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Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremprdsgsum 19101* 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 19102* 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 19103* 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 19104* 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 19105* 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 19106* 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 19107* 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 19108* Lemma for telgsumfzs 19109 (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 19109* 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 19110* Telescoping group sum ranging over a finite set of sequential integers, using implicit substitution, analogous to telfsum 15159. (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 19111* 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 19112* Telescoping finite group sum ranging over nonnegative integers, using implicit substitution, analogous to telfsum 15159. (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 19113* 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 19114* 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)

10.2.14.5  Internal direct products

Syntaxcdprd 19115 Internal direct product of a family of subgroups.
class DProd

Syntaxcdpj 19116 Projection operator for a direct product.
class dProj

Definitiondf-dprd 19117* 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 19118* 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 19119 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 19120* 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 19121* 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 19122 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 19123 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 19124 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 19125* 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 19126* 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 19127 Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝐺dom DProd 𝑆𝐺 ∈ Grp)

Theoremdprdf 19128 The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝐺dom DProd 𝑆𝑆:dom 𝑆⟶(SubGrp‘𝐺))

Theoremdprdf2 19129 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 19130 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 19131 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 19132* 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 19133* 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 19134* 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 19135* 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 19136* 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 19137* 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 19138 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 19139* 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 19140* 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 19141* 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 19142* 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 19143* 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 19144* 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 19145* 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 19146 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 19147 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 19148* 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 19149 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 19150 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 19151* 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 19152* 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 19153 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 19154 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 19155 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 19156 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 19157 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 19158 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 19159* Lemma for dmdprdsplit 19169. (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 19160 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 19161 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 19162* 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 19163* 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 19164* 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 19165* 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 19166* 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 19167 Lemma for dmdprdsplit 19169. (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 19168 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 19169 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 19170 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 19171 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⟩, ⟨1o, 𝑇⟩} ↔ (𝑆 ⊆ (𝑍𝑇) ∧ (𝑆𝑇) = { 0 })))

Theoremdprdpr 19172 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⟩, ⟨1o, 𝑇⟩}) = (𝑆 𝑇))

Theoremdpjlem 19173 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 19174 The two subgroups that appear in dpjval 19178 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 19175 The two subgroups that appear in dpjval 19178 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 19176 The two subgroups that appear in dpjval 19178 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 19177* 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 19178 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 19179 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 19180* 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 19181* 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 19182* 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 19183 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 19184 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 19185 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 19186 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))))

10.2.14.6  The Fundamental Theorem of Abelian Groups

Theoremablfacrplem 19187* Lemma for ablfacrp2 19189. (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 19188* 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 19189* The factors 𝐾, 𝐿 of ablfacrp 19188 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 19190* Lemma for ablfac1b 19192. 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 19191* The factors of ablfac1b 19192 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 19192* 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 19193* The factors of ablfac1b 19192 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)

Theoremablfac1eulem 19194* Lemma for ablfac1eu 19195. (Contributed by Mario Carneiro, 27-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 → (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝐵))    &   (𝜑 → dom 𝑇 = 𝐴)    &   ((𝜑𝑞𝐴) → 𝐶 ∈ ℕ0)    &   ((𝜑𝑞𝐴) → (♯‘(𝑇𝑞)) = (𝑞𝐶))    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ Fin)       (𝜑 → ¬ 𝑃 ∥ (♯‘(𝐺 DProd (𝑇 ↾ (𝐴 ∖ {𝑃})))))

Theoremablfac1eu 19195* The factorization of ablfac1b 19192 is unique, in that any other factorization into prime power factors (even if the exponents are different) must be equal to 𝑆. (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 → (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝐵))    &   (𝜑 → dom 𝑇 = 𝐴)    &   ((𝜑𝑞𝐴) → 𝐶 ∈ ℕ0)    &   ((𝜑𝑞𝐴) → (♯‘(𝑇𝑞)) = (𝑞𝐶))       (𝜑𝑇 = 𝑆)

Theorempgpfac1lem1 19196* Lemma for pgpfac1 19202. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐾 = (mrCls‘(SubGrp‘𝐺))    &   𝑆 = (𝐾‘{𝐴})    &   𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    0 = (0g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → (𝑂𝐴) = 𝐸)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑆𝑊) = { 0 })    &   (𝜑 → (𝑆 𝑊) ⊆ 𝑈)    &   (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ (𝑆 𝑊) ⊊ 𝑤))       ((𝜑𝐶 ∈ (𝑈 ∖ (𝑆 𝑊))) → ((𝑆 𝑊) (𝐾‘{𝐶})) = 𝑈)

Theorempgpfac1lem2 19197* Lemma for pgpfac1 19202. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐾 = (mrCls‘(SubGrp‘𝐺))    &   𝑆 = (𝐾‘{𝐴})    &   𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    0 = (0g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → (𝑂𝐴) = 𝐸)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑆𝑊) = { 0 })    &   (𝜑 → (𝑆 𝑊) ⊆ 𝑈)    &   (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ (𝑆 𝑊) ⊊ 𝑤))    &   (𝜑𝐶 ∈ (𝑈 ∖ (𝑆 𝑊)))    &    · = (.g𝐺)       (𝜑 → (𝑃 · 𝐶) ∈ (𝑆 𝑊))

Theorempgpfac1lem3a 19198* Lemma for pgpfac1 19202. (Contributed by Mario Carneiro, 4-Jun-2016.)
𝐾 = (mrCls‘(SubGrp‘𝐺))    &   𝑆 = (𝐾‘{𝐴})    &   𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    0 = (0g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → (𝑂𝐴) = 𝐸)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑆𝑊) = { 0 })    &   (𝜑 → (𝑆 𝑊) ⊆ 𝑈)    &   (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ (𝑆 𝑊) ⊊ 𝑤))    &   (𝜑𝐶 ∈ (𝑈 ∖ (𝑆 𝑊)))    &    · = (.g𝐺)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ((𝑃 · 𝐶)(+g𝐺)(𝑀 · 𝐴)) ∈ 𝑊)       (𝜑 → (𝑃𝐸𝑃𝑀))

Theorempgpfac1lem3 19199* Lemma for pgpfac1 19202. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐾 = (mrCls‘(SubGrp‘𝐺))    &   𝑆 = (𝐾‘{𝐴})    &   𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    0 = (0g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → (𝑂𝐴) = 𝐸)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑆𝑊) = { 0 })    &   (𝜑 → (𝑆 𝑊) ⊆ 𝑈)    &   (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ (𝑆 𝑊) ⊊ 𝑤))    &   (𝜑𝐶 ∈ (𝑈 ∖ (𝑆 𝑊)))    &    · = (.g𝐺)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ((𝑃 · 𝐶)(+g𝐺)(𝑀 · 𝐴)) ∈ 𝑊)    &   𝐷 = (𝐶(+g𝐺)((𝑀 / 𝑃) · 𝐴))       (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))

Theorempgpfac1lem4 19200* Lemma for pgpfac1 19202. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐾 = (mrCls‘(SubGrp‘𝐺))    &   𝑆 = (𝐾‘{𝐴})    &   𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    0 = (0g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → (𝑂𝐴) = 𝐸)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑆𝑊) = { 0 })    &   (𝜑 → (𝑆 𝑊) ⊆ 𝑈)    &   (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ (𝑆 𝑊) ⊊ 𝑤))    &   (𝜑𝐶 ∈ (𝑈 ∖ (𝑆 𝑊)))    &    · = (.g𝐺)       (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))

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