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Theorem List for Metamath Proof Explorer - 19901-20000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlsmsubg2 19901 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 19902 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 19903 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 19904 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 19905 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 19906 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 19907 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 19908 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 19909 Abelian groups (and therefore also groups and monoids) exist. (Contributed by AV, 29-Apr-2019.)
Abel ≠ ∅
 
Theoremcnaddablx 19910 The complex numbers are an Abelian group under addition. This version of cnaddabl 19911 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 19911 instead. (New usage is discouraged.) (Contributed by NM, 18-Oct-2012.)
𝐺 = {⟨1, ℂ⟩, ⟨2, + ⟩}       𝐺 ∈ Abel
 
Theoremcnaddabl 19911 The complex numbers are an Abelian group under addition. This version of cnaddablx 19910 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 21426. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.)
𝐺 = {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩}       𝐺 ∈ Abel
 
Theoremcnaddid 19912 The group identity element of complex number addition is zero. See also cnfld0 21428. (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 19913 Value of the group inverse of complex number addition. See also cnfldneg 21431. (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 19914 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 21461 instead. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)
𝐺 = {⟨1, ℤ⟩, ⟨2, + ⟩}       𝐺 ∈ Abel
 
Theoremfrgpnabllem1 19915* Lemma for frgpnabl 19917. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 25-Apr-2024.)
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)    &   (𝜑𝐼𝑉)    &   (𝜑𝐴𝐼)    &   (𝜑𝐵𝐼)       (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈𝐴) + (𝑈𝐵))))
 
Theoremfrgpnabllem2 19916* Lemma for frgpnabl 19917. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 25-Apr-2024.)
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)    &   (𝜑𝐼𝑉)    &   (𝜑𝐴𝐼)    &   (𝜑𝐵𝐼)    &   (𝜑 → ((𝑈𝐴) + (𝑈𝐵)) = ((𝑈𝐵) + (𝑈𝐴)))       (𝜑𝐴 = 𝐵)
 
Theoremfrgpnabl 19917 The free group on two or more generators is not abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐺 = (freeGrp‘𝐼)       (1o𝐼 → ¬ 𝐺 ∈ Abel)
 
Theoremimasabl 19918* The image structure of an abelian group is an abelian group (imasgrp 19096 analog). (Contributed by AV, 22-Feb-2025.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑𝑅 ∈ Abel)    &    0 = (0g𝑅)       (𝜑 → (𝑈 ∈ Abel ∧ (𝐹0 ) = (0g𝑈)))
 
10.2.14.2  Cyclic groups
 
Syntaxccyg 19919 Cyclic group.
class CycGrp
 
Definitiondf-cyg 19920* 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 19921* 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 19922* 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 19923* 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 19924* 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 19925* 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 19926* 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 19927* 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 19928* 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 19929* 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 19930 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 19931* 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 19932 A cyclic group is a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝐺 ∈ CycGrp → 𝐺 ∈ Grp)
 
Theoremcygabl 19933 A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 20-Jan-2024.)
(𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
 
Theoremcygctb 19934 A cyclic group is countable. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ CycGrp → 𝐵 ≼ ω)
 
Theorem0cyg 19935 The trivial group is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) → 𝐺 ∈ CycGrp)
 
Theoremprmcyg 19936 A group with prime order is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐵 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ (♯‘𝐵) ∈ ℙ) → 𝐺 ∈ CycGrp)
 
Theoremlt6abl 19937 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 19938 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 19939 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 19940 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 19941 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 19942 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 19943* 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 19944 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 19945* 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 19946* The group sum as defined in gsumval3a 19945 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 19947* Lemma 1 for gsumval3 19949. (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 19948* Lemma 2 for gsumval3 19949. (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 19949 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 19950* Lemma for gsumcl 19957 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 19951 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 19952 Closure of a finite group sum. This theorem has a weaker hypothesis than gsumzcl 19953, 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 19953 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 19954 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 19955 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 19956 Closure of a finite group sum. This theorem has a weaker hypothesis than gsumcl 19957, 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 19957 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 19958 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 19959 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 19960 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 19961 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 19962 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 19963* 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 19964 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 19965 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 19966* 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 19967* 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 19968* 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 19969 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 19970 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 19971* 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 19972* 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 19973* 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 19974* 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 19975* 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 19976* 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 19977* 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 19978* 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 19979 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 19980 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 19981* 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 19982* 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 19983* Lemma for gsummulg 19984 and gsummulgz 19985. (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 19984* 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 19985* 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 19986 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 19987 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 19988 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 19989* 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 19990 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 19991* 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)))
 
Theoremgsummptfidmsub 19992* The difference of two group sums expressed as mappings with finite domain. (Contributed by AV, 7-Nov-2019.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑥𝐴) → 𝐷𝐵)    &   𝐹 = (𝑥𝐴𝐶)    &   𝐻 = (𝑥𝐴𝐷)       (𝜑 → (𝐺 Σg (𝑥𝐴 ↦ (𝐶 𝐷))) = ((𝐺 Σg 𝐹) (𝐺 Σg 𝐻)))
 
Theoremgsumsnfd 19993* Group sum of a singleton, deduction form, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Thierry Arnoux, 28-Mar-2018.) (Revised by AV, 11-Dec-2019.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝑀𝑉)    &   (𝜑𝐶𝐵)    &   ((𝜑𝑘 = 𝑀) → 𝐴 = 𝐶)    &   𝑘𝜑    &   𝑘𝐶       (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)
 
Theoremgsumsnd 19994* Group sum of a singleton, deduction form. (Contributed by Thierry Arnoux, 30-Jan-2017.) (Proof shortened by AV, 11-Dec-2019.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝑀𝑉)    &   (𝜑𝐶𝐵)    &   ((𝜑𝑘 = 𝑀) → 𝐴 = 𝐶)       (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)
 
Theoremgsumsnf 19995* Group sum of a singleton, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Thierry Arnoux, 28-Mar-2018.) (Proof shortened by AV, 11-Dec-2019.)
𝑘𝐶    &   𝐵 = (Base‘𝐺)    &   (𝑘 = 𝑀𝐴 = 𝐶)       ((𝐺 ∈ Mnd ∧ 𝑀𝑉𝐶𝐵) → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)
 
Theoremgsumsn 19996* Group sum of a singleton. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Proof shortened by AV, 11-Dec-2019.)
𝐵 = (Base‘𝐺)    &   (𝑘 = 𝑀𝐴 = 𝐶)       ((𝐺 ∈ Mnd ∧ 𝑀𝑉𝐶𝐵) → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)
 
Theoremgsumpr 19997* Group sum of a pair. (Contributed by AV, 6-Dec-2018.) (Proof shortened by AV, 28-Jul-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝑘 = 𝑀𝐴 = 𝐶)    &   (𝑘 = 𝑁𝐴 = 𝐷)       ((𝐺 ∈ CMnd ∧ (𝑀𝑉𝑁𝑊𝑀𝑁) ∧ (𝐶𝐵𝐷𝐵)) → (𝐺 Σg (𝑘 ∈ {𝑀, 𝑁} ↦ 𝐴)) = (𝐶 + 𝐷))
 
Theoremgsumzunsnd 19998* Append an element to a finite group sum, more general version of gsumunsnd 20000. (Contributed by AV, 7-Oct-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   𝐹 = (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑 → ¬ 𝑀𝐴)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘 = 𝑀) → 𝑋 = 𝑌)       (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝑘𝐴𝑋)) + 𝑌))
 
Theoremgsumunsnfd 19999* Append an element to a finite group sum, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 11-Dec-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑 → ¬ 𝑀𝐴)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘 = 𝑀) → 𝑋 = 𝑌)    &   𝑘𝑌       (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘𝐴𝑋)) + 𝑌))
 
Theoremgsumunsnd 20000* Append an element to a finite group sum. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 2-Jan-2019.) (Proof shortened by AV, 11-Dec-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑 → ¬ 𝑀𝐴)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘 = 𝑀) → 𝑋 = 𝑌)       (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘𝐴𝑋)) + 𝑌))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48600 487 48601-48700 488 48701-48800 489 48801-48899
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