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Theorem List for Metamath Proof Explorer - 18901-19000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremodadd 18901 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)    &    + = (+g𝐺)       (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 1) → (𝑂‘(𝐴 + 𝐵)) = ((𝑂𝐴) · (𝑂𝐵)))
 
Theoremgex2abl 18902 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)
 
Theoremgexexlem 18903* Lemma for gexex 18904. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝑂 = (od‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐸 ∈ ℕ)    &   (𝜑𝐴𝑋)    &   ((𝜑𝑦𝑋) → (𝑂𝑦) ≤ (𝑂𝐴))       (𝜑 → (𝑂𝐴) = 𝐸)
 
Theoremgexex 18904* 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) → ∃𝑥𝑋 (𝑂𝑥) = 𝐸)
 
Theoremtorsubg 18905 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))
 
Theoremoddvdssubg 18906* 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) → {𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑁} ∈ (SubGrp‘𝐺))
 
Theoremlsmcomx 18907 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 ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = (𝑈 𝑇))
 
Theoremablcntzd 18908 All subgroups in an abelian group commute. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))       (𝜑𝑇 ⊆ (𝑍𝑈))
 
Theoremlsmcom 18909 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))
 
Theoremlsmsubg2 18910 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))
 
Theoremlsm4 18911 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)))
 
Theoremprdscmnd 18912 The product of a family of commutative monoids is commutative. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶CMnd)       (𝜑𝑌 ∈ CMnd)
 
Theoremprdsabld 18913 The product of a family of Abelian groups is an Abelian group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Abel)       (𝜑𝑌 ∈ Abel)
 
Theorempwscmn 18914 The structure power on a commutative monoid is commutative. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ CMnd ∧ 𝐼𝑉) → 𝑌 ∈ CMnd)
 
Theorempwsabl 18915 The structure power on an Abelian group is Abelian. (Contributed by Mario Carneiro, 21-Jan-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ Abel ∧ 𝐼𝑉) → 𝑌 ∈ Abel)
 
Theoremqusabl 18916 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)
 
Theoremabl1 18917 The (smallest) structure representing a trivial abelian group. (Contributed by AV, 28-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}       (𝐼𝑉𝑀 ∈ Abel)
 
Theoremabln0 18918 Abelian groups (and therefore also groups and monoids) exist. (Contributed by AV, 29-Apr-2019.)
Abel ≠ ∅
 
Theoremcnaddablx 18919 The complex numbers are an Abelian group under addition. This version of cnaddabl 18920 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 18920 instead. (New usage is discouraged.) (Contributed by NM, 18-Oct-2012.)
𝐺 = {⟨1, ℂ⟩, ⟨2, + ⟩}       𝐺 ∈ Abel
 
Theoremcnaddabl 18920 The complex numbers are an Abelian group under addition. This version of cnaddablx 18919 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 20497. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.)
𝐺 = {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩}       𝐺 ∈ Abel
 
Theoremcnaddid 18921 The group identity element of complex number addition is zero. See also cnfld0 20499. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 26-Aug-2021.) (New usage is discouraged.)
𝐺 = {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩}       (0g𝐺) = 0
 
Theoremcnaddinv 18922 Value of the group inverse of complex number addition. See also cnfldneg 20501. (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 → ((invg𝐺)‘𝐴) = -𝐴)
 
Theoremzaddablx 18923 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 20528 instead. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)
𝐺 = {⟨1, ℤ⟩, ⟨2, + ⟩}       𝐺 ∈ Abel
 
Theoremfrgpnabllem1 18924* Lemma for frgpnabl 18926. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐺 = (freeGrp‘𝐼)    &   𝑊 = ( I ‘Word (𝐼 × 2o))    &    = ( ~FG𝐼)    &    + = (+g𝐺)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑈 = (varFGrp𝐼)    &   (𝜑𝐼 ∈ V)    &   (𝜑𝐴𝐼)    &   (𝜑𝐵𝐼)       (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈𝐴) + (𝑈𝐵))))
 
Theoremfrgpnabllem2 18925* Lemma for frgpnabl 18926. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐺 = (freeGrp‘𝐼)    &   𝑊 = ( I ‘Word (𝐼 × 2o))    &    = ( ~FG𝐼)    &    + = (+g𝐺)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑈 = (varFGrp𝐼)    &   (𝜑𝐼 ∈ V)    &   (𝜑𝐴𝐼)    &   (𝜑𝐵𝐼)    &   (𝜑 → ((𝑈𝐴) + (𝑈𝐵)) = ((𝑈𝐵) + (𝑈𝐴)))       (𝜑𝐴 = 𝐵)
 
Theoremfrgpnabl 18926 The free group on two or more generators is not abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐺 = (freeGrp‘𝐼)       (1o𝐼 → ¬ 𝐺 ∈ Abel)
 
10.2.14.2  Cyclic groups
 
Syntaxccyg 18927 Cyclic group.
class CycGrp
 
Definitiondf-cyg 18928* 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(.g𝑔)𝑥)) = (Base‘𝑔)}
 
Theoremiscyg 18929* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))
 
Theoremiscyggen 18930* The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}       (𝑋𝐸 ↔ (𝑋𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
 
Theoremiscyggen2 18931* The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}       (𝐺 ∈ Grp → (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋))))
 
Theoremiscyg2 18932* A cyclic group is a group which contains a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}       (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ 𝐸 ≠ ∅))
 
Theoremcyggeninv 18933* The inverse of a cyclic generator is a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐸) → (𝑁𝑋) ∈ 𝐸)
 
Theoremcyggenod 18934* 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)    &    · = (.g𝐺)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → (𝑋𝐸 ↔ (𝑋𝐵 ∧ (𝑂𝑋) = (♯‘𝐵))))
 
Theoremcyggenod2 18935* 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)    &    · = (.g𝐺)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐸) → (𝑂𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0))
 
Theoremiscyg3 18936* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥𝐵𝑦𝐵𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑥)))
 
Theoremiscygd 18937* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋𝐵)    &   ((𝜑𝑦𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋))       (𝜑𝐺 ∈ CycGrp)
 
Theoremiscygodd 18938 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)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋𝐵)    &   (𝜑 → (𝑂𝑋) = (♯‘𝐵))       (𝜑𝐺 ∈ CycGrp)
 
Theoremcycsubmcmn 18939* 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)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   𝐶 = ran 𝐹       ((𝐺 ∈ Mnd ∧ 𝐴𝐵) → (𝐺s 𝐶) ∈ CMnd)
 
Theoremcyggrp 18940 A cyclic group is a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝐺 ∈ CycGrp → 𝐺 ∈ Grp)
 
Theoremcygabl 18941 A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 20-Jan-2024.)
(𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
 
TheoremcygablOLD 18942 Obsolete proof of cygabl 18941 as of 20-Jan-2024. A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
 
Theoremcygctb 18943 A cyclic group is countable. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ CycGrp → 𝐵 ≼ ω)
 
Theorem0cyg 18944 The trivial group is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) → 𝐺 ∈ CycGrp)
 
Theoremprmcyg 18945 A group with prime order is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐵 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ (♯‘𝐵) ∈ ℙ) → 𝐺 ∈ CycGrp)
 
Theoremlt6abl 18946 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)
 
Theoremghmcyg 18947 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:𝐵onto𝐶) → (𝐺 ∈ CycGrp → 𝐻 ∈ CycGrp))
 
Theoremcyggex2 18948 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))
 
Theoremcyggex 18949 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))
 
Theoremcyggexb 18950 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)))
 
Theoremgiccyg 18951 Cyclicity is a group property, i.e. it is preserved under isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝐺𝑔 𝐻 → (𝐺 ∈ CycGrp → 𝐻 ∈ CycGrp))
 
Theoremcycsubgcyg 18952* The cyclic subgroup generated by 𝐴 is a cyclic group. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝑆 = ran (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐺s 𝑆) ∈ CycGrp)
 
Theoremcycsubgcyg2 18953 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 ∧ 𝐴𝐵) → (𝐺s (𝐾‘{𝐴})) ∈ CycGrp)
 
10.2.14.3  Group sum operation
 
Theoremgsumval3a 18954* 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 = (0g𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝑊 ∈ Fin)    &   (𝜑𝑊 ≠ ∅)    &   𝑊 = (𝐹 supp 0 )    &   (𝜑 → ¬ 𝐴 ∈ ran ...)       (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))))
 
Theoremgsumval3eu 18955* The group sum as defined in gsumval3a 18954 is uniquely defined. (Contributed by Mario Carneiro, 8-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝑊 ∈ Fin)    &   (𝜑𝑊 ≠ ∅)    &   (𝜑𝑊𝐴)       (𝜑 → ∃!𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))
 
Theoremgsumval3lem1 18956* Lemma 1 for gsumval3 18958. (Contributed by AV, 31-May-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐻:(1...𝑀)–1-1𝐴)    &   (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)    &   𝑊 = ((𝐹𝐻) supp 0 )       (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻𝑓):(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))
 
Theoremgsumval3lem2 18957* Lemma 2 for gsumval3 18958. (Contributed by AV, 31-May-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐻:(1...𝑀)–1-1𝐴)    &   (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)    &   𝑊 = ((𝐹𝐻) supp 0 )       (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ (𝐻𝑓)))‘(♯‘𝑊)))
 
Theoremgsumval3 18958 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 = (0g𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐻:(1...𝑀)–1-1𝐴)    &   (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)    &   𝑊 = ((𝐹𝐻) supp 0 )       (𝜑 → (𝐺 Σg 𝐹) = (seq1( + , (𝐹𝐻))‘𝑀))
 
Theoremgsumcllem 18959* Lemma for gsumcl 18966 and related theorems. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 31-May-2019.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝑍𝑈)    &   (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)       ((𝜑𝑊 = ∅) → 𝐹 = (𝑘𝐴𝑍))
 
Theoremgsumzres 18960 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 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹))
 
Theoremgsumzcl2 18961 Closure of a finite group sum. This theorem has a weaker hypothesis than gsumzcl 18962, 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 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑 → (𝐹 supp 0 ) ∈ Fin)       (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵)
 
Theoremgsumzcl 18962 Closure of a finite group sum. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 1-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵)
 
Theoremgsumzf1o 18963 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 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝐹 finSupp 0 )    &   (𝜑𝐻:𝐶1-1-onto𝐴)       (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹𝐻)))
 
Theoremgsumres 18964 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 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹))
 
Theoremgsumcl2 18965 Closure of a finite group sum. This theorem has a weaker hypothesis than gsumcl 18966, 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 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → (𝐹 supp 0 ) ∈ Fin)       (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵)
 
Theoremgsumcl 18966 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 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵)
 
Theoremgsumf1o 18967 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 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )    &   (𝜑𝐻:𝐶1-1-onto𝐴)       (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹𝐻)))
 
Theoremgsumreidx 18968 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 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐹:(𝑀...𝑁)⟶𝐵)    &   (𝜑𝐻:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))       (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹𝐻)))
 
Theoremgsumzsubmcl 18969 Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.)
0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝑆 ∈ (SubMnd‘𝐺))    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆)
 
Theoremgsumsubmcl 18970 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 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝑆 ∈ (SubMnd‘𝐺))    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆)
 
Theoremgsumsubgcl 18971 Closure of a group sum in a subgroup. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 3-Jun-2019.)
0 = (0g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐴𝑉)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆)
 
Theoremgsumzaddlem 18972* The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹 finSupp 0 )    &   (𝜑𝐻 finSupp 0 )    &   𝑊 = ((𝐹𝐻) supp 0 )    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐻:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑 → ran 𝐻 ⊆ (𝑍‘ran 𝐻))    &   (𝜑 → ran (𝐹f + 𝐻) ⊆ (𝑍‘ran (𝐹f + 𝐻)))    &   ((𝜑 ∧ (𝑥𝐴𝑘 ∈ (𝐴𝑥))) → (𝐹𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻𝑥))}))       (𝜑 → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
 
Theoremgsumzadd 18973 The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹 finSupp 0 )    &   (𝜑𝐻 finSupp 0 )    &   (𝜑𝑆 ∈ (SubMnd‘𝐺))    &   (𝜑𝑆 ⊆ (𝑍𝑆))    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)       (𝜑 → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
 
Theoremgsumadd 18974 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 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐻:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )    &   (𝜑𝐻 finSupp 0 )       (𝜑 → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
 
Theoremgsummptfsadd 18975* 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 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑥𝐴) → 𝐷𝐵)    &   (𝜑𝐹 = (𝑥𝐴𝐶))    &   (𝜑𝐻 = (𝑥𝐴𝐷))    &   (𝜑𝐹 finSupp 0 )    &   (𝜑𝐻 finSupp 0 )       (𝜑 → (𝐺 Σg (𝑥𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
 
Theoremgsummptfidmadd 18976* The sum of two group sums expressed as mappings with finite domain. (Contributed by AV, 23-Jul-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑥𝐴) → 𝐷𝐵)    &   𝐹 = (𝑥𝐴𝐶)    &   𝐻 = (𝑥𝐴𝐷)       (𝜑 → (𝐺 Σg (𝑥𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
 
Theoremgsummptfidmadd2 18977* 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)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑥𝐴) → 𝐷𝐵)    &   𝐹 = (𝑥𝐴𝐶)    &   𝐻 = (𝑥𝐴𝐷)       (𝜑 → (𝐺 Σg (𝐹f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))
 
Theoremgsumzsplit 18978 Split a group sum into two parts. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝐹 finSupp 0 )    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐴 = (𝐶𝐷))       (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹𝐶)) + (𝐺 Σg (𝐹𝐷))))
 
Theoremgsumsplit 18979 Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 5-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐴 = (𝐶𝐷))       (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹𝐶)) + (𝐺 Σg (𝐹𝐷))))
 
Theoremgsumsplit2 18980* Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 5-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐴 = (𝐶𝐷))       (𝜑 → (𝐺 Σg (𝑘𝐴𝑋)) = ((𝐺 Σg (𝑘𝐶𝑋)) + (𝐺 Σg (𝑘𝐷𝑋))))
 
Theoremgsummptfidmsplit 18981* Split a group sum expressed as mapping with a finite domain into two parts. (Contributed by AV, 23-Jul-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝑌𝐵)    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐴 = (𝐶𝐷))       (𝜑 → (𝐺 Σg (𝑘𝐴𝑌)) = ((𝐺 Σg (𝑘𝐶𝑌)) + (𝐺 Σg (𝑘𝐷𝑌))))
 
Theoremgsummptfidmsplitres 18982* 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)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝑌𝐵)    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐴 = (𝐶𝐷))    &   𝐹 = (𝑘𝐴𝑌)       (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹𝐶)) + (𝐺 Σg (𝐹𝐷))))
 
Theoremgsummptfzsplit 18983* 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)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑘 ∈ (0...(𝑁 + 1))) → 𝑌𝐵)       (𝜑 → (𝐺 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ 𝑌)) = ((𝐺 Σg (𝑘 ∈ (0...𝑁) ↦ 𝑌)) + (𝐺 Σg (𝑘 ∈ {(𝑁 + 1)} ↦ 𝑌))))
 
Theoremgsummptfzsplitl 18984* 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)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑘 ∈ (0...𝑁)) → 𝑌𝐵)       (𝜑 → (𝐺 Σg (𝑘 ∈ (0...𝑁) ↦ 𝑌)) = ((𝐺 Σg (𝑘 ∈ (1...𝑁) ↦ 𝑌)) + (𝐺 Σg (𝑘 ∈ {0} ↦ 𝑌))))
 
Theoremgsumconst 18985* Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
 
Theoremgsumconstf 18986* Sum of a constant series. (Contributed by Thierry Arnoux, 5-Jul-2017.)
𝑘𝑋    &   𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
 
Theoremgsummptshft 18987* Index shift of a finite group sum over a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴𝐵)    &   (𝑗 = (𝑘𝐾) → 𝐴 = 𝐶)       (𝜑 → (𝐺 Σg (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴)) = (𝐺 Σg (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ 𝐶)))
 
Theoremgsumzmhm 18988 Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 6-Jun-2019.)
𝐵 = (Base‘𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐻 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐾 ∈ (𝐺 MndHom 𝐻))    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &    0 = (0g𝐺)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
 
Theoremgsummhm 18989 Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 6-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐻 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐾 ∈ (𝐺 MndHom 𝐻))    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
 
Theoremgsummhm2 18990* Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.) (Revised by AV, 6-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐻 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑 → (𝑥𝐵𝐶) ∈ (𝐺 MndHom 𝐻))    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )    &   (𝑥 = 𝑋𝐶 = 𝐷)    &   (𝑥 = (𝐺 Σg (𝑘𝐴𝑋)) → 𝐶 = 𝐸)       (𝜑 → (𝐻 Σg (𝑘𝐴𝐷)) = 𝐸)
 
Theoremgsummptmhm 18991* Apply a group homomorphism to a group sum expressed with a mapping. (Contributed by Thierry Arnoux, 7-Sep-2018.) (Revised by AV, 8-Sep-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐻 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐾 ∈ (𝐺 MndHom 𝐻))    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   (𝜑 → (𝑥𝐴𝐶) finSupp 0 )       (𝜑 → (𝐻 Σg (𝑥𝐴 ↦ (𝐾𝐶))) = (𝐾‘(𝐺 Σg (𝑥𝐴𝐶))))
 
Theoremgsummulglem 18992* Lemma for gsummulg 18993 and gsummulgz 18994. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 6-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑 → (𝐺 ∈ Abel ∨ 𝑁 ∈ ℕ0))       (𝜑 → (𝐺 Σg (𝑘𝐴 ↦ (𝑁 · 𝑋))) = (𝑁 · (𝐺 Σg (𝑘𝐴𝑋))))
 
Theoremgsummulg 18993* Nonnegative multiple of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 6-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐺 Σg (𝑘𝐴 ↦ (𝑁 · 𝑋))) = (𝑁 · (𝐺 Σg (𝑘𝐴𝑋))))
 
Theoremgsummulgz 18994* Integer multiple of a group sum. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 6-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐺 Σg (𝑘𝐴 ↦ (𝑁 · 𝑋))) = (𝑁 · (𝐺 Σg (𝑘𝐴𝑋))))
 
Theoremgsumzoppg 18995 The opposite of a group sum is the same as the original. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   𝑂 = (oppg𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
 
Theoremgsumzinv 18996 Inverse of a group sum. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   𝐼 = (invg𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg (𝐼𝐹)) = (𝐼‘(𝐺 Σg 𝐹)))
 
Theoremgsuminv 18997 Inverse of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 4-May-2015.) (Revised by AV, 6-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝐼 = (invg𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg (𝐼𝐹)) = (𝐼‘(𝐺 Σg 𝐹)))
 
Theoremgsummptfidminv 18998* Inverse of a group sum expressed as mapping with a finite domain. (Contributed by AV, 23-Jul-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝐼 = (invg𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   𝐹 = (𝑥𝐴𝐶)       (𝜑 → (𝐺 Σg (𝐼𝐹)) = (𝐼‘(𝐺 Σg 𝐹)))
 
Theoremgsumsub 18999 The difference of two group sums. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 6-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐻:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )    &   (𝜑𝐻 finSupp 0 )       (𝜑 → (𝐺 Σg (𝐹f 𝐻)) = ((𝐺 Σg 𝐹) (𝐺 Σg 𝐻)))
 
Theoremgsummptfssub 19000* The difference of two group sums expressed as mappings. (Contributed by AV, 7-Nov-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑥𝐴) → 𝐷𝐵)    &   (𝜑𝐹 = (𝑥𝐴𝐶))    &   (𝜑𝐻 = (𝑥𝐴𝐷))    &   (𝜑𝐹 finSupp 0 )    &   (𝜑𝐻 finSupp 0 )       (𝜑 → (𝐺 Σg (𝑥𝐴 ↦ (𝐶 𝐷))) = ((𝐺 Σg 𝐹) (𝐺 Σg 𝐻)))
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