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Type | Label | Description |
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Statement | ||
Theorem | dprdwd 18801* | 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 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ (𝑆‘𝑥)) & ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) finSupp 0 ) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ 𝑊) | ||
Theorem | dprdff 18802* | 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 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) | ||
Theorem | dprdfcl 18803* | 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 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐼) → (𝐹‘𝑋) ∈ (𝑆‘𝑋)) | ||
Theorem | dprdffsupp 18804* | 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 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐹 finSupp 0 ) | ||
Theorem | dprdfcntz 18805* | 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 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) & ⊢ 𝑍 = (Cntz‘𝐺) ⇒ ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) | ||
Theorem | dprdssv 18806 | 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 | ||
Theorem | dprdfid 18807* | 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 = (0g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝐴 ∈ (𝑆‘𝑋)) & ⊢ 𝐹 = (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) ⇒ ⊢ (𝜑 → (𝐹 ∈ 𝑊 ∧ (𝐺 Σg 𝐹) = 𝐴)) | ||
Theorem | eldprdi 18808* | 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 = (0g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆)) | ||
Theorem | dprdfinv 18809* | 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 = (0g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ (𝜑 → ((𝑁 ∘ 𝐹) ∈ 𝑊 ∧ (𝐺 Σg (𝑁 ∘ 𝐹)) = (𝑁‘(𝐺 Σg 𝐹)))) | ||
Theorem | dprdfadd 18810* | 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 = (0g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) & ⊢ (𝜑 → 𝐻 ∈ 𝑊) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝜑 → ((𝐹 ∘𝑓 + 𝐻) ∈ 𝑊 ∧ (𝐺 Σg (𝐹 ∘𝑓 + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))) | ||
Theorem | dprdfsub 18811* | 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 = (0g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) & ⊢ (𝜑 → 𝐻 ∈ 𝑊) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝜑 → ((𝐹 ∘𝑓 − 𝐻) ∈ 𝑊 ∧ (𝐺 Σg (𝐹 ∘𝑓 − 𝐻)) = ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻)))) | ||
Theorem | dprdfeq0 18812* | 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 = (0g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) ⇒ ⊢ (𝜑 → ((𝐺 Σg 𝐹) = 0 ↔ 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ))) | ||
Theorem | dprdf11 18813* | 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 = (0g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) & ⊢ (𝜑 → 𝐻 ∈ 𝑊) ⇒ ⊢ (𝜑 → ((𝐺 Σg 𝐹) = (𝐺 Σg 𝐻) ↔ 𝐹 = 𝐻)) | ||
Theorem | dprdsubg 18814 | 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)) | ||
Theorem | dprdub 18815 | Each factor is a subset of the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑆‘𝑋) ⊆ (𝐺 DProd 𝑆)) | ||
Theorem | dprdlub 18816* | The direct product is smaller than any subgroup which contains the factors. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑆‘𝑘) ⊆ 𝑇) ⇒ ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝑇) | ||
Theorem | dprdspan 18817 | 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)) | ||
Theorem | dprdres 18818 | Restriction of a direct product (dropping factors). (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐴 ⊆ 𝐼) ⇒ ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ 𝐴) ∧ (𝐺 DProd (𝑆 ↾ 𝐴)) ⊆ (𝐺 DProd 𝑆))) | ||
Theorem | dprdss 18819* | Create a direct product by finding subgroups inside each factor of another direct product. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑇) & ⊢ (𝜑 → dom 𝑇 = 𝐼) & ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑆‘𝑘) ⊆ (𝑇‘𝑘)) ⇒ ⊢ (𝜑 → (𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) ⊆ (𝐺 DProd 𝑇))) | ||
Theorem | dprdz 18820* | 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 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐼 ∈ 𝑉) → (𝐺dom DProd (𝑥 ∈ 𝐼 ↦ { 0 }) ∧ (𝐺 DProd (𝑥 ∈ 𝐼 ↦ { 0 })) = { 0 })) | ||
Theorem | dprd0 18821 | The empty family is an internal direct product, the product of which is the trivial subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = { 0 })) | ||
Theorem | dprdf1o 18822 | Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐹:𝐽–1-1-onto→𝐼) ⇒ ⊢ (𝜑 → (𝐺dom DProd (𝑆 ∘ 𝐹) ∧ (𝐺 DProd (𝑆 ∘ 𝐹)) = (𝐺 DProd 𝑆))) | ||
Theorem | dprdf1 18823 | Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐹:𝐽–1-1→𝐼) ⇒ ⊢ (𝜑 → (𝐺dom DProd (𝑆 ∘ 𝐹) ∧ (𝐺 DProd (𝑆 ∘ 𝐹)) ⊆ (𝐺 DProd 𝑆))) | ||
Theorem | subgdmdprd 18824 | A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ 𝐻 = (𝐺 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴))) | ||
Theorem | subgdprd 18825 | A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ (𝜑 → 𝐴 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 𝐴) ⇒ ⊢ (𝜑 → (𝐻 DProd 𝑆) = (𝐺 DProd 𝑆)) | ||
Theorem | dprdsn 18826 | A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {〈𝐴, 𝑆〉} ∧ (𝐺 DProd {〈𝐴, 𝑆〉}) = 𝑆)) | ||
Theorem | dmdprdsplitlem 18827* | Lemma for dmdprdsplit 18837. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.) |
⊢ 0 = (0g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐴 ⊆ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) & ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd (𝑆 ↾ 𝐴))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) → (𝐹‘𝑋) = 0 ) | ||
Theorem | dprdcntz2 18828 | The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐶 ⊆ 𝐼) & ⊢ (𝜑 → 𝐷 ⊆ 𝐼) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ 𝑍 = (Cntz‘𝐺) ⇒ ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷)))) | ||
Theorem | dprddisj2 18829 | The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.) (Revised by AV, 14-Jul-2019.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐶 ⊆ 𝐼) & ⊢ (𝜑 → 𝐷 ⊆ 𝐼) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 }) | ||
Theorem | dprd2dlem2 18830* | The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → Rel 𝐴) & ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺)) & ⊢ (𝜑 → dom 𝐴 ⊆ 𝐼) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) & ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝑆‘𝑋) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st ‘𝑋)}) ↦ ((1st ‘𝑋)𝑆𝑗)))) | ||
Theorem | dprd2dlem1 18831* | The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → Rel 𝐴) & ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺)) & ⊢ (𝜑 → dom 𝐴 ⊆ 𝐼) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) & ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐼) ⇒ ⊢ (𝜑 → (𝐾‘∪ (𝑆 “ (𝐴 ↾ 𝐶))) = (𝐺 DProd (𝑖 ∈ 𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))) | ||
Theorem | dprd2da 18832* | The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → Rel 𝐴) & ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺)) & ⊢ (𝜑 → dom 𝐴 ⊆ 𝐼) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) & ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) ⇒ ⊢ (𝜑 → 𝐺dom DProd 𝑆) | ||
Theorem | dprd2db 18833* | The direct product of a collection of direct products. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝜑 → Rel 𝐴) & ⊢ (𝜑 → 𝑆:𝐴⟶(SubGrp‘𝐺)) & ⊢ (𝜑 → dom 𝐴 ⊆ 𝐼) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) & ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) ⇒ ⊢ (𝜑 → (𝐺 DProd 𝑆) = (𝐺 DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))) | ||
Theorem | dprd2d2 18834* | The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽)) → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝐺dom DProd (𝑗 ∈ 𝐽 ↦ 𝑆)) & ⊢ (𝜑 → 𝐺dom DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ 𝑆)))) ⇒ ⊢ (𝜑 → (𝐺dom DProd (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆) ∧ (𝐺 DProd (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ 𝑆)) = (𝐺 DProd (𝑖 ∈ 𝐼 ↦ (𝐺 DProd (𝑗 ∈ 𝐽 ↦ 𝑆)))))) | ||
Theorem | dmdprdsplit2lem 18835 | Lemma for dmdprdsplit 18837. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐼 = (𝐶 ∪ 𝐷)) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐶)) & ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐷)) & ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷)))) & ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 }) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑌 ∈ 𝐼 → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))) ∧ ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) ⊆ { 0 })) | ||
Theorem | dmdprdsplit2 18836 | The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐼 = (𝐶 ∪ 𝐷)) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐶)) & ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐷)) & ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷)))) & ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 }) ⇒ ⊢ (𝜑 → 𝐺dom DProd 𝑆) | ||
Theorem | dmdprdsplit 18837 | The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐼 = (𝐶 ∪ 𝐷)) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝜑 → (𝐺dom DProd 𝑆 ↔ ((𝐺dom DProd (𝑆 ↾ 𝐶) ∧ 𝐺dom DProd (𝑆 ↾ 𝐷)) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 }))) | ||
Theorem | dprdsplit 18838 | The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐼 = (𝐶 ∪ 𝐷)) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝐺dom DProd 𝑆) ⇒ ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝐺 DProd (𝑆 ↾ 𝐶)) ⊕ (𝐺 DProd (𝑆 ↾ 𝐷)))) | ||
Theorem | dmdprdpr 18839 | 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 = (0g‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) ⇒ ⊢ (𝜑 → (𝐺dom DProd ◡({𝑆} +𝑐 {𝑇}) ↔ (𝑆 ⊆ (𝑍‘𝑇) ∧ (𝑆 ∩ 𝑇) = { 0 }))) | ||
Theorem | dprdpr 18840 | 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 = (0g‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑆 ⊆ (𝑍‘𝑇)) & ⊢ (𝜑 → (𝑆 ∩ 𝑇) = { 0 }) ⇒ ⊢ (𝜑 → (𝐺 DProd ◡({𝑆} +𝑐 {𝑇})) = (𝑆 ⊕ 𝑇)) | ||
Theorem | dpjlem 18841 | Lemma for theorems about direct product projection. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ {𝑋})) = (𝑆‘𝑋)) | ||
Theorem | dpjcntz 18842 | The two subgroups that appear in dpjval 18846 commute. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ 𝑍 = (Cntz‘𝐺) ⇒ ⊢ (𝜑 → (𝑆‘𝑋) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) | ||
Theorem | dpjdisj 18843 | The two subgroups that appear in dpjval 18846 are disjoint. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = { 0 }) | ||
Theorem | dpjlsm 18844 | The two subgroups that appear in dpjval 18846 add to the full direct product. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝑆‘𝑋) ⊕ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) | ||
Theorem | dpjfval 18845* | Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ 𝑃 = (𝐺dProj𝑆) & ⊢ 𝑄 = (proj1‘𝐺) ⇒ ⊢ (𝜑 → 𝑃 = (𝑖 ∈ 𝐼 ↦ ((𝑆‘𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖})))))) | ||
Theorem | dpjval 18846 | Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ 𝑃 = (𝐺dProj𝑆) & ⊢ 𝑄 = (proj1‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) | ||
Theorem | dpjf 18847 | The 𝑋-th index projection is a function from the direct product to the 𝑋-th factor. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ 𝑃 = (𝐺dProj𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑃‘𝑋):(𝐺 DProd 𝑆)⟶(𝑆‘𝑋)) | ||
Theorem | dpjidcl 18848* | 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.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ 𝑃 = (𝐺dProj𝑆) & ⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆)) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))))) | ||
Theorem | dpjeq 18849* | Decompose a group sum into projections. (Contributed by Mario Carneiro, 26-Apr-2016.) (Revised by AV, 14-Jul-2019.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ 𝑃 = (𝐺dProj𝑆) & ⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆)) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐶) ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 𝐶)) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = 𝐶)) | ||
Theorem | dpjid 18850* | The key property of projections: the sum of all the projections of 𝐴 is 𝐴. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ 𝑃 = (𝐺dProj𝑆) & ⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆)) ⇒ ⊢ (𝜑 → 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)))) | ||
Theorem | dpjlid 18851 | The 𝑋-th index projection acts as the identity on elements of the 𝑋-th factor. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ 𝑃 = (𝐺dProj𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝐴 ∈ (𝑆‘𝑋)) ⇒ ⊢ (𝜑 → ((𝑃‘𝑋)‘𝐴) = 𝐴) | ||
Theorem | dpjrid 18852 | The 𝑌-th index projection annihilates elements of other factors. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ 𝑃 = (𝐺dProj𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝐴 ∈ (𝑆‘𝑋)) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ (𝜑 → 𝑌 ≠ 𝑋) ⇒ ⊢ (𝜑 → ((𝑃‘𝑌)‘𝐴) = 0 ) | ||
Theorem | dpjghm 18853 | The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ 𝑃 = (𝐺dProj𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑃‘𝑋) ∈ ((𝐺 ↾s (𝐺 DProd 𝑆)) GrpHom 𝐺)) | ||
Theorem | dpjghm2 18854 | The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ 𝑃 = (𝐺dProj𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑃‘𝑋) ∈ ((𝐺 ↾s (𝐺 DProd 𝑆)) GrpHom (𝐺 ↾s (𝑆‘𝑋)))) | ||
Theorem | ablfacrplem 18855* | Lemma for ablfacrp2 18857. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐾 = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ 𝑀} & ⊢ 𝐿 = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ 𝑁} & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) & ⊢ (𝜑 → (♯‘𝐵) = (𝑀 · 𝑁)) ⇒ ⊢ (𝜑 → ((♯‘𝐾) gcd 𝑁) = 1) | ||
Theorem | ablfacrp 18856* | 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 = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ 𝑀} & ⊢ 𝐿 = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ 𝑁} & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) & ⊢ (𝜑 → (♯‘𝐵) = (𝑀 · 𝑁)) & ⊢ 0 = (0g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝜑 → ((𝐾 ∩ 𝐿) = { 0 } ∧ (𝐾 ⊕ 𝐿) = 𝐵)) | ||
Theorem | ablfacrp2 18857* | The factors 𝐾, 𝐿 of ablfacrp 18856 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 = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ 𝑀} & ⊢ 𝐿 = {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ 𝑁} & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) & ⊢ (𝜑 → (♯‘𝐵) = (𝑀 · 𝑁)) ⇒ ⊢ (𝜑 → ((♯‘𝐾) = 𝑀 ∧ (♯‘𝐿) = 𝑁)) | ||
Theorem | ablfac1lem 18858* | Lemma for ablfac1b 18860. Satisfy the assumptions of ablfacrp. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ ℙ) & ⊢ 𝑀 = (𝑃↑(𝑃 pCnt (♯‘𝐵))) & ⊢ 𝑁 = ((♯‘𝐵) / 𝑀) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑀 gcd 𝑁) = 1 ∧ (♯‘𝐵) = (𝑀 · 𝑁))) | ||
Theorem | ablfac1a 18859* | The factors of ablfac1b 18860 are of prime power order. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ ℙ) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → (♯‘(𝑆‘𝑃)) = (𝑃↑(𝑃 pCnt (♯‘𝐵)))) | ||
Theorem | ablfac1b 18860* | 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 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ ℙ) ⇒ ⊢ (𝜑 → 𝐺dom DProd 𝑆) | ||
Theorem | ablfac1c 18861* | The factors of ablfac1b 18860 cover the entire group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ ℙ) & ⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} & ⊢ (𝜑 → 𝐷 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝐵) | ||
Theorem | ablfac1eulem 18862* | Lemma for ablfac1eu 18863. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ ℙ) & ⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} & ⊢ (𝜑 → 𝐷 ⊆ 𝐴) & ⊢ (𝜑 → (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝐵)) & ⊢ (𝜑 → dom 𝑇 = 𝐴) & ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ∈ ℕ0) & ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) = (𝑞↑𝐶)) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝐴 ∈ Fin) ⇒ ⊢ (𝜑 → ¬ 𝑃 ∥ (♯‘(𝐺 DProd (𝑇 ↾ (𝐴 ∖ {𝑃}))))) | ||
Theorem | ablfac1eu 18863* | The factorization of ablfac1b 18860 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 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ ℙ) & ⊢ 𝐷 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} & ⊢ (𝜑 → 𝐷 ⊆ 𝐴) & ⊢ (𝜑 → (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝐵)) & ⊢ (𝜑 → dom 𝑇 = 𝐴) & ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ∈ ℕ0) & ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (♯‘(𝑇‘𝑞)) = (𝑞↑𝐶)) ⇒ ⊢ (𝜑 → 𝑇 = 𝑆) | ||
Theorem | pgpfac1lem1 18864* | Lemma for pgpfac1 18870. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) & ⊢ 𝑆 = (𝐾‘{𝐴}) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 }) & ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈) & ⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤)) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) = 𝑈) | ||
Theorem | pgpfac1lem2 18865* | Lemma for pgpfac1 18870. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) & ⊢ 𝑆 = (𝐾‘{𝐴}) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 }) & ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈) & ⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤)) & ⊢ (𝜑 → 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝜑 → (𝑃 · 𝐶) ∈ (𝑆 ⊕ 𝑊)) | ||
Theorem | pgpfac1lem3a 18866* | Lemma for pgpfac1 18870. (Contributed by Mario Carneiro, 4-Jun-2016.) |
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) & ⊢ 𝑆 = (𝐾‘{𝐴}) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 }) & ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈) & ⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤)) & ⊢ (𝜑 → 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) & ⊢ · = (.g‘𝐺) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → ((𝑃 · 𝐶)(+g‘𝐺)(𝑀 · 𝐴)) ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝑃 ∥ 𝐸 ∧ 𝑃 ∥ 𝑀)) | ||
Theorem | pgpfac1lem3 18867* | Lemma for pgpfac1 18870. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) & ⊢ 𝑆 = (𝐾‘{𝐴}) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 }) & ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈) & ⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤)) & ⊢ (𝜑 → 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) & ⊢ · = (.g‘𝐺) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → ((𝑃 · 𝐶)(+g‘𝐺)(𝑀 · 𝐴)) ∈ 𝑊) & ⊢ 𝐷 = (𝐶(+g‘𝐺)((𝑀 / 𝑃) · 𝐴)) ⇒ ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)) | ||
Theorem | pgpfac1lem4 18868* | Lemma for pgpfac1 18870. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) & ⊢ 𝑆 = (𝐾‘{𝐴}) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 }) & ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈) & ⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤)) & ⊢ (𝜑 → 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)) | ||
Theorem | pgpfac1lem5 18869* | Lemma for pgpfac1 18870. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) & ⊢ 𝑆 = (𝐾‘{𝐴}) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))) ⇒ ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)) | ||
Theorem | pgpfac1 18870* | 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 = (0g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝐵)) | ||
Theorem | pgpfaclem1 18871* | Lemma for pgpfac 18874. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))) & ⊢ 𝐻 = (𝐺 ↾s 𝑈) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐻)) & ⊢ 𝑂 = (od‘𝐻) & ⊢ 𝐸 = (gEx‘𝐻) & ⊢ 0 = (0g‘𝐻) & ⊢ ⊕ = (LSSum‘𝐻) & ⊢ (𝜑 → 𝐸 ≠ 1) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → (𝑂‘𝑋) = 𝐸) & ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐻)) & ⊢ (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 }) & ⊢ (𝜑 → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈) & ⊢ (𝜑 → 𝑆 ∈ Word 𝐶) & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → (𝐺 DProd 𝑆) = 𝑊) & ⊢ 𝑇 = (𝑆 ++ 〈“(𝐾‘{𝑋})”〉) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈)) | ||
Theorem | pgpfaclem2 18872* | Lemma for pgpfac 18874. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))) & ⊢ 𝐻 = (𝐺 ↾s 𝑈) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐻)) & ⊢ 𝑂 = (od‘𝐻) & ⊢ 𝐸 = (gEx‘𝐻) & ⊢ 0 = (0g‘𝐻) & ⊢ ⊕ = (LSSum‘𝐻) & ⊢ (𝜑 → 𝐸 ≠ 1) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → (𝑂‘𝑋) = 𝐸) & ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐻)) & ⊢ (𝜑 → ((𝐾‘{𝑋}) ∩ 𝑊) = { 0 }) & ⊢ (𝜑 → ((𝐾‘{𝑋}) ⊕ 𝑊) = 𝑈) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈)) | ||
Theorem | pgpfaclem3 18873* | Lemma for pgpfac 18874. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡))) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈)) | ||
Theorem | pgpfac 18874* | Full factorization of a finite abelian p-group, by iterating pgpfac1 18870. 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) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝐵 ∈ Fin) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) | ||
Theorem | ablfaclem1 18875* | Lemma for ablfac 18878. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} & ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) & ⊢ 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)}) ⇒ ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝑊‘𝑈) = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈)}) | ||
Theorem | ablfaclem2 18876* | Lemma for ablfac 18878. (Contributed by Mario Carneiro, 27-Apr-2016.) (Proof shortened by Mario Carneiro, 3-May-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} & ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) & ⊢ 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)}) & ⊢ (𝜑 → 𝐹:𝐴⟶Word 𝐶) & ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) ∈ (𝑊‘(𝑆‘𝑦))) & ⊢ 𝐿 = ∪ 𝑦 ∈ 𝐴 ({𝑦} × dom (𝐹‘𝑦)) & ⊢ (𝜑 → 𝐻:(0..^(♯‘𝐿))–1-1-onto→𝐿) ⇒ ⊢ (𝜑 → (𝑊‘𝐵) ≠ ∅) | ||
Theorem | ablfaclem3 18877* | Lemma for ablfac 18878. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} & ⊢ 𝑆 = (𝑝 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) & ⊢ 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)}) ⇒ ⊢ (𝜑 → (𝑊‘𝐵) ≠ ∅) | ||
Theorem | ablfac 18878* | 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) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝐵)) | ||
Theorem | ablfac2 18879* | Choose generators for each cyclic group in ablfac 18878. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ · = (.g‘𝐺) & ⊢ 𝑆 = (𝑘 ∈ dom 𝑤 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑤‘𝑘)))) ⇒ ⊢ (𝜑 → ∃𝑤 ∈ Word 𝐵(𝑆:dom 𝑤⟶𝐶 ∧ 𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) = 𝐵)) | ||
Syntax | cmgp 18880 | Multiplicative group. |
class mulGrp | ||
Definition | df-mgp 18881 | 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 19058 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 18944) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 10220). (Contributed by Mario Carneiro, 21-Dec-2014.) |
⊢ mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet 〈(+g‘ndx), (.r‘𝑤)〉)) | ||
Theorem | fnmgp 18882 | The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.) |
⊢ mulGrp Fn V | ||
Theorem | mgpval 18883 | Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.) |
⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ 𝑀 = (𝑅 sSet 〈(+g‘ndx), · 〉) | ||
Theorem | mgpplusg 18884 | Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) |
⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ · = (+g‘𝑀) | ||
Theorem | mgplem 18885 | Lemma for mgpbas 18886. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ 𝑁 ≠ 2 ⇒ ⊢ (𝐸‘𝑅) = (𝐸‘𝑀) | ||
Theorem | mgpbas 18886 | 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) | ||
Theorem | mgpsca 18887 | The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 19009. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑆 = (Scalar‘𝑅) ⇒ ⊢ 𝑆 = (Scalar‘𝑀) | ||
Theorem | mgptset 18888 | Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ (TopSet‘𝑅) = (TopSet‘𝑀) | ||
Theorem | mgptopn 18889 | Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑅) ⇒ ⊢ 𝐽 = (TopOpen‘𝑀) | ||
Theorem | mgpds 18890 | Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝐵 = (dist‘𝑅) ⇒ ⊢ 𝐵 = (dist‘𝑀) | ||
Theorem | mgpress 18891 | Subgroup commutes with the multiplication group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (mulGrp‘𝑆)) | ||
Syntax | cur 18892 | Extend class notation with ring unit. |
class 1r | ||
Definition | df-ur 18893 | 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 18881). See also dfur2 18895, 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) | ||
Theorem | ringidval 18894 | 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 = (1r‘𝑅) ⇒ ⊢ 1 = (0g‘𝐺) | ||
Theorem | dfur2 18895* | 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) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ 1 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥))) | ||
Syntax | csrg 18896 | Extend class notation with the class of all semirings. |
class SRing | ||
Definition | df-srg 18897* | 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][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡][(0g‘𝑓) / 𝑛]∀𝑥 ∈ 𝑟 (∀𝑦 ∈ 𝑟 ∀𝑧 ∈ 𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))} | ||
Theorem | issrg 18898* | The predicate "is a semiring." (Contributed by Thierry Arnoux, 21-Mar-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))) | ||
Theorem | srgcmn 18899 | A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.) |
⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd) | ||
Theorem | srgmnd 18900 | A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.) |
⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) |
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