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Theorem List for Metamath Proof Explorer - 19001-19100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcygabl 19001 A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 20-Jan-2024.)
(𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
 
TheoremcygablOLD 19002 Obsolete proof of cygabl 19001 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 19003 A cyclic group is countable. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ CycGrp → 𝐵 ≼ ω)
 
Theorem0cyg 19004 The trivial group is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) → 𝐺 ∈ CycGrp)
 
Theoremprmcyg 19005 A group with prime order is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐵 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ (♯‘𝐵) ∈ ℙ) → 𝐺 ∈ CycGrp)
 
Theoremlt6abl 19006 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 19007 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 19008 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 19009 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 19010 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 19011 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 19012* 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 19013 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 19014* 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 19015* The group sum as defined in gsumval3a 19014 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 19016* Lemma 1 for gsumval3 19018. (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 19017* Lemma 2 for gsumval3 19018. (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 19018 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 19019* Lemma for gsumcl 19026 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 19020 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 19021 Closure of a finite group sum. This theorem has a weaker hypothesis than gsumzcl 19022, 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 19022 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 19023 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 19024 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 19025 Closure of a finite group sum. This theorem has a weaker hypothesis than gsumcl 19026, 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 19026 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 19027 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 19028 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 19029 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 19030 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 19031 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 19032* 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 19033 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 19034 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 19035* 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 19036* 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 19037* 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 19038 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 19039 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 19040* 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 19041* 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 19042* 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 19043* 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 19044* 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 19045* 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 19046* 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 19047* 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 19048 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 19049 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 19050* 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 19051* 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 19052* Lemma for gsummulg 19053 and gsummulgz 19054. (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 19053* 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 19054* 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 19055 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 19056 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 19057 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 19058* 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 19059 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 19060* 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 19061* 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 19062* 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 19063* 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 19064* 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 19065* 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 19066* 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 19067* Append an element to a finite group sum, more general version of gsumunsnd 19069. (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 19068* 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 19069* 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))
 
Theoremgsumunsnf 19070* 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 Thierry Arnoux, 28-Mar-2018.) (Proof shortened by AV, 11-Dec-2019.)
𝑘𝑌    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑 → ¬ 𝑀𝐴)    &   (𝜑𝑌𝐵)    &   (𝑘 = 𝑀𝑋 = 𝑌)       (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘𝐴𝑋)) + 𝑌))
 
Theoremgsumunsn 19071* Append an element to a finite group sum. (Contributed by Mario Carneiro, 19-Dec-2014.) (Proof shortened by AV, 8-Mar-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑 → ¬ 𝑀𝐴)    &   (𝜑𝑌𝐵)    &   (𝑘 = 𝑀𝑋 = 𝑌)       (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘𝐴𝑋)) + 𝑌))
 
Theoremgsumdifsnd 19072* Extract a summand from a finitely supported group sum. (Contributed by AV, 21-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑊)    &   (𝜑 → (𝑘𝐴𝑋) finSupp (0g𝐺))    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑𝑀𝐴)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘 = 𝑀) → 𝑋 = 𝑌)       (𝜑 → (𝐺 Σg (𝑘𝐴𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌))
 
Theoremgsumpt 19073 Sum of a family that is nonzero at at most one point. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → (𝐹 supp 0 ) ⊆ {𝑋})       (𝜑 → (𝐺 Σg 𝐹) = (𝐹𝑋))
 
Theoremgsummptf1o 19074* Re-index a finite group sum using a bijection. (Contributed by Thierry Arnoux, 29-Mar-2018.)
𝑥𝐻    &   𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝑥 = 𝐸𝐶 = 𝐻)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐹𝐵)    &   ((𝜑𝑥𝐴) → 𝐶𝐹)    &   ((𝜑𝑦𝐷) → 𝐸𝐴)    &   ((𝜑𝑥𝐴) → ∃!𝑦𝐷 𝑥 = 𝐸)       (𝜑 → (𝐺 Σg (𝑥𝐴𝐶)) = (𝐺 Σg (𝑦𝐷𝐻)))
 
Theoremgsummptun 19075* Group sum of a disjoint union, whereas sums are expressed as mappings. (Contributed by Thierry Arnoux, 28-Mar-2018.) (Proof shortened by AV, 11-Dec-2019.)
𝐵 = (Base‘𝑊)    &    + = (+g𝑊)    &   (𝜑𝑊 ∈ CMnd)    &   (𝜑 → (𝐴𝐶) ∈ Fin)    &   (𝜑 → (𝐴𝐶) = ∅)    &   ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝐷𝐵)       (𝜑 → (𝑊 Σg (𝑥 ∈ (𝐴𝐶) ↦ 𝐷)) = ((𝑊 Σg (𝑥𝐴𝐷)) + (𝑊 Σg (𝑥𝐶𝐷))))
 
Theoremgsummpt1n0 19076* If only one summand in a finite group sum is not zero, the whole sum equals this summand. More general version of gsummptif1n0 19077. (Contributed by AV, 11-Oct-2019.)
0 = (0g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐼𝑊)    &   (𝜑𝑋𝐼)    &   𝐹 = (𝑛𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 ))    &   (𝜑 → ∀𝑛𝐼 𝐴 ∈ (Base‘𝐺))       (𝜑 → (𝐺 Σg 𝐹) = 𝑋 / 𝑛𝐴)
 
Theoremgsummptif1n0 19077* If only one summand in a finite group sum is not zero, the whole sum equals this summand. (Contributed by AV, 17-Feb-2019.) (Proof shortened by AV, 11-Oct-2019.)
0 = (0g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐼𝑊)    &   (𝜑𝑋𝐼)    &   𝐹 = (𝑛𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 ))    &   (𝜑𝐴 ∈ (Base‘𝐺))       (𝜑 → (𝐺 Σg 𝐹) = 𝐴)
 
Theoremgsummptcl 19078* Closure of a finite group sum over a finite set as map. (Contributed by AV, 29-Dec-2018.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑 → ∀𝑖𝑁 𝑋𝐵)       (𝜑 → (𝐺 Σg (𝑖𝑁𝑋)) ∈ 𝐵)
 
Theoremgsummptfif1o 19079* Re-index a finite group sum as map, using a bijection. (Contributed by by AV, 23-Jul-2019.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑 → ∀𝑖𝑁 𝑋𝐵)    &   𝐹 = (𝑖𝑁𝑋)    &   (𝜑𝐻:𝐶1-1-onto𝑁)       (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹𝐻)))
 
Theoremgsummptfzcl 19080* Closure of a finite group sum over a finite set of sequential integers as map. (Contributed by AV, 14-Dec-2018.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐼 = (𝑀...𝑁))    &   (𝜑 → ∀𝑖𝐼 𝑋𝐵)       (𝜑 → (𝐺 Σg (𝑖𝐼𝑋)) ∈ 𝐵)
 
Theoremgsum2dlem1 19081* Lemma 1 for gsum2d 19083. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑 → Rel 𝐴)    &   (𝜑𝐷𝑊)    &   (𝜑 → dom 𝐴𝐷)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)
 
Theoremgsum2dlem2 19082* Lemma for gsum2d 19083. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑 → Rel 𝐴)    &   (𝜑𝐷𝑊)    &   (𝜑 → dom 𝐴𝐷)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))
 
Theoremgsum2d 19083* Write a sum over a two-dimensional region as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑 → Rel 𝐴)    &   (𝜑𝐷𝑊)    &   (𝜑 → dom 𝐴𝐷)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑗𝐷 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))
 
Theoremgsum2d2lem 19084* Lemma for gsum2d2 19085: show the function is finitely supported. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 9-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐶𝑊)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )       (𝜑 → (𝑗𝐴, 𝑘𝐶𝑋) finSupp 0 )
 
Theoremgsum2d2 19085* Write a group sum over a two-dimensional region as a double sum. Note that 𝐶(𝑗) is a function of 𝑗. (Contributed by Mario Carneiro, 28-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐶𝑊)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )       (𝜑 → (𝐺 Σg (𝑗𝐴, 𝑘𝐶𝑋)) = (𝐺 Σg (𝑗𝐴 ↦ (𝐺 Σg (𝑘𝐶𝑋)))))
 
Theoremgsumcom2 19086* Two-dimensional commutation of a group sum. Note that while 𝐴 and 𝐷 are constants w.r.t. 𝑗, 𝑘, 𝐶(𝑗) and 𝐸(𝑘) are not. (Contributed by Mario Carneiro, 28-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐶𝑊)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )    &   (𝜑𝐷𝑌)    &   (𝜑 → ((𝑗𝐴𝑘𝐶) ↔ (𝑘𝐷𝑗𝐸)))       (𝜑 → (𝐺 Σg (𝑗𝐴, 𝑘𝐶𝑋)) = (𝐺 Σg (𝑘𝐷, 𝑗𝐸𝑋)))
 
Theoremgsumxp 19087* Write a group sum over a cartesian product as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 9-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑𝐹:(𝐴 × 𝐶)⟶𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑗𝐴 ↦ (𝐺 Σg (𝑘𝐶 ↦ (𝑗𝐹𝑘))))))
 
Theoremgsumcom 19088* Commute the arguments of a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )       (𝜑 → (𝐺 Σg (𝑗𝐴, 𝑘𝐶𝑋)) = (𝐺 Σg (𝑘𝐶, 𝑗𝐴𝑋)))
 
Theoremgsumcom3 19089* A commutative law for finitely supported iterated sums. (Contributed by Stefan O'Rear, 2-Nov-2015.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )       (𝜑 → (𝐺 Σg (𝑗𝐴 ↦ (𝐺 Σg (𝑘𝐶𝑋)))) = (𝐺 Σg (𝑘𝐶 ↦ (𝐺 Σg (𝑗𝐴𝑋)))))
 
Theoremgsumcom3fi 19090* A commutative law for finite iterated sums. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)       (𝜑 → (𝐺 Σg (𝑗𝐴 ↦ (𝐺 Σg (𝑘𝐶𝑋)))) = (𝐺 Σg (𝑘𝐶 ↦ (𝐺 Σg (𝑗𝐴𝑋)))))
 
Theoremgsumxp2 19091* Write a group sum over a cartesian product as a double sum in two ways. This corresponds to the first equation in [Lang] p. 6. (Contributed by AV, 27-Dec-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑𝐹:(𝐴 × 𝐶)⟶𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg (𝑘𝐶 ↦ (𝐺 Σg (𝑗𝐴 ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗𝐴 ↦ (𝐺 Σg (𝑘𝐶 ↦ (𝑗𝐹𝑘))))))
 
Theoremprdsgsum 19092* Finite commutative sums in a product structure are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2015.) (Revised by AV, 9-Jun-2019.)
𝑌 = (𝑆Xs(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑌)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑆𝑋)    &   ((𝜑𝑥𝐼) → 𝑅 ∈ CMnd)    &   ((𝜑 ∧ (𝑥𝐼𝑦𝐽)) → 𝑈𝐵)    &   (𝜑 → (𝑦𝐽 ↦ (𝑥𝐼𝑈)) finSupp 0 )       (𝜑 → (𝑌 Σg (𝑦𝐽 ↦ (𝑥𝐼𝑈))) = (𝑥𝐼 ↦ (𝑅 Σg (𝑦𝐽𝑈))))
 
Theorempwsgsum 19093* Finite commutative sums in a power structure are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2015.) (Revised by AV, 9-Jun-2019.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑌)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ CMnd)    &   ((𝜑 ∧ (𝑥𝐼𝑦𝐽)) → 𝑈𝐵)    &   (𝜑 → (𝑦𝐽 ↦ (𝑥𝐼𝑈)) finSupp 0 )       (𝜑 → (𝑌 Σg (𝑦𝐽 ↦ (𝑥𝐼𝑈))) = (𝑥𝐼 ↦ (𝑅 Σg (𝑦𝐽𝑈))))
 
10.2.14.4  Group sums over (ranges of) integers
 
Theoremfsfnn0gsumfsffz 19094* Replacing a finitely supported function over the nonnegative integers by a function over a finite set of sequential integers in a finite group sum. (Contributed by AV, 9-Oct-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐹 ∈ (𝐵m0))    &   (𝜑𝑆 ∈ ℕ0)    &   𝐻 = (𝐹 ↾ (0...𝑆))       (𝜑 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg 𝐻)))
 
Theoremnn0gsumfz 19095* Replacing a finitely supported function over the nonnegative integers by a function over a finite set of sequential integers in a finite group sum. (Contributed by AV, 9-Oct-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐹 ∈ (𝐵m0))    &   (𝜑𝐹 finSupp 0 )       (𝜑 → ∃𝑠 ∈ ℕ0𝑓 ∈ (𝐵m (0...𝑠))(𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)))
 
Theoremnn0gsumfz0 19096* Replacing a finitely supported function over the nonnegative integers by a function over a finite set of sequential integers in a finite group sum. (Contributed by AV, 9-Oct-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐹 ∈ (𝐵m0))    &   (𝜑𝐹 finSupp 0 )       (𝜑 → ∃𝑠 ∈ ℕ0𝑓 ∈ (𝐵m (0...𝑠))(𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
 
Theoremgsummptnn0fz 19097* A final group sum over a function over the nonnegative integers (given as mapping) is equal to a final group sum over a finite interval of nonnegative integers. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 3-Jul-2022.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑 → ∀𝑘 ∈ ℕ0 𝐶𝐵)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘𝐶 = 0 ))       (𝜑 → (𝐺 Σg (𝑘 ∈ ℕ0𝐶)) = (𝐺 Σg (𝑘 ∈ (0...𝑆) ↦ 𝐶)))
 
Theoremgsummptnn0fzfv 19098* A final group sum over a function over the nonnegative integers (given as mapping to its function values) is equal to a final group sum over a finite interval of nonnegative integers. (Contributed by AV, 10-Oct-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐹 ∈ (𝐵m0))    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 0 ))       (𝜑 → (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (𝐹𝑘))) = (𝐺 Σg (𝑘 ∈ (0...𝑆) ↦ (𝐹𝑘))))
 
Theoremtelgsumfzslem 19099* Lemma for telgsumfzs 19100 (induction step). (Contributed by AV, 23-Nov-2019.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &    = (-g𝐺)       ((𝑦 ∈ (ℤ𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵)) → ((𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑦 + 1) / 𝑘𝐶) → (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 ((𝑦 + 1) + 1) / 𝑘𝐶)))
 
Theoremtelgsumfzs 19100* Telescoping group sum ranging over a finite set of sequential integers, using explicit substitution. (Contributed by AV, 23-Nov-2019.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &    = (-g𝐺)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶𝐵)       (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶))
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