P. 203 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20201-20300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	gsumsn 20201*	Group sum of a singleton. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Proof shortened by AV, 11-Dec-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐶)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ 𝑉 ∧ 𝐶 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)
Theorem	gsumpr 20202*	Group sum of a pair. (Contributed by AV, 6-Dec-2018.) (Proof shortened by AV, 28-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐶)    &   ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐷)    ⇒   ⊢ ((𝐺 ∈ CMnd ∧ (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊 ∧ 𝑀 ≠ 𝑁) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → (𝐺 Σg (𝑘 ∈ {𝑀, 𝑁} ↦ 𝐴)) = (𝐶 + 𝐷))
Theorem	gsumzunsnd 20203*	Append an element to a finite group sum, more general version of gsumunsnd 20205. (Contributed by AV, 7-Oct-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ 𝐹 = (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌))
Theorem	gsumunsnfd 20204*	Append an element to a finite group sum, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 11-Dec-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌)    &   ⊢ Ⅎ𝑘𝑌    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌))
Theorem	gsumunsnd 20205*	Append an element to a finite group sum. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 2-Jan-2019.) (Proof shortened by AV, 11-Dec-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌))
Theorem	gsumunsnf 20206*	Append an element to a finite group sum, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Thierry Arnoux, 28-Mar-2018.) (Proof shortened by AV, 11-Dec-2019.)
⊢ Ⅎ𝑘𝑌    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝑘 = 𝑀 → 𝑋 = 𝑌)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌))
Theorem	gsumunsn 20207*	Append an element to a finite group sum. (Contributed by Mario Carneiro, 19-Dec-2014.) (Proof shortened by AV, 8-Mar-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝑘 = 𝑀 → 𝑋 = 𝑌)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌))
Theorem	gsumdifsnd 20208*	Extract a summand from a finitely supported group sum. (Contributed by AV, 21-Apr-2019.) (Revised by AV, 28-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp (0g‘𝐺))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌))
Theorem	gsumpt 20209	Sum of a family that is nonzero at at most one point. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ {𝑋})    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐹‘𝑋))
Theorem	gsummptf1o 20210*	Re-index a finite group sum using a bijection. (Contributed by Thierry Arnoux, 29-Mar-2018.)
⊢ Ⅎ𝑥𝐻    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝑥 = 𝐸 → 𝐶 = 𝐻)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐹)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐸 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑦 ∈ 𝐷 ↦ 𝐻)))
Theorem	gsummptun 20211*	Group sum of a disjoint union, whereas sums are expressed as mappings. (Contributed by Thierry Arnoux, 28-Mar-2018.) (Proof shortened by AV, 11-Dec-2019.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ CMnd)    &   ⊢ (𝜑 → (𝐴 ∪ 𝐶) ∈ Fin)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ 𝐶)) → 𝐷 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ (𝐴 ∪ 𝐶) ↦ 𝐷)) = ((𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐷)) + (𝑊 Σg (𝑥 ∈ 𝐶 ↦ 𝐷))))
Theorem	gsummpt1n0 20212*	If only one summand in a finite group sum is not zero, the whole sum equals this summand. More general version of gsummptif1n0 20213. (Contributed by AV, 11-Oct-2019.)
⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ 𝐹 = (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 ))    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝐼 𝐴 ∈ (Base‘𝐺))    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = ⦋𝑋 / 𝑛⦌𝐴)
Theorem	gsummptif1n0 20213*	If only one summand in a finite group sum is not zero, the whole sum equals this summand. (Contributed by AV, 17-Feb-2019.) (Proof shortened by AV, 11-Oct-2019.)
⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ 𝐹 = (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 ))    &   ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐺))    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = 𝐴)
Theorem	gsummptcl 20214*	Closure of a finite group sum over a finite set as map. (Contributed by AV, 29-Dec-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → ∀𝑖 ∈ 𝑁 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ 𝑁 ↦ 𝑋)) ∈ 𝐵)
Theorem	gsummptfif1o 20215*	Re-index a finite group sum as map, using a bijection. (Contributed by by AV, 23-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → ∀𝑖 ∈ 𝑁 𝑋 ∈ 𝐵)    &   ⊢ 𝐹 = (𝑖 ∈ 𝑁 ↦ 𝑋)    &   ⊢ (𝜑 → 𝐻:𝐶⟶1-1onto𝑁)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))
Theorem	gsummptfzcl 20216*	Closure of a finite group sum over a finite set of sequential integers as map. (Contributed by AV, 14-Dec-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐼 = (𝑀...𝑁))    &   ⊢ (𝜑 → ∀𝑖 ∈ 𝐼 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ 𝐼 ↦ 𝑋)) ∈ 𝐵)
Theorem	gsum2dlem1 20217*	Lemma 1 for gsum2d 20219. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → Rel 𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → dom 𝐴 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)
Theorem	gsum2dlem2 20218*	Lemma for gsum2d 20219. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → Rel 𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → dom 𝐴 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))
Theorem	gsum2d 20219*	Write a sum over a two-dimensional region as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → Rel 𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → dom 𝐴 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑗 ∈ 𝐷 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))
Theorem	gsum2d2lem 20220*	Lemma for gsum2d2 20221: show the function is finitely supported. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 9-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )    ⇒   ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) finSupp 0 )
Theorem	gsum2d2 20221*	Write a group sum over a two-dimensional region as a double sum. Note that 𝐶(𝑗) is a function of 𝑗. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)))))
Theorem	gsumcom2 20222*	Two-dimensional commutation of a group sum. Note that while 𝐴 and 𝐷 are constants w.r.t. 𝑗, 𝑘, 𝐶(𝑗) and 𝐸(𝑘) are not. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ↔ (𝑘 ∈ 𝐷 ∧ 𝑗 ∈ 𝐸)))    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ 𝐷, 𝑗 ∈ 𝐸 ↦ 𝑋)))
Theorem	gsumxp 20223*	Write a group sum over a cartesian product as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 9-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))))
Theorem	gsumcom 20224*	Commute the arguments of a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ 𝐶, 𝑗 ∈ 𝐴 ↦ 𝑋)))
Theorem	gsumcom3 20225*	A commutative law for finitely supported iterated sums. (Contributed by Stefan O'Rear, 2-Nov-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)))) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ 𝑋)))))
Theorem	gsumcom3fi 20226*	A commutative law for finite iterated sums. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ Fin)    &   ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)))) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ 𝑋)))))
Theorem	gsumxp2 20227*	Write a group sum over a cartesian product as a double sum in two ways. This corresponds to the first equation in [Lang] p. 6. (Contributed by AV, 27-Dec-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))))
Theorem	prdsgsum 20228*	Finite commutative sums in a product structure are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2015.) (Revised by AV, 9-Jun-2019.)
⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ CMnd)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽)) → 𝑈 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈)) finSupp 0 )    ⇒   ⊢ (𝜑 → (𝑌 Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (𝑥 ∈ 𝐼 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈))))
Theorem	pwsgsum 20229*	Finite commutative sums in a power structure are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2015.) (Revised by AV, 9-Jun-2019.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CMnd)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽)) → 𝑈 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈)) finSupp 0 )    ⇒   ⊢ (𝜑 → (𝑌 Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (𝑥 ∈ 𝐼 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈))))
10.2.14.4  Group sums over (ranges of) integers
Theorem	fsfnn0gsumfsffz 20230*	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 → 𝑆 ∈ ℕ0)    &   ⊢ 𝐻 = (𝐹 ↾ (0...𝑆))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg 𝐻)))
Theorem	nn0gsumfz 20231*	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))(𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)))
Theorem	nn0gsumfz0 20232*	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 20233*	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 20234*	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 20235*	Lemma for telgsumfzs 20236 (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 20236*	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 20237*	Telescoping group sum ranging over a finite set of sequential integers, using implicit substitution, analogous to telfsum 16003. (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 20238*	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 20239*	Telescoping finite group sum ranging over nonnegative integers, using implicit substitution, analogous to telfsum 16003. (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 20240*	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 20241*	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)
10.2.14.5  Internal direct products
Syntax	cdprd 20242	Internal direct product of a family of subgroups.
class DProd
Syntax	cdpj 20243	Projection operator for a direct product.
class dProj
Definition	df-dprd 20244*	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 20245*	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 20246	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 20247*	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 20248*	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 20249	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 20250	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 20251	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 20252*	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 20253*	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 20254	Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
Theorem	dprdf 20255	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 20256	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 20257	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 20258	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 20259*	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 20260*	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 20261*	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 20262*	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 20263*	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 20264*	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 20265	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 20266*	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 20267*	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 20268*	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 20269*	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 20270*	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 20271*	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 20272*	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 20273	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 20274	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 20275*	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 20276	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 20277	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 20278*	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 20279*	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 20280	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 20281	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⟶1-1onto𝐼)    ⇒   ⊢ (𝜑 → (𝐺dom DProd (𝑆 ∘ 𝐹) ∧ (𝐺 DProd (𝑆 ∘ 𝐹)) = (𝐺 DProd 𝑆)))
Theorem	dprdf1 20282	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⟶1-1𝐼)    ⇒   ⊢ (𝜑 → (𝐺dom DProd (𝑆 ∘ 𝐹) ∧ (𝐺 DProd (𝑆 ∘ 𝐹)) ⊆ (𝐺 DProd 𝑆)))
Theorem	subgdmdprd 20283	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 20284	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 20285	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 20286*	Lemma for dmdprdsplit 20296. (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 20287	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 20288	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 20289*	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 20290*	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 20291*	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 20292*	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 20293*	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 20294	Lemma for dmdprdsplit 20296. (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 20295	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 20296	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 20297	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 20298	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 20299	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 20300	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))