	odadd1 19715	The order of a product in an abelian group divides the LCM of the orders of the factors. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)))
Theorem	odadd2 19716	The order of a product in an abelian group is divisible by the LCM of the orders of the factors divided by the GCD. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)))
Theorem	odadd 19717	The order of a product is the product of the orders, if the factors have coprime order. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (𝑂‘(𝐴 + 𝐵)) = ((𝑂‘𝐴) · (𝑂‘𝐵)))
Theorem	gex2abl 19718	A group with exponent 2 (or 1) is abelian. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) → 𝐺 ∈ Abel)
Theorem	gexexlem 19719*	Lemma for gexex 19720. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐸 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ≤ (𝑂‘𝐴))    ⇒   ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)
Theorem	gexex 19720*	In an abelian group with finite exponent, there is an element in the group with order equal to the exponent. In other words, all orders of elements divide the largest order of an element of the group. This fails if 𝐸 = 0, for example in an infinite p-group, where there are elements of arbitrarily large orders (so 𝐸 is zero) but no elements of infinite order. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = 𝐸)
Theorem	torsubg 19721	The set of all elements of finite order forms a subgroup of any abelian group, called the torsion subgroup. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ (𝐺 ∈ Abel → (◡𝑂 “ ℕ) ∈ (SubGrp‘𝐺))
Theorem	oddvdssubg 19722*	The set of all elements whose order divides a fixed integer is a subgroup of any abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.)
⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝑁 ∈ ℤ) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ 𝑁} ∈ (SubGrp‘𝐺))
Theorem	lsmcomx 19723	Subgroup sum commutes (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
Theorem	ablcntzd 19724	All subgroups in an abelian group commute. (Contributed by Mario Carneiro, 19-Apr-2016.)
⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    ⇒   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))
Theorem	lsmcom 19725	Subgroup sum commutes. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
Theorem	lsmsubg2 19726	The sum of two subgroups is a subgroup. (Contributed by NM, 4-Feb-2014.) (Proof shortened by Mario Carneiro, 19-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺))
Theorem	lsm4 19727	Commutative/associative law for subgroup sum. (Contributed by NM, 26-Sep-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ (𝑄 ∈ (SubGrp‘𝐺) ∧ 𝑅 ∈ (SubGrp‘𝐺)) ∧ (𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺))) → ((𝑄 ⊕ 𝑅) ⊕ (𝑇 ⊕ 𝑈)) = ((𝑄 ⊕ 𝑇) ⊕ (𝑅 ⊕ 𝑈)))
Theorem	prdscmnd 19728	The product of a family of commutative monoids is commutative. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶CMnd)    ⇒   ⊢ (𝜑 → 𝑌 ∈ CMnd)
Theorem	prdsabld 19729	The product of a family of Abelian groups is an Abelian group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Abel)    ⇒   ⊢ (𝜑 → 𝑌 ∈ Abel)
Theorem	pwscmn 19730	The structure power on a commutative monoid is commutative. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    ⇒   ⊢ ((𝑅 ∈ CMnd ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ CMnd)
Theorem	pwsabl 19731	The structure power on an Abelian group is Abelian. (Contributed by Mario Carneiro, 21-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    ⇒   ⊢ ((𝑅 ∈ Abel ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Abel)
Theorem	qusabl 19732	If 𝑌 is a subgroup of the abelian group 𝐺, then 𝐻 = 𝐺 / 𝑌 is an abelian group. (Contributed by Mario Carneiro, 26-Apr-2016.)
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)
Theorem	abl1 19733	The (smallest) structure representing a trivial abelian group. (Contributed by AV, 28-Apr-2019.)
⊢ 𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Abel)
Theorem	abln0 19734	Abelian groups (and therefore also groups and monoids) exist. (Contributed by AV, 29-Apr-2019.)
⊢ Abel ≠ ∅
Theorem	cnaddablx 19735	The complex numbers are an Abelian group under addition. This version of cnaddabl 19736 shows the explicit structure "scaffold" we chose for the definition for Abelian groups. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use cnaddabl 19736 instead. (New usage is discouraged.) (Contributed by NM, 18-Oct-2012.)
⊢ 𝐺 = {⟨1, ℂ⟩, ⟨2, + ⟩}    ⇒   ⊢ 𝐺 ∈ Abel
Theorem	cnaddabl 19736	The complex numbers are an Abelian group under addition. This version of cnaddablx 19735 hides the explicit structure indices i.e. is "scaffold-independent". Note that the proof also does not reference explicit structure indices. The actual structure is dependent on how Base and +g is defined. This theorem should not be referenced in any proof. For the group/ring properties of the complex numbers, see cnring 20966. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ 𝐺 ∈ Abel
Theorem	cnaddid 19737	The group identity element of complex number addition is zero. See also cnfld0 20968. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 26-Aug-2021.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ (0g‘𝐺) = 0
Theorem	cnaddinv 19738	Value of the group inverse of complex number addition. See also cnfldneg 20970. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 26-Aug-2021.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ (𝐴 ∈ ℂ → ((invg‘𝐺)‘𝐴) = -𝐴)
Theorem	zaddablx 19739	The integers are an Abelian group under addition. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. Use zsubrg 20997 instead. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)
⊢ 𝐺 = {⟨1, ℤ⟩, ⟨2, + ⟩}    ⇒   ⊢ 𝐺 ∈ Abel
Theorem	frgpnabllem1 19740*	Lemma for frgpnabl 19742. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 25-Apr-2024.)
⊢ 𝐺 = (freeGrp‘𝐼)    &   ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑈 = (varFGrp‘𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐼)    ⇒   ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈‘𝐴) + (𝑈‘𝐵))))
Theorem	frgpnabllem2 19741*	Lemma for frgpnabl 19742. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 25-Apr-2024.)
⊢ 𝐺 = (freeGrp‘𝐼)    &   ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑈 = (varFGrp‘𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐼)    &   ⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = ((𝑈‘𝐵) + (𝑈‘𝐴)))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	frgpnabl 19742	The free group on two or more generators is not abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐺 = (freeGrp‘𝐼)    ⇒   ⊢ (1o ≺ 𝐼 → ¬ 𝐺 ∈ Abel)
Theorem	imasabl 19743*	The image structure of an abelian group is an abelian group (imasgrp 18938 analog). (Contributed by AV, 22-Feb-2025.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → + = (+g‘𝑅))    &   ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   ⊢ (𝜑 → 𝑅 ∈ Abel)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘ 0 ) = (0g‘𝑈)))
10.2.14.2  Cyclic groups
Syntax	ccyg 19744	Cyclic group.
class CycGrp
Definition	df-cyg 19745*	Define a cyclic group, which is a group with an element 𝑥, called the generator of the group, such that all elements in the group are multiples of 𝑥. A generator is usually not unique. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔)}
Theorem	iscyg 19746*	Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
Theorem	iscyggen 19747*	The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    ⇒   ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
Theorem	iscyggen2 19748*	The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    ⇒   ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋))))
Theorem	iscyg2 19749*	A cyclic group is a group which contains a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    ⇒   ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ 𝐸 ≠ ∅))
Theorem	cyggeninv 19750*	The inverse of a cyclic generator is a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   ⊢ 𝑁 = (invg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸) → (𝑁‘𝑋) ∈ 𝐸)
Theorem	cyggenod 19751*	An element is the generator of a finite group iff the order of the generator equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ (𝑂‘𝑋) = (♯‘𝐵))))
Theorem	cyggenod2 19752*	In an infinite cyclic group, the generator must have infinite order, but this property no longer characterizes the generators. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸) → (𝑂‘𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0))
Theorem	iscyg3 19753*	Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑥)))
Theorem	iscygd 19754*	Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋))    ⇒   ⊢ (𝜑 → 𝐺 ∈ CycGrp)
Theorem	iscygodd 19755	Show that a group with an element the same order as the group is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑂‘𝑋) = (♯‘𝐵))    ⇒   ⊢ (𝜑 → 𝐺 ∈ CycGrp)
Theorem	cycsubmcmn 19756*	The set of nonnegative integer powers of an element 𝐴 of a monoid forms a commutative monoid. (Contributed by AV, 20-Jan-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   ⊢ 𝐶 = ran 𝐹    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → (𝐺 ↾s 𝐶) ∈ CMnd)
Theorem	cyggrp 19757	A cyclic group is a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)
Theorem	cygabl 19758	A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 20-Jan-2024.)
⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
Theorem	cygctb 19759	A cyclic group is countable. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ CycGrp → 𝐵 ≼ ω)
Theorem	0cyg 19760	The trivial group is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) → 𝐺 ∈ CycGrp)
Theorem	prmcyg 19761	A group with prime order is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ (♯‘𝐵) ∈ ℙ) → 𝐺 ∈ CycGrp)
Theorem	lt6abl 19762	A group with fewer than 6 elements is abelian. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ (♯‘𝐵) < 6) → 𝐺 ∈ Abel)
Theorem	ghmcyg 19763	The image of a cyclic group under a surjective group homomorphism is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐶 = (Base‘𝐻)    ⇒   ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵–onto→𝐶) → (𝐺 ∈ CycGrp → 𝐻 ∈ CycGrp))
Theorem	cyggex2 19764	The exponent of a cyclic group is 0 if the group is infinite, otherwise it equals the order of the group. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    ⇒   ⊢ (𝐺 ∈ CycGrp → 𝐸 = if(𝐵 ∈ Fin, (♯‘𝐵), 0))
Theorem	cyggex 19765	The exponent of a finite cyclic group is the order of the group. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ CycGrp ∧ 𝐵 ∈ Fin) → 𝐸 = (♯‘𝐵))
Theorem	cyggexb 19766	A finite abelian group is cyclic iff the exponent equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) → (𝐺 ∈ CycGrp ↔ 𝐸 = (♯‘𝐵)))
Theorem	giccyg 19767	Cyclicity is a group property, i.e. it is preserved under isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ (𝐺 ≃𝑔 𝐻 → (𝐺 ∈ CycGrp → 𝐻 ∈ CycGrp))
Theorem	cycsubgcyg 19768*	The cyclic subgroup generated by 𝐴 is a cyclic group. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝑆 = ran (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐺 ↾s 𝑆) ∈ CycGrp)
Theorem	cycsubgcyg2 19769	The cyclic subgroup generated by 𝐴 is a cyclic group. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → (𝐺 ↾s (𝐾‘{𝐴})) ∈ CycGrp)
10.2.14.3  Group sum operation
Theorem	gsumval3a 19770*	Value of the group sum operation over an index set with finite support. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by AV, 29-May-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ⊢ (𝜑 → 𝑊 ∈ Fin)    &   ⊢ (𝜑 → 𝑊 ≠ ∅)    &   ⊢ 𝑊 = (𝐹 supp 0 )    &   ⊢ (𝜑 → ¬ 𝐴 ∈ ran ...)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))))
Theorem	gsumval3eu 19771*	The group sum as defined in gsumval3a 19770 is uniquely defined. (Contributed by Mario Carneiro, 8-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ⊢ (𝜑 → 𝑊 ∈ Fin)    &   ⊢ (𝜑 → 𝑊 ≠ ∅)    &   ⊢ (𝜑 → 𝑊 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → ∃!𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))
Theorem	gsumval3lem1 19772*	Lemma 1 for gsumval3 19774. (Contributed by AV, 31-May-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1→𝐴)    &   ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)    &   ⊢ 𝑊 = ((𝐹 ∘ 𝐻) supp 0 )    ⇒   ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))
Theorem	gsumval3lem2 19773*	Lemma 2 for gsumval3 19774. (Contributed by AV, 31-May-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1→𝐴)    &   ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)    &   ⊢ 𝑊 = ((𝐹 ∘ 𝐻) supp 0 )    ⇒   ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊)))
Theorem	gsumval3 19774	Value of the group sum operation over an arbitrary finite set. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 31-May-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1→𝐴)    &   ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)    &   ⊢ 𝑊 = ((𝐹 ∘ 𝐻) supp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀))
Theorem	gsumcllem 19775*	Lemma for gsumcl 19782 and related theorems. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 31-May-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)    ⇒   ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑍))
Theorem	gsumzres 19776	Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 31-May-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑊)) = (𝐺 Σg 𝐹))
Theorem	gsumzcl2 19777	Closure of a finite group sum. This theorem has a weaker hypothesis than gsumzcl 19778, because it is not required that 𝐹 is a function (actually, the hypothesis always holds for any proper class 𝐹). (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 1-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵)
Theorem	gsumzcl 19778	Closure of a finite group sum. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 1-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵)
Theorem	gsumzf1o 19779	Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 2-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    &   ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))
Theorem	gsumres 19780	Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑊)) = (𝐺 Σg 𝐹))
Theorem	gsumcl2 19781	Closure of a finite group sum. This theorem has a weaker hypothesis than gsumcl 19782, because it is not required that 𝐹 is a function (actually, the hypothesis always holds for any proper class 𝐹). (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵)
Theorem	gsumcl 19782	Closure of a finite group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵)
Theorem	gsumf1o 19783	Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    &   ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))
Theorem	gsumreidx 19784	Re-index a finite group sum using a bijection. Corresponds to the first equation in [Lang] p. 5 with 𝑀 = 1. (Contributed by AV, 26-Dec-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    &   ⊢ (𝜑 → 𝐻:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))
Theorem	gsumzsubmcl 19785	Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.)
⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺))    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆)
Theorem	gsumsubmcl 19786	Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.)
⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺))    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆)
Theorem	gsumsubgcl 19787	Closure of a group sum in a subgroup. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 3-Jun-2019.)
⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆)
Theorem	gsumzaddlem 19788*	The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    &   ⊢ (𝜑 → 𝐻 finSupp 0 )    &   ⊢ 𝑊 = ((𝐹 ∪ 𝐻) supp 0 )    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ⊢ (𝜑 → ran 𝐻 ⊆ (𝑍‘ran 𝐻))    &   ⊢ (𝜑 → ran (𝐹 ∘f + 𝐻) ⊆ (𝑍‘ran (𝐹 ∘f + 𝐻)))    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑘 ∈ (𝐴 ∖ 𝑥))) → (𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))}))    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
Theorem	gsumzadd 19789	The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    &   ⊢ (𝜑 → 𝐻 finSupp 0 )    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺))    &   ⊢ (𝜑 → 𝑆 ⊆ (𝑍‘𝑆))    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
Theorem	gsumadd 19790	The sum of two group sums. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    &   ⊢ (𝜑 → 𝐻 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
Theorem	gsummptfsadd 19791*	The sum of two group sums expressed as mappings. (Contributed by AV, 4-Apr-2019.) (Revised by AV, 9-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶))    &   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷))    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    &   ⊢ (𝜑 → 𝐻 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
Theorem	gsummptfidmadd 19792*	The sum of two group sums expressed as mappings with finite domain. (Contributed by AV, 23-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
Theorem	gsummptfidmadd2 19793*	The sum of two group sums expressed as mappings with finite domain, using a function operation. (Contributed by AV, 23-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
Theorem	gsumzsplit 19794	Split a group sum into two parts. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    &   ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)    &   ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷))    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷))))
Theorem	gsumsplit 19795	Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 5-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    &   ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)    &   ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷))    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷))))
Theorem	gsumsplit2 19796*	Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 5-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 )    &   ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)    &   ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷))    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋))))
Theorem	gsummptfidmsplit 19797*	Split a group sum expressed as mapping with a finite domain into two parts. (Contributed by AV, 23-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)    &   ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷))    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑌)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑌)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑌))))
Theorem	gsummptfidmsplitres 19798*	Split a group sum expressed as mapping with a finite domain into two parts using restrictions. (Contributed by AV, 23-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)    &   ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷))    &   ⊢ 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑌)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷))))
Theorem	gsummptfzsplit 19799*	Split a group sum expressed as mapping with a finite set of sequential integers as domain into two parts, extracting a singleton from the right. (Contributed by AV, 25-Oct-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ 𝑌)) = ((𝐺 Σg (𝑘 ∈ (0...𝑁) ↦ 𝑌)) + (𝐺 Σg (𝑘 ∈ {(𝑁 + 1)} ↦ 𝑌))))
Theorem	gsummptfzsplitl 19800*	Split a group sum expressed as mapping with a finite set of sequential integers as domain into two parts, , extracting a singleton from the left. (Contributed by AV, 7-Nov-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (0...𝑁) ↦ 𝑌)) = ((𝐺 Σg (𝑘 ∈ (1...𝑁) ↦ 𝑌)) + (𝐺 Σg (𝑘 ∈ {0} ↦ 𝑌))))