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Theorem List for Metamath Proof Explorer - 18401-18500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdprdcntz 18401 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 18402 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 18403* 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 18404* 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 18405* 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 18406* 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 18407* 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 18408* 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 18409 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 18410* 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 18411* 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 18412* 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 18413* 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 18414* 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 18415* 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 18416* 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 18417 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 18418 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 18419* 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 18420 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 18421 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 18422* 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 18423* 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 18424 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 18425 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 18426 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 18427 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 18428 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 18429 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 18430* Lemma for dmdprdsplit 18440. (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 18431 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 18432 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 18433* 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 18434* 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 18435* 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 18436* 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 18437* 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 18438 Lemma for dmdprdsplit 18440. (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 18439 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 18440 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 18441 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 18442 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 18443 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 18444 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 18445 The two subgroups that appear in dpjval 18449 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 18446 The two subgroups that appear in dpjval 18449 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 18447 The two subgroups that appear in dpjval 18449 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 18448* 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 18449 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 18450 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 18451* 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 18452* 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 18453* 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 18454 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 18455 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 18456 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 18457 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 18458* Lemma for ablfacrp2 18460. (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 18459* 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 18460* The factors 𝐾, 𝐿 of ablfacrp 18459 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 18461* Lemma for ablfac1b 18463. 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 18462* The factors of ablfac1b 18463 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 18463* 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 18464* The factors of ablfac1b 18463 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 18465* Lemma for ablfac1eu 18466. (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 18466* The factorization of ablfac1b 18463 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 18467* Lemma for pgpfac1 18473. (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 18468* Lemma for pgpfac1 18473. (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 18469* Lemma for pgpfac1 18473. (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 18470* Lemma for pgpfac1 18473. (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 18471* Lemma for pgpfac1 18473. (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))

Theorempgpfac1lem5 18472* Lemma for pgpfac1 18473. (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)    &   (𝜑 → ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠𝑈𝐴𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠)))       (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))

Theorempgpfac1 18473* Factorization of a finite abelian p-group. There is a direct product decomposition of any abelian group of prime-power order where one of the factors is cyclic and generated by an element of maximal order. (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)    &   (𝜑𝐴𝐵)       (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝐵))

Theorempgpfaclem1 18474* Lemma for pgpfac 18477. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))    &   𝐻 = (𝐺s 𝑈)    &   𝐾 = (mrCls‘(SubGrp‘𝐻))    &   𝑂 = (od‘𝐻)    &   𝐸 = (gEx‘𝐻)    &    0 = (0g𝐻)    &    = (LSSum‘𝐻)    &   (𝜑𝐸 ≠ 1)    &   (𝜑𝑋𝑈)    &   (𝜑 → (𝑂𝑋) = 𝐸)    &   (𝜑𝑊 ∈ (SubGrp‘𝐻))    &   (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })    &   (𝜑 → ((𝐾‘{𝑋}) 𝑊) = 𝑈)    &   (𝜑𝑆 ∈ Word 𝐶)    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → (𝐺 DProd 𝑆) = 𝑊)    &   𝑇 = (𝑆 ++ ⟨“(𝐾‘{𝑋})”⟩)       (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))

Theorempgpfaclem2 18475* Lemma for pgpfac 18477. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))    &   𝐻 = (𝐺s 𝑈)    &   𝐾 = (mrCls‘(SubGrp‘𝐻))    &   𝑂 = (od‘𝐻)    &   𝐸 = (gEx‘𝐻)    &    0 = (0g𝐻)    &    = (LSSum‘𝐻)    &   (𝜑𝐸 ≠ 1)    &   (𝜑𝑋𝑈)    &   (𝜑 → (𝑂𝑋) = 𝐸)    &   (𝜑𝑊 ∈ (SubGrp‘𝐻))    &   (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 })    &   (𝜑 → ((𝐾‘{𝑋}) 𝑊) = 𝑈)       (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))

Theorempgpfaclem3 18476* Lemma for pgpfac 18477. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))       (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))

Theorempgpfac 18477* Full factorization of a finite abelian p-group, by iterating pgpfac1 18473. There is a direct product decomposition of any abelian group of prime-power order into cyclic subgroups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))

Theoremablfaclem1 18478* Lemma for ablfac 18481. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   𝑂 = (od‘𝐺)    &   𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (#‘𝐵)}    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))})    &   𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})       (𝑈 ∈ (SubGrp‘𝐺) → (𝑊𝑈) = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈)})

Theoremablfaclem2 18479* Lemma for ablfac 18481. (Contributed by Mario Carneiro, 27-Apr-2016.) (Proof shortened by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   𝑂 = (od‘𝐺)    &   𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (#‘𝐵)}    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))})    &   𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})    &   (𝜑𝐹:𝐴⟶Word 𝐶)    &   (𝜑 → ∀𝑦𝐴 (𝐹𝑦) ∈ (𝑊‘(𝑆𝑦)))    &   𝐿 = 𝑦𝐴 ({𝑦} × dom (𝐹𝑦))    &   (𝜑𝐻:(0..^(#‘𝐿))–1-1-onto𝐿)       (𝜑 → (𝑊𝐵) ≠ ∅)

Theoremablfaclem3 18480* Lemma for ablfac 18481. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &   𝑂 = (od‘𝐺)    &   𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (#‘𝐵)}    &   𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (#‘𝐵)))})    &   𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})       (𝜑 → (𝑊𝐵) ≠ ∅)

Theoremablfac 18481* The Fundamental Theorem of (finite) Abelian Groups. Any finite abelian group is a direct product of cyclic p-groups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵))

Theoremablfac2 18482* Choose generators for each cyclic group in ablfac 18481. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐵 ∈ Fin)    &    · = (.g𝐺)    &   𝑆 = (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤𝑘))))       (𝜑 → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤𝐶𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵))

10.4  Rings

10.4.1  Multiplicative Group

Syntaxcmgp 18483 Multiplicative group.
class mulGrp

Definitiondf-mgp 18484 Define a structure that puts the multiplication operation of a ring in the addition slot. Note that this will not actually be a group for the average ring, or even for a field, but it will be a monoid, and unitgrp 18661 shows that we get a group if we restrict to the elements that have inverses. This allows us to formalize such notions as "the multiplication operation of a ring is a monoid" (ringmgp 18547) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 9880). (Contributed by Mario Carneiro, 21-Dec-2014.)
mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r𝑤)⟩))

Theoremfnmgp 18485 The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
mulGrp Fn V

Theoremmgpval 18486 Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
𝑀 = (mulGrp‘𝑅)    &    · = (.r𝑅)       𝑀 = (𝑅 sSet ⟨(+g‘ndx), · ⟩)

Theoremmgpplusg 18487 Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
𝑀 = (mulGrp‘𝑅)    &    · = (.r𝑅)        · = (+g𝑀)

Theoremmgplem 18488 Lemma for mgpbas 18489. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 ≠ 2       (𝐸𝑅) = (𝐸𝑀)

Theoremmgpbas 18489 Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝐵 = (Base‘𝑅)       𝐵 = (Base‘𝑀)

Theoremmgpsca 18490 The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 18612. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝑆 = (Scalar‘𝑅)       𝑆 = (Scalar‘𝑀)

Theoremmgptset 18491 Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)       (TopSet‘𝑅) = (TopSet‘𝑀)

Theoremmgptopn 18492 Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝐽 = (TopOpen‘𝑅)       𝐽 = (TopOpen‘𝑀)

Theoremmgpds 18493 Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝐵 = (dist‘𝑅)       𝐵 = (dist‘𝑀)

Theoremmgpress 18494 Subgroup commutes with the multiplication group operator. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑆 = (𝑅s 𝐴)    &   𝑀 = (mulGrp‘𝑅)       ((𝑅𝑉𝐴𝑊) → (𝑀s 𝐴) = (mulGrp‘𝑆))

10.4.2  Ring unit

Syntaxcur 18495 Extend class notation with ring unit.
class 1r

Definitiondf-ur 18496 Define the multiplicative neutral element of a ring. This definition works by extracting the 0g element, i.e. the neutral element in a group or monoid, and transferring it to the multiplicative monoid via the mulGrp function (df-mgp 18484). See also dfur2 18498, which derives the "traditional" definition as the unique element of a ring which is left- and right-neutral under multiplication. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
1r = (0g ∘ mulGrp)

Theoremringidval 18497 The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐺 = (mulGrp‘𝑅)    &    1 = (1r𝑅)        1 = (0g𝐺)

Theoremdfur2 18498* The multiplicative identity is the unique element of the ring that is left- and right-neutral on all elements under multiplication. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)        1 = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥)))

10.4.2.1  Semirings

Syntaxcsrg 18499 Extend class notation with the class of all semirings.
class SRing

Definitiondf-srg 18500* Define class of all semirings. A semiring is a set equipped with two everywhere-defined internal operations, whose first one is an additive commutative monoid structure and the second one is a multiplicative monoid structure, and where multiplication is (left- and right-) distributive over addition. Compared to the definition of a ring, this definition also adds that the additive identity is an absorbing element of the multiplicative law, as this cannot be deduced from distributivity alone. Definition of [Golan] p. 1. Note that our semirings are unital. Such semirings are sometimes called "rigs", being "rings without negatives". (Contributed by Thierry Arnoux, 21-Mar-2018.)
SRing = {𝑓 ∈ CMnd ∣ ((mulGrp‘𝑓) ∈ Mnd ∧ [(Base‘𝑓) / 𝑟][(+g𝑓) / 𝑝][(.r𝑓) / 𝑡][(0g𝑓) / 𝑛]𝑥𝑟 (∀𝑦𝑟𝑧𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))}

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