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Type | Label | Description |
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Statement | ||
Theorem | nn0gsumfz0 19901* | 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 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m ℕ0)) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∃𝑓 ∈ (𝐵 ↑m (0...𝑠))(𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)) | ||
Theorem | gsummptnn0fz 19902* | 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 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 )) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ ℕ0 ↦ 𝐶)) = (𝐺 Σg (𝑘 ∈ (0...𝑆) ↦ 𝐶))) | ||
Theorem | gsummptnn0fzfv 19903* | 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 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m ℕ0)) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (𝐹‘𝑘))) = (𝐺 Σg (𝑘 ∈ (0...𝑆) ↦ (𝐹‘𝑘)))) | ||
Theorem | telgsumfzslem 19904* | Lemma for telgsumfzs 19905 (induction step). (Contributed by AV, 23-Nov-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝑦 ∈ (ℤ≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ((𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑦 + 1) / 𝑘⦌𝐶) → (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))) | ||
Theorem | telgsumfzs 19905* | Telescoping group sum ranging over a finite set of sequential integers, using explicit substitution. (Contributed by AV, 23-Nov-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑁 + 1) / 𝑘⦌𝐶)) | ||
Theorem | telgsumfz 19906* | Telescoping group sum ranging over a finite set of sequential integers, using implicit substitution, analogous to telfsum 15757. (Contributed by AV, 23-Nov-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 ∈ 𝐵) & ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐿) & ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐶) & ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷) & ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 − 𝐶))) = (𝐷 − 𝐸)) | ||
Theorem | telgsumfz0s 19907* | 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) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → ∀𝑘 ∈ (0...(𝑆 + 1))𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋0 / 𝑘⦌𝐶 − ⦋(𝑆 + 1) / 𝑘⦌𝐶)) | ||
Theorem | telgsumfz0 19908* | Telescoping finite group sum ranging over nonnegative integers, using implicit substitution, analogous to telfsum 15757. (Contributed by AV, 23-Nov-2019.) |
⊢ 𝐾 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → ∀𝑘 ∈ (0...(𝑆 + 1))𝐴 ∈ 𝐾) & ⊢ (𝑘 = 𝑖 → 𝐴 = 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)) | ||
Theorem | telgsums 19909* | Telescoping finitely supported group sum ranging over nonnegative integers, using explicit substitution. (Contributed by AV, 24-Oct-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ − = (-g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 )) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = ⦋0 / 𝑘⦌𝐶) | ||
Theorem | telgsum 19910* | Telescoping finitely supported group sum ranging over nonnegative integers, using implicit substitution. (Contributed by AV, 31-Dec-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ − = (-g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐴 = 0 )) & ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐶) & ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐷) & ⊢ (𝑘 = 0 → 𝐴 = 𝐸) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (𝐶 − 𝐷))) = 𝐸) | ||
Syntax | cdprd 19911 | Internal direct product of a family of subgroups. |
class DProd | ||
Syntax | cdpj 19912 | Projection operator for a direct product. |
class dProj | ||
Definition | df-dprd 19913* | 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 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)}))} ↦ ran (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓))) | ||
Definition | df-dpj 19914* | 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 ↦ ((𝑠‘𝑖)(proj1‘𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖})))))) | ||
Theorem | reldmdprd 19915 | 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 | ||
Theorem | dmdprd 19916* | 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 = (0g‘𝐺) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })))) | ||
Theorem | dmdprdd 19917* | 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 = (0g‘𝐺) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 }) ⇒ ⊢ (𝜑 → 𝐺dom DProd 𝑆) | ||
Theorem | dprddomprc 19918 | 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) | ||
Theorem | dprddomcld 19919 | 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.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) ⇒ ⊢ (𝜑 → 𝐼 ∈ V) | ||
Theorem | dprdval0prc 19920 | 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) = ∅) | ||
Theorem | dprdval 19921* | 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 = (0g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } ⇒ ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓))) | ||
Theorem | eldprd 19922* | 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 = (0g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } ⇒ ⊢ (dom 𝑆 = 𝐼 → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)))) | ||
Theorem | dprdgrp 19923 | Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | ||
Theorem | dprdf 19924 | The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝐺dom DProd 𝑆 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺)) | ||
Theorem | dprdf2 19925 | The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) ⇒ ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) | ||
Theorem | dprdcntz 19926 | The function 𝑆 is a family having pairwise commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ 𝑍 = (Cntz‘𝐺) ⇒ ⊢ (𝜑 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌))) | ||
Theorem | dprddisj 19927 | The function 𝑆 is a family having trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) ⇒ ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 }) | ||
Theorem | dprdw 19928* | 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 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) ⇒ ⊢ (𝜑 → (𝐹 ∈ 𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) ∧ 𝐹 finSupp 0 ))) | ||
Theorem | dprdwd 19929* | 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 19930* | 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 19931* | 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 19932* | 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 19933* | 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 19934 | 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 19935* | 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 19936* | 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 19937* | 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 19938* | 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 19939* | 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 19940* | 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 19941* | 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 19942 | 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 19943 | 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 19944* | 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 19945 | 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 19946 | 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 19947* | 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 19948* | 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 19949 | 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 19950 | 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 19951 | 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 19952 | 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 19953 | 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 19954 | 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 19955* | Lemma for dmdprdsplit 19965. (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 19956 | 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 19957 | 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 19958* | 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 19959* | 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 19960* | 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 19961* | 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 19962* | 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 19963 | Lemma for dmdprdsplit 19965. (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 19964 | 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 19965 | 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 19966 | 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 19967 | 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〉, 〈1o, 𝑇〉} ↔ (𝑆 ⊆ (𝑍‘𝑇) ∧ (𝑆 ∩ 𝑇) = { 0 }))) | ||
Theorem | dprdpr 19968 | 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〉, 〈1o, 𝑇〉}) = (𝑆 ⊕ 𝑇)) | ||
Theorem | dpjlem 19969 | 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 19970 | The two subgroups that appear in dpjval 19974 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 19971 | The two subgroups that appear in dpjval 19974 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 19972 | The two subgroups that appear in dpjval 19974 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 19973* | 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 19974 | 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 19975 | 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 19976* | 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 19977* | 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 19978* | 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 19979 | 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 19980 | 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 19981 | 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 19982 | 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 19983* | Lemma for ablfacrp2 19985. (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 19984* | 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 19985* | The factors 𝐾, 𝐿 of ablfacrp 19984 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 19986* | Lemma for ablfac1b 19988. 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 19987* | The factors of ablfac1b 19988 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 19988* | 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 19989* | The factors of ablfac1b 19988 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 19990* | Lemma for ablfac1eu 19991. (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 19991* | The factorization of ablfac1b 19988 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 19992* | Lemma for pgpfac1 19998. (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 19993* | Lemma for pgpfac1 19998. (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 19994* | Lemma for pgpfac1 19998. (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 19995* | Lemma for pgpfac1 19998. (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 19996* | Lemma for pgpfac1 19998. (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 19997* | Lemma for pgpfac1 19998. (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 19998* | 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 19999* | Lemma for pgpfac 20002. (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 20000* | Lemma for pgpfac 20002. (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)) |
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