P. 193 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19201-19300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dmdprd 19201*	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 19202*	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 19203	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 19204	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 19205	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 19206*	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 19207*	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 19208	Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
Theorem	dprdf 19209	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 19210	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 19211	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 19212	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 19213*	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 19214*	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 19215*	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 19216*	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 19217*	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 19218*	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 19219	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 19220*	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 19221*	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 19222*	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 19223*	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 19224*	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 19225*	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 19226*	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 19227	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 19228	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 19229*	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 19230	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 19231	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 19232*	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 19233*	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 19234	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 19235	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 19236	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 19237	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 19238	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 19239	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 19240*	Lemma for dmdprdsplit 19250. (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 19241	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 19242	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 19243*	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 19244*	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 19245*	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 19246*	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 19247*	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 19248	Lemma for dmdprdsplit 19250. (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 19249	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 19250	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 19251	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 19252	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 19253	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 19254	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 19255	The two subgroups that appear in dpjval 19259 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 19256	The two subgroups that appear in dpjval 19259 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 19257	The two subgroups that appear in dpjval 19259 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 19258*	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 19259	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 19260	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 19261*	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 19262*	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 19263*	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 19264	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 19265	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 19266	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 19267	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 19268*	Lemma for ablfacrp2 19270. (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 19269*	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 19270*	The factors 𝐾, 𝐿 of ablfacrp 19269 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 19271*	Lemma for ablfac1b 19273. 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 19272*	The factors of ablfac1b 19273 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 19273*	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 19274*	The factors of ablfac1b 19273 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 19275*	Lemma for ablfac1eu 19276. (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 19276*	The factorization of ablfac1b 19273 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 19277*	Lemma for pgpfac1 19283. (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 19278*	Lemma for pgpfac1 19283. (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 19279*	Lemma for pgpfac1 19283. (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 19280*	Lemma for pgpfac1 19283. (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 19281*	Lemma for pgpfac1 19283. (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 19282*	Lemma for pgpfac1 19283. (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 19283*	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 19284*	Lemma for pgpfac 19287. (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 19285*	Lemma for pgpfac 19287. (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 19286*	Lemma for pgpfac 19287. (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 19287*	Full factorization of a finite abelian p-group, by iterating pgpfac1 19283. 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 19288*	Lemma for ablfac 19291. (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 19289*	Lemma for ablfac 19291. (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 19290*	Lemma for ablfac 19291. (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 19291*	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 19292*	Choose generators for each cyclic group in ablfac 19291. (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))
10.2.15  Simple groups
10.2.15.1  Definition and basic properties
Syntax	csimpg 19293	Extend class notation with the class of simple groups.
class SimpGrp
Definition	df-simpg 19294	Define class of all simple groups. A simple group is a group (df-grp 18185) with exactly two normal subgroups. These are always the subgroup of all elements and the subgroup containing only the identity (simpgnsgbid 19306). (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ SimpGrp = {𝑔 ∈ Grp ∣ (NrmSGrp‘𝑔) ≈ 2o}
Theorem	issimpg 19295	The predicate "is a simple group". (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
Theorem	issimpgd 19296	Deduce a simple group from its properties. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → (NrmSGrp‘𝐺) ≈ 2o)    ⇒   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)
Theorem	simpggrp 19297	A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)
Theorem	simpggrpd 19298	A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → 𝐺 ∈ SimpGrp)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Grp)
Theorem	simpg2nsg 19299	A simple group has two normal subgroups. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝐺 ∈ SimpGrp → (NrmSGrp‘𝐺) ≈ 2o)
Theorem	trivnsimpgd 19300	Trivial groups are not simple. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 = { 0 })    ⇒   ⊢ (𝜑 → ¬ 𝐺 ∈ SimpGrp)