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Theorem List for Metamath Proof Explorer - 18101-18200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgsum2d 18101* 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 18102* Lemma for gsum2d2 18103: 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 18103* 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 18104* 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 18105* 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 18106* 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)))
 
Theoremprdsgsum 18107* 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 18108* 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))))
 
10.3.4  Group sums over (ranges of) integers
 
Theoremfsfnn0gsumfsffz 18109* 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 18110* 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 18111* 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 18112* 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.)
𝑘𝜑    &   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)))
 
Theoremgsummptnn0fzv 18113* 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.)
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 18114* 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 18115* Lemma for telgsumfzs 18116 (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 18116* 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))
 
Theoremtelgsumfz 18117* Telescoping group sum ranging over a finite set of sequential integers, using implicit substitution, analogous to telfsum 14246. (Contributed by AV, 23-Nov-2019.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &    = (-g𝐺)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐴𝐵)    &   (𝑘 = 𝑖𝐴 = 𝐿)    &   (𝑘 = (𝑖 + 1) → 𝐴 = 𝐶)    &   (𝑘 = 𝑀𝐴 = 𝐷)    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸)       (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 𝐶))) = (𝐷 𝐸))
 
Theoremtelgsumfz0s 18118* Telescoping finite group sum ranging over nonnegative integers, using explicit substitution. (Contributed by AV, 24-Oct-2019.) (Proof shortened by AV, 25-Nov-2019.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &    = (-g𝐺)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → ∀𝑘 ∈ (0...(𝑆 + 1))𝐶𝐵)       (𝜑 → (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (0 / 𝑘𝐶 (𝑆 + 1) / 𝑘𝐶))
 
Theoremtelgsumfz0 18119* Telescoping finite group sum ranging over nonnegative integers, using implicit substitution, analogous to telfsum 14246. (Contributed by AV, 23-Nov-2019.)
𝐾 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &    = (-g𝐺)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → ∀𝑘 ∈ (0...(𝑆 + 1))𝐴𝐾)    &   (𝑘 = 𝑖𝐴 = 𝐵)    &   (𝑘 = (𝑖 + 1) → 𝐴 = 𝐶)    &   (𝑘 = 0 → 𝐴 = 𝐷)    &   (𝑘 = (𝑆 + 1) → 𝐴 = 𝐸)       (𝜑 → (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (𝐵 𝐶))) = (𝐷 𝐸))
 
Theoremtelgsums 18120* Telescoping finitely supported group sum ranging over nonnegative integers, using explicit substitution. (Contributed by AV, 24-Oct-2019.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &    = (-g𝐺)    &    0 = (0g𝐺)    &   (𝜑 → ∀𝑘 ∈ ℕ0 𝐶𝐵)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘𝐶 = 0 ))       (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = 0 / 𝑘𝐶)
 
Theoremtelgsum 18121* Telescoping finitely supported group sum ranging over nonnegative integers, using implicit substitution. (Contributed by AV, 31-Dec-2019.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &    = (-g𝐺)    &    0 = (0g𝐺)    &   (𝜑 → ∀𝑘 ∈ ℕ0 𝐴𝐵)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘𝐴 = 0 ))    &   (𝑘 = 𝑖𝐴 = 𝐶)    &   (𝑘 = (𝑖 + 1) → 𝐴 = 𝐷)    &   (𝑘 = 0 → 𝐴 = 𝐸)       (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (𝐶 𝐷))) = 𝐸)
 
10.3.5  Internal direct products
 
Syntaxcdprd 18122 Internal direct product of a family of subgroups.
class DProd
 
Syntaxcdpj 18123 Projection operator for a direct product.
class dProj
 
Definitiondf-dprd 18124* Define the internal direct product of a family of subgroups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 11-Jul-2019.)
DProd = (𝑔 ∈ Grp, 𝑠 ∈ { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))} ↦ ran (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)))
 
Definitiondf-dpj 18125* Define the projection operator for a direct product. (Contributed by Mario Carneiro, 21-Apr-2016.)
dProj = (𝑔 ∈ Grp, 𝑠 ∈ (dom DProd “ {𝑔}) ↦ (𝑖 ∈ dom 𝑠 ↦ ((𝑠𝑖)(proj1𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))))))
 
Theoremreldmdprd 18126 The domain of the internal direct product operation is a relation. (Contributed by Mario Carneiro, 25-Apr-2016.) (Proof shortened by AV, 11-Jul-2019.)
Rel dom DProd
 
Theoremdmdprd 18127* The domain of definition of the internal direct product, which states that 𝑆 is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) (Proof shortened by AV, 11-Jul-2019.)
𝑍 = (Cntz‘𝐺)    &    0 = (0g𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐼𝑉 ∧ dom 𝑆 = 𝐼) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
 
Theoremdmdprdd 18128* Show that a given family is a direct product decomposition. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
𝑍 = (Cntz‘𝐺)    &    0 = (0g𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))    &   ((𝜑 ∧ (𝑥𝐼𝑦𝐼𝑥𝑦)) → (𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)))    &   ((𝜑𝑥𝐼) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 })       (𝜑𝐺dom DProd 𝑆)
 
Theoremdprddomprc 18129 A family of subgroups indexed by a proper class cannot be a family of subgroups for an internal direct product. (Contributed by AV, 13-Jul-2019.)
(dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆)
 
Theoremdprddomcld 18130 If a family of subgroups is a family of subgroups for an internal direct product, then it is indexed by a set. (Contributed by AV, 13-Jul-2019.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)       (𝜑𝐼 ∈ V)
 
Theoremdprdval0prc 18131 The internal direct product of a family of subgroups indexed by a proper class is empty. (Contributed by AV, 13-Jul-2019.)
(dom 𝑆 ∉ V → (𝐺 DProd 𝑆) = ∅)
 
Theoremdprdval 18132* The value of the internal direct product operation, which is a function mapping the (infinite, but finitely supported) cartesian product of subgroups (which mutually commute and have trivial intersections) to its (group) sum . (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
0 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }       ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))
 
Theoremeldprd 18133* A class 𝐴 is an internal direct product iff it is the (group) sum of an infinite, but finitely supported cartesian product of subgroups (which mutually commute and have trivial intersections). (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
0 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }       (dom 𝑆 = 𝐼 → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓))))
 
Theoremdprdgrp 18134 Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝐺dom DProd 𝑆𝐺 ∈ Grp)
 
Theoremdprdf 18135 The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝐺dom DProd 𝑆𝑆:dom 𝑆⟶(SubGrp‘𝐺))
 
Theoremdprdf2 18136 The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)       (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))
 
Theoremdprdcntz 18137 The function 𝑆 is a family having pairwise commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)    &   (𝜑𝑌𝐼)    &   (𝜑𝑋𝑌)    &   𝑍 = (Cntz‘𝐺)       (𝜑 → (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑌)))
 
Theoremdprddisj 18138 The function 𝑆 is a family having trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)    &    0 = (0g𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       (𝜑 → ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 })
 
Theoremdprdw 18139* The property of being a finitely supported function in the family 𝑆. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)       (𝜑 → (𝐹𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥) ∧ 𝐹 finSupp 0 )))
 
Theoremdprdwd 18140* A mapping being a finitely supported function in the family 𝑆. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.) (Proof shortened by OpenAI, 30-Mar-2020.)
𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   ((𝜑𝑥𝐼) → 𝐴 ∈ (𝑆𝑥))    &   (𝜑 → (𝑥𝐼𝐴) finSupp 0 )       (𝜑 → (𝑥𝐼𝐴) ∈ 𝑊)
 
Theoremdprdff 18141* A finitely supported function in 𝑆 is a function into the base. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)    &   𝐵 = (Base‘𝐺)       (𝜑𝐹:𝐼𝐵)
 
Theoremdprdfcl 18142* A finitely supported function in 𝑆 has its 𝑋-th element in 𝑆(𝑋). (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)       ((𝜑𝑋𝐼) → (𝐹𝑋) ∈ (𝑆𝑋))
 
Theoremdprdffsupp 18143* A finitely supported function in 𝑆 is a finitely supported function. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)       (𝜑𝐹 finSupp 0 )
 
Theoremdprdfcntz 18144* A function on the elements of an internal direct product has pairwise commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)    &   𝑍 = (Cntz‘𝐺)       (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
 
Theoremdprdssv 18145 The internal direct product of a family of subgroups is a subset of the base. (Contributed by Mario Carneiro, 25-Apr-2016.)
𝐵 = (Base‘𝐺)       (𝐺 DProd 𝑆) ⊆ 𝐵
 
Theoremdprdfid 18146* A function mapping all but one arguments to zero sums to the value of this argument in a direct product. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
0 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)    &   (𝜑𝐴 ∈ (𝑆𝑋))    &   𝐹 = (𝑛𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 ))       (𝜑 → (𝐹𝑊 ∧ (𝐺 Σg 𝐹) = 𝐴))
 
Theoremeldprdi 18147* The domain of definition of the internal direct product, which states that 𝑆 is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
0 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)       (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆))
 
Theoremdprdfinv 18148* Take the inverse of a group sum over a family of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
0 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)    &   𝑁 = (invg𝐺)       (𝜑 → ((𝑁𝐹) ∈ 𝑊 ∧ (𝐺 Σg (𝑁𝐹)) = (𝑁‘(𝐺 Σg 𝐹))))
 
Theoremdprdfadd 18149* Take the sum of group sums over two families of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
0 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)    &   (𝜑𝐻𝑊)    &    + = (+g𝐺)       (𝜑 → ((𝐹𝑓 + 𝐻) ∈ 𝑊 ∧ (𝐺 Σg (𝐹𝑓 + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))))
 
Theoremdprdfsub 18150* Take the difference of group sums over two families of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
0 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)    &   (𝜑𝐻𝑊)    &    = (-g𝐺)       (𝜑 → ((𝐹𝑓 𝐻) ∈ 𝑊 ∧ (𝐺 Σg (𝐹𝑓 𝐻)) = ((𝐺 Σg 𝐹) (𝐺 Σg 𝐻))))
 
Theoremdprdfeq0 18151* The zero function is the only function that sums to zero in a direct product. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
0 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)       (𝜑 → ((𝐺 Σg 𝐹) = 0𝐹 = (𝑥𝐼0 )))
 
Theoremdprdf11 18152* Two group sums over a direct product that give the same value are equal as functions. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
0 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)    &   (𝜑𝐻𝑊)       (𝜑 → ((𝐺 Σg 𝐹) = (𝐺 Σg 𝐻) ↔ 𝐹 = 𝐻))
 
Theoremdprdsubg 18153 The internal direct product of a family of subgroups is a subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺))
 
Theoremdprdub 18154 Each factor is a subset of the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑆𝑋) ⊆ (𝐺 DProd 𝑆))
 
Theoremdprdlub 18155* The direct product is smaller than any subgroup which contains the factors. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   ((𝜑𝑘𝐼) → (𝑆𝑘) ⊆ 𝑇)       (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝑇)
 
Theoremdprdspan 18156 The direct product is the span of the union of the factors. (Contributed by Mario Carneiro, 25-Apr-2016.)
𝐾 = (mrCls‘(SubGrp‘𝐺))       (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = (𝐾 ran 𝑆))
 
Theoremdprdres 18157 Restriction of a direct product (dropping factors). (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐴𝐼)       (𝜑 → (𝐺dom DProd (𝑆𝐴) ∧ (𝐺 DProd (𝑆𝐴)) ⊆ (𝐺 DProd 𝑆)))
 
Theoremdprdss 18158* Create a direct product by finding subgroups inside each factor of another direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑇)    &   (𝜑 → dom 𝑇 = 𝐼)    &   (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))    &   ((𝜑𝑘𝐼) → (𝑆𝑘) ⊆ (𝑇𝑘))       (𝜑 → (𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) ⊆ (𝐺 DProd 𝑇)))
 
Theoremdprdz 18159* A family consisting entirely of trivial groups is an internal direct product, the product of which is the trivial subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐼𝑉) → (𝐺dom DProd (𝑥𝐼 ↦ { 0 }) ∧ (𝐺 DProd (𝑥𝐼 ↦ { 0 })) = { 0 }))
 
Theoremdprd0 18160 The empty family is an internal direct product, the product of which is the trivial subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
0 = (0g𝐺)       (𝐺 ∈ Grp → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = { 0 }))
 
Theoremdprdf1o 18161 Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹:𝐽1-1-onto𝐼)       (𝜑 → (𝐺dom DProd (𝑆𝐹) ∧ (𝐺 DProd (𝑆𝐹)) = (𝐺 DProd 𝑆)))
 
Theoremdprdf1 18162 Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹:𝐽1-1𝐼)       (𝜑 → (𝐺dom DProd (𝑆𝐹) ∧ (𝐺 DProd (𝑆𝐹)) ⊆ (𝐺 DProd 𝑆)))
 
Theoremsubgdmdprd 18163 A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐻 = (𝐺s 𝐴)       (𝐴 ∈ (SubGrp‘𝐺) → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴)))
 
Theoremsubgdprd 18164 A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐻 = (𝐺s 𝐴)    &   (𝜑𝐴 ∈ (SubGrp‘𝐺))    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → ran 𝑆 ⊆ 𝒫 𝐴)       (𝜑 → (𝐻 DProd 𝑆) = (𝐺 DProd 𝑆))
 
Theoremdprdsn 18165 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {⟨𝐴, 𝑆⟩} ∧ (𝐺 DProd {⟨𝐴, 𝑆⟩}) = 𝑆))
 
Theoremdmdprdsplitlem 18166* Lemma for dmdprdsplit 18176. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
0 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐴𝐼)    &   (𝜑𝐹𝑊)    &   (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd (𝑆𝐴)))       ((𝜑𝑋 ∈ (𝐼𝐴)) → (𝐹𝑋) = 0 )
 
Theoremdprdcntz2 18167 The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐶𝐼)    &   (𝜑𝐷𝐼)    &   (𝜑 → (𝐶𝐷) = ∅)    &   𝑍 = (Cntz‘𝐺)       (𝜑 → (𝐺 DProd (𝑆𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆𝐷))))
 
Theoremdprddisj2 18168 The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.) (Revised by AV, 14-Jul-2019.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐶𝐼)    &   (𝜑𝐷𝐼)    &   (𝜑 → (𝐶𝐷) = ∅)    &    0 = (0g𝐺)       (𝜑 → ((𝐺 DProd (𝑆𝐶)) ∩ (𝐺 DProd (𝑆𝐷))) = { 0 })
 
Theoremdprd2dlem2 18169* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑 → Rel 𝐴)    &   (𝜑𝑆:𝐴⟶(SubGrp‘𝐺))    &   (𝜑 → dom 𝐴𝐼)    &   ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))    &   (𝜑𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝜑𝑋𝐴) → (𝑆𝑋) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))))
 
Theoremdprd2dlem1 18170* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑 → Rel 𝐴)    &   (𝜑𝑆:𝐴⟶(SubGrp‘𝐺))    &   (𝜑 → dom 𝐴𝐼)    &   ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))    &   (𝜑𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))    &   𝐾 = (mrCls‘(SubGrp‘𝐺))    &   (𝜑𝐶𝐼)       (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) = (𝐺 DProd (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
 
Theoremdprd2da 18171* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑 → Rel 𝐴)    &   (𝜑𝑆:𝐴⟶(SubGrp‘𝐺))    &   (𝜑 → dom 𝐴𝐼)    &   ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))    &   (𝜑𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       (𝜑𝐺dom DProd 𝑆)
 
Theoremdprd2db 18172* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑 → Rel 𝐴)    &   (𝜑𝑆:𝐴⟶(SubGrp‘𝐺))    &   (𝜑 → dom 𝐴𝐼)    &   ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))    &   (𝜑𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       (𝜑 → (𝐺 DProd 𝑆) = (𝐺 DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
 
Theoremdprd2d2 18173* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
((𝜑 ∧ (𝑖𝐼𝑗𝐽)) → 𝑆 ∈ (SubGrp‘𝐺))    &   ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗𝐽𝑆))    &   (𝜑𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗𝐽𝑆))))       (𝜑 → (𝐺dom DProd (𝑖𝐼, 𝑗𝐽𝑆) ∧ (𝐺 DProd (𝑖𝐼, 𝑗𝐽𝑆)) = (𝐺 DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗𝐽𝑆))))))
 
Theoremdmdprdsplit2lem 18174 Lemma for dmdprdsplit 18176. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝑆:𝐼⟶(SubGrp‘𝐺))    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐼 = (𝐶𝐷))    &   𝑍 = (Cntz‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺dom DProd (𝑆𝐶))    &   (𝜑𝐺dom DProd (𝑆𝐷))    &   (𝜑 → (𝐺 DProd (𝑆𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆𝐷))))    &   (𝜑 → ((𝐺 DProd (𝑆𝐶)) ∩ (𝐺 DProd (𝑆𝐷))) = { 0 })    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝜑𝑋𝐶) → ((𝑌𝐼 → (𝑋𝑌 → (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑌)))) ∧ ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))) ⊆ { 0 }))
 
Theoremdmdprdsplit2 18175 The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝑆:𝐼⟶(SubGrp‘𝐺))    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐼 = (𝐶𝐷))    &   𝑍 = (Cntz‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺dom DProd (𝑆𝐶))    &   (𝜑𝐺dom DProd (𝑆𝐷))    &   (𝜑 → (𝐺 DProd (𝑆𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆𝐷))))    &   (𝜑 → ((𝐺 DProd (𝑆𝐶)) ∩ (𝐺 DProd (𝑆𝐷))) = { 0 })       (𝜑𝐺dom DProd 𝑆)
 
Theoremdmdprdsplit 18176 The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝑆:𝐼⟶(SubGrp‘𝐺))    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐼 = (𝐶𝐷))    &   𝑍 = (Cntz‘𝐺)    &    0 = (0g𝐺)       (𝜑 → (𝐺dom DProd 𝑆 ↔ ((𝐺dom DProd (𝑆𝐶) ∧ 𝐺dom DProd (𝑆𝐷)) ∧ (𝐺 DProd (𝑆𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆𝐷))) ∧ ((𝐺 DProd (𝑆𝐶)) ∩ (𝐺 DProd (𝑆𝐷))) = { 0 })))
 
Theoremdprdsplit 18177 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝑆:𝐼⟶(SubGrp‘𝐺))    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐼 = (𝐶𝐷))    &    = (LSSum‘𝐺)    &   (𝜑𝐺dom DProd 𝑆)       (𝜑 → (𝐺 DProd 𝑆) = ((𝐺 DProd (𝑆𝐶)) (𝐺 DProd (𝑆𝐷))))
 
Theoremdmdprdpr 18178 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
𝑍 = (Cntz‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))       (𝜑 → (𝐺dom DProd ({𝑆} +𝑐 {𝑇}) ↔ (𝑆 ⊆ (𝑍𝑇) ∧ (𝑆𝑇) = { 0 })))
 
Theoremdprdpr 18179 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 26-Apr-2016.)
𝑍 = (Cntz‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &    = (LSSum‘𝐺)    &   (𝜑𝑆 ⊆ (𝑍𝑇))    &   (𝜑 → (𝑆𝑇) = { 0 })       (𝜑 → (𝐺 DProd ({𝑆} +𝑐 {𝑇})) = (𝑆 𝑇))
 
Theoremdpjlem 18180 Lemma for theorems about direct product projection. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)       (𝜑 → (𝐺 DProd (𝑆 ↾ {𝑋})) = (𝑆𝑋))
 
Theoremdpjcntz 18181 The two subgroups that appear in dpjval 18185 commute. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)    &   𝑍 = (Cntz‘𝐺)       (𝜑 → (𝑆𝑋) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
 
Theoremdpjdisj 18182 The two subgroups that appear in dpjval 18185 are disjoint. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)    &    0 = (0g𝐺)       (𝜑 → ((𝑆𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = { 0 })
 
Theoremdpjlsm 18183 The two subgroups that appear in dpjval 18185 add to the full direct product. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)    &    = (LSSum‘𝐺)       (𝜑 → (𝐺 DProd 𝑆) = ((𝑆𝑋) (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
 
Theoremdpjfval 18184* Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   𝑄 = (proj1𝐺)       (𝜑𝑃 = (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))
 
Theoremdpjval 18185 Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   𝑄 = (proj1𝐺)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑃𝑋) = ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
 
Theoremdpjf 18186 The 𝑋-th index projection is a function from the direct product to the 𝑋-th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑃𝑋):(𝐺 DProd 𝑆)⟶(𝑆𝑋))
 
Theoremdpjidcl 18187* The key property of projections: the sum of all the projections of 𝐴 is 𝐴. (Contributed by Mario Carneiro, 26-Apr-2016.) (Revised by AV, 14-Jul-2019.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝐴 ∈ (𝐺 DProd 𝑆))    &    0 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }       (𝜑 → ((𝑥𝐼 ↦ ((𝑃𝑥)‘𝐴)) ∈ 𝑊𝐴 = (𝐺 Σg (𝑥𝐼 ↦ ((𝑃𝑥)‘𝐴)))))
 
Theoremdpjeq 18188* Decompose a group sum into projections. (Contributed by Mario Carneiro, 26-Apr-2016.) (Revised by AV, 14-Jul-2019.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝐴 ∈ (𝐺 DProd 𝑆))    &    0 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑 → (𝑥𝐼𝐶) ∈ 𝑊)       (𝜑 → (𝐴 = (𝐺 Σg (𝑥𝐼𝐶)) ↔ ∀𝑥𝐼 ((𝑃𝑥)‘𝐴) = 𝐶))
 
Theoremdpjid 18189* The key property of projections: the sum of all the projections of 𝐴 is 𝐴. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝐴 ∈ (𝐺 DProd 𝑆))       (𝜑𝐴 = (𝐺 Σg (𝑥𝐼 ↦ ((𝑃𝑥)‘𝐴))))
 
Theoremdpjlid 18190 The 𝑋-th index projection acts as the identity on elements of the 𝑋-th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝑋𝐼)    &   (𝜑𝐴 ∈ (𝑆𝑋))       (𝜑 → ((𝑃𝑋)‘𝐴) = 𝐴)
 
Theoremdpjrid 18191 The 𝑌-th index projection annihilates elements of other factors. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝑋𝐼)    &   (𝜑𝐴 ∈ (𝑆𝑋))    &    0 = (0g𝐺)    &   (𝜑𝑌𝐼)    &   (𝜑𝑌𝑋)       (𝜑 → ((𝑃𝑌)‘𝐴) = 0 )
 
Theoremdpjghm 18192 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑃𝑋) ∈ ((𝐺s (𝐺 DProd 𝑆)) GrpHom 𝐺))
 
Theoremdpjghm2 18193 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑃𝑋) ∈ ((𝐺s (𝐺 DProd 𝑆)) GrpHom (𝐺s (𝑆𝑋))))
 
10.3.6  The Fundamental Theorem of Abelian Groups
 
Theoremablfacrplem 18194* Lemma for ablfacrp2 18196. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = {𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑀}    &   𝐿 = {𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑁}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   (𝜑 → (#‘𝐵) = (𝑀 · 𝑁))       (𝜑 → ((#‘𝐾) gcd 𝑁) = 1)
 
Theoremablfacrp 18195* A finite abelian group whose order factors into relatively prime integers, itself "factors" into two subgroups 𝐾, 𝐿 that have trivial intersection and whose product is the whole group. Lemma 6.1C.2 of [Shapiro], p. 199. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = {𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑀}    &   𝐿 = {𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑁}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   (𝜑 → (#‘𝐵) = (𝑀 · 𝑁))    &    0 = (0g𝐺)    &    = (LSSum‘𝐺)       (𝜑 → ((𝐾𝐿) = { 0 } ∧ (𝐾 𝐿) = 𝐵))
 
Theoremablfacrp2 18196* The factors 𝐾, 𝐿 of ablfacrp 18195 have the expected orders (which allows for repeated application to decompose 𝐺 into subgroups of prime-power order). Lemma 6.1C.2 of [Shapiro], p. 199. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = {𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑀}    &   𝐿 = {𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑁}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   (𝜑 → (#‘𝐵) = (𝑀 · 𝑁))       (𝜑 → ((#‘𝐾) = 𝑀 ∧ (#‘𝐿) = 𝑁))
 
Theoremablfac1lem 18197* Lemma for ablfac1b 18199. Satisfy the assumptions of ablfacrp. (Contributed by Mario Carneiro, 26-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))})    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐴 ⊆ ℙ)    &   𝑀 = (𝑃↑(𝑃 pCnt (#‘𝐵)))    &   𝑁 = ((#‘𝐵) / 𝑀)       ((𝜑𝑃𝐴) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀 gcd 𝑁) = 1 ∧ (#‘𝐵) = (𝑀 · 𝑁)))
 
Theoremablfac1a 18198* The factors of ablfac1b 18199 are of prime power order. (Contributed by Mario Carneiro, 26-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))})    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐴 ⊆ ℙ)       ((𝜑𝑃𝐴) → (#‘(𝑆𝑃)) = (𝑃↑(𝑃 pCnt (#‘𝐵))))
 
Theoremablfac1b 18199* Any abelian group is the direct product of factors of prime power order (with the exact order further matching the prime factorization of the group order). (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))})    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐴 ⊆ ℙ)       (𝜑𝐺dom DProd 𝑆)
 
Theoremablfac1c 18200* The factors of ablfac1b 18199 cover the entire group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))})    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐴 ⊆ ℙ)    &   𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (#‘𝐵)}    &   (𝜑𝐷𝐴)       (𝜑 → (𝐺 DProd 𝑆) = 𝐵)
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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 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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 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