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Theorem List for Metamath Proof Explorer - 18301-18400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgexexlem 18301* Lemma for gexex 18302. (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 18302* 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 18303 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 18304* 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 18305 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 18306 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 18307 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 18308 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 18309 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 18310 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 18311 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 18312 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 18313 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 18314 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 18315 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 18316 Abelian groups (and therefore also groups and monoids) exist. (Contributed by AV, 29-Apr-2019.)
Abel ≠ ∅

Theoremcnaddablx 18317 The complex numbers are an Abelian group under addition. This version of cnaddabl 18318 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 18318 instead. (New usage is discouraged.) (Contributed by NM, 18-Oct-2012.)
𝐺 = {⟨1, ℂ⟩, ⟨2, + ⟩}       𝐺 ∈ Abel

Theoremcnaddabl 18318 The complex numbers are an Abelian group under addition. This version of cnaddablx 18317 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 19816. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.)
𝐺 = {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩}       𝐺 ∈ Abel

Theoremcnaddid 18319 The group identity element of complex number addition is zero. See also cnfld0 19818. (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 18320 Value of the group inverse of complex number addition. See also cnfldneg 19820. (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 18321 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 19847 instead. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)
𝐺 = {⟨1, ℤ⟩, ⟨2, + ⟩}       𝐺 ∈ Abel

Theoremfrgpnabllem1 18322* Lemma for frgpnabl 18324. (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 18323* Lemma for frgpnabl 18324. (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 18324 The free group on two or more generators is not abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐺 = (freeGrp‘𝐼)       (1𝑜𝐼 → ¬ 𝐺 ∈ Abel)

10.3.2  Cyclic groups

Syntaxccyg 18325 Cyclic group.
class CycGrp

Definitiondf-cyg 18326* 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 18327* 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 18328* 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 18329* 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 18330* 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 18331* 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 18332* 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 18333* 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 18334* 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 18335* 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 18336 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)

Theoremcyggrp 18337 A cyclic group is a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝐺 ∈ CycGrp → 𝐺 ∈ Grp)

Theoremcygabl 18338 A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝐺 ∈ CycGrp → 𝐺 ∈ Abel)

Theoremcygctb 18339 A cyclic group is countable. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ CycGrp → 𝐵 ≼ ω)

Theorem0cyg 18340 The trivial group is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1𝑜) → 𝐺 ∈ CycGrp)

Theoremprmcyg 18341 A group with prime order is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐵 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ ℙ) → 𝐺 ∈ CycGrp)

Theoremlt6abl 18342 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 18343 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 18344 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 18345 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 18346 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 18347 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 18348* 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 18349 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)

10.3.3  Group sum operation

Theoremgsumval3a 18350* 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 18351* The group sum as defined in gsumval3a 18350 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 18352* Lemma 1 for gsumval3 18354. (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 18353* Lemma 2 for gsumval3 18354. (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 18354 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 18355* Lemma for gsumcl 18362 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 18356 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 18357 Closure of a finite group sum. This theorem has a weaker hypothesis than gsumzcl 18358, 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 18358 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 18359 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 18360 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 18361 Closure of a finite group sum. This theorem has a weaker hypothesis than gsumcl 18362, 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 18362 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 18363 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)))

Theoremgsumzsubmcl 18364 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 18365 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 18366 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 18367* 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 18368 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 18369 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 18370* 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 18371* 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 18372* 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 18373 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 18374 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 18375* 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 18376* 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 18377* 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 18378* 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 18379* 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 18380* 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 18381* 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 18382* 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 18383 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 18384 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 18385* 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 18386* 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 18387* Lemma for gsummulg 18388 and gsummulgz 18389. (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 18388* 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 18389* 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 18390 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 18391 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 18392 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 18393* 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 18394 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 18395* 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 18396* 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 18397* 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 18398* 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 18399* 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 18400* 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)

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