P. 200 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 19901-20000 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	telgsumfz0s 19901*	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 Σ_g (𝑖 ∈ (0...𝑆) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋0 / 𝑘⦌𝐶 − ⦋(𝑆 + 1) / 𝑘⦌𝐶))

Theorem	telgsumfz0 19902*	Telescoping finite group sum ranging over nonnegative integers, using implicit substitution, analogous to telfsum 15755. (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 Σ_g (𝑖 ∈ (0...𝑆) ↦ (𝐵 − 𝐶))) = (𝐷 − 𝐸))

Theorem	telgsums 19903*	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 = (0_g‘𝐺) & ⊢ (𝜑 → ∀𝑘 ∈ ℕ₀ 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ ℕ₀) & ⊢ (𝜑 → ∀𝑘 ∈ ℕ₀ (𝑆 < 𝑘 → 𝐶 = 0 )) ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝑖 ∈ ℕ₀ ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = ⦋0 / 𝑘⦌𝐶)

Theorem	telgsum 19904*	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 = (0_g‘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 Σ_g (𝑖 ∈ ℕ₀ ↦ (𝐶 − 𝐷))) = 𝐸)

10.2.14.5 Internal direct products

Syntax	cdprd 19905	Internal direct product of a family of subgroups.
class DProd

Syntax	cdpj 19906	Projection operator for a direct product.
class dProj

Definition	df-dprd 19907*	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})))) = {(0_g‘𝑔)}))} ↦ ran (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0_g‘𝑔)} ↦ (𝑔 Σ_g 𝑓)))

Definition	df-dpj 19908*	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 19909	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 19910*	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 = (0_g‘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 19911*	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 = (0_g‘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 19912	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 19913	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 19914	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 19915*	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 = (0_g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } ⇒ ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σ_g 𝑓)))

Theorem	eldprd 19916*	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 = (0_g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } ⇒ ⊢ (dom 𝑆 = 𝐼 → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σ_g 𝑓))))

Theorem	dprdgrp 19917	Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)

Theorem	dprdf 19918	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 19919	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 19920	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 19921	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 = (0_g‘𝐺) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) ⇒ ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 })

Theorem	dprdw 19922*	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 19923*	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 19924*	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 19925*	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 19926*	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 19927*	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 19928	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 19929*	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 = (0_g‘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 Σ_g 𝐹) = 𝐴))

Theorem	eldprdi 19930*	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 = (0_g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐺 Σ_g 𝐹) ∈ (𝐺 DProd 𝑆))

Theorem	dprdfinv 19931*	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 = (0_g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) & ⊢ 𝑁 = (inv_g‘𝐺) ⇒ ⊢ (𝜑 → ((𝑁 ∘ 𝐹) ∈ 𝑊 ∧ (𝐺 Σ_g (𝑁 ∘ 𝐹)) = (𝑁‘(𝐺 Σ_g 𝐹))))

Theorem	dprdfadd 19932*	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 = (0_g‘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 Σ_g (𝐹 ∘_f + 𝐻)) = ((𝐺 Σ_g 𝐹) + (𝐺 Σ_g 𝐻))))

Theorem	dprdfsub 19933*	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 = (0_g‘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 Σ_g (𝐹 ∘_f − 𝐻)) = ((𝐺 Σ_g 𝐹) − (𝐺 Σ_g 𝐻))))

Theorem	dprdfeq0 19934*	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 = (0_g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) ⇒ ⊢ (𝜑 → ((𝐺 Σ_g 𝐹) = 0 ↔ 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 )))

Theorem	dprdf11 19935*	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 = (0_g‘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 Σ_g 𝐹) = (𝐺 Σ_g 𝐻) ↔ 𝐹 = 𝐻))

Theorem	dprdsubg 19936	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 19937	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 19938*	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 19939	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 19940	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 19941*	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 19942*	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 = (0_g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐼 ∈ 𝑉) → (𝐺dom DProd (𝑥 ∈ 𝐼 ↦ { 0 }) ∧ (𝐺 DProd (𝑥 ∈ 𝐼 ↦ { 0 })) = { 0 }))

Theorem	dprd0 19943	The empty family is an internal direct product, the product of which is the trivial subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
⊢ 0 = (0_g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = { 0 }))

Theorem	dprdf1o 19944	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 19945	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 19946	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 19947	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 19948	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 19949*	Lemma for dmdprdsplit 19959. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
⊢ 0 = (0_g‘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 Σ_g 𝐹) ∈ (𝐺 DProd (𝑆 ↾ 𝐴))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) → (𝐹‘𝑋) = 0 )

Theorem	dprdcntz2 19950	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 19951	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 = (0_g‘𝐺) ⇒ ⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 })

Theorem	dprd2dlem2 19952*	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 “ {(1^st ‘𝑋)}) ↦ ((1^st ‘𝑋)𝑆𝑗))))

Theorem	dprd2dlem1 19953*	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 19954*	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 19955*	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 19956*	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 19957	Lemma for dmdprdsplit 19959. (Contributed by Mario Carneiro, 26-Apr-2016.)
⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐼 = (𝐶 ∪ 𝐷)) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ 0 = (0_g‘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 19958	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 = (0_g‘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 19959	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 = (0_g‘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 19960	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 19961	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 = (0_g‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) ⇒ ⊢ (𝜑 → (𝐺dom DProd {⟨∅, 𝑆⟩, ⟨1_o, 𝑇⟩} ↔ (𝑆 ⊆ (𝑍‘𝑇) ∧ (𝑆 ∩ 𝑇) = { 0 })))

Theorem	dprdpr 19962	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 = (0_g‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑆 ⊆ (𝑍‘𝑇)) & ⊢ (𝜑 → (𝑆 ∩ 𝑇) = { 0 }) ⇒ ⊢ (𝜑 → (𝐺 DProd {⟨∅, 𝑆⟩, ⟨1_o, 𝑇⟩}) = (𝑆 ⊕ 𝑇))

Theorem	dpjlem 19963	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 19964	The two subgroups that appear in dpjval 19968 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 19965	The two subgroups that appear in dpjval 19968 are disjoint. (Contributed by Mario Carneiro, 26-Apr-2016.)
⊢ (𝜑 → 𝐺dom DProd 𝑆) & ⊢ (𝜑 → dom 𝑆 = 𝐼) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ 0 = (0_g‘𝐺) ⇒ ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = { 0 })

Theorem	dpjlsm 19966	The two subgroups that appear in dpjval 19968 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 19967*	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 19968	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 19969	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 19970*	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 = (0_g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊 ∧ 𝐴 = (𝐺 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)))))

Theorem	dpjeq 19971*	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 = (0_g‘𝐺) & ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } & ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐶) ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐴 = (𝐺 Σ_g (𝑥 ∈ 𝐼 ↦ 𝐶)) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = 𝐶))

Theorem	dpjid 19972*	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 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))))

Theorem	dpjlid 19973	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 19974	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 = (0_g‘𝐺) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ (𝜑 → 𝑌 ≠ 𝑋) ⇒ ⊢ (𝜑 → ((𝑃‘𝑌)‘𝐴) = 0 )

Theorem	dpjghm 19975	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 19976	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))))

10.2.14.6 The Fundamental Theorem of Abelian Groups

Theorem	ablfacrplem 19977*	Lemma for ablfacrp2 19979. (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 19978*	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 = (0_g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝜑 → ((𝐾 ∩ 𝐿) = { 0 } ∧ (𝐾 ⊕ 𝐿) = 𝐵))

Theorem	ablfacrp2 19979*	The factors 𝐾, 𝐿 of ablfacrp 19978 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 19980*	Lemma for ablfac1b 19982. 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 19981*	The factors of ablfac1b 19982 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 19982*	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 19983*	The factors of ablfac1b 19982 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 19984*	Lemma for ablfac1eu 19985. (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 19985*	The factorization of ablfac1b 19982 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 19986*	Lemma for pgpfac1 19992. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) & ⊢ 𝑆 = (𝐾‘{𝐴}) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 0 = (0_g‘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 19987*	Lemma for pgpfac1 19992. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) & ⊢ 𝑆 = (𝐾‘{𝐴}) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 0 = (0_g‘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 19988*	Lemma for pgpfac1 19992. (Contributed by Mario Carneiro, 4-Jun-2016.)
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) & ⊢ 𝑆 = (𝐾‘{𝐴}) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 0 = (0_g‘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 19989*	Lemma for pgpfac1 19992. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) & ⊢ 𝑆 = (𝐾‘{𝐴}) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 0 = (0_g‘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 19990*	Lemma for pgpfac1 19992. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) & ⊢ 𝑆 = (𝐾‘{𝐴}) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 0 = (0_g‘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 19991*	Lemma for pgpfac1 19992. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) & ⊢ 𝑆 = (𝐾‘{𝐴}) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 0 = (0_g‘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 19992*	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 = (0_g‘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 19993*	Lemma for pgpfac 19996. (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 = (0_g‘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 19994*	Lemma for pgpfac 19996. (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 = (0_g‘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 19995*	Lemma for pgpfac 19996. (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 19996*	Full factorization of a finite abelian p-group, by iterating pgpfac1 19992. 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 19997*	Lemma for ablfac 20000. (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 19998*	Lemma for ablfac 20000. (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 19999*	Lemma for ablfac 20000. (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 20000*	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))

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45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-47941

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