P. 200 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	gsummulg 19901*	Nonnegative multiple of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 6-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 )    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ (𝑁 · 𝑋))) = (𝑁 · (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋))))
Theorem	gsummulgz 19902*	Integer multiple of a group sum. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 6-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 )    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ (𝑁 · 𝑋))) = (𝑁 · (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋))))
Theorem	gsumzoppg 19903	The opposite of a group sum is the same as the original. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ 𝑂 = (oppg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
Theorem	gsumzinv 19904	Inverse of a group sum. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐼 ∘ 𝐹)) = (𝐼‘(𝐺 Σg 𝐹)))
Theorem	gsuminv 19905	Inverse of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 4-May-2015.) (Revised by AV, 6-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐼 ∘ 𝐹)) = (𝐼‘(𝐺 Σg 𝐹)))
Theorem	gsummptfidminv 19906*	Inverse of a group sum expressed as mapping with a finite domain. (Contributed by AV, 23-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐼 ∘ 𝐹)) = (𝐼‘(𝐺 Σg 𝐹)))
Theorem	gsumsub 19907	The difference of two group sums. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 6-Jun-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    &   ⊢ (𝜑 → 𝐻 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f − 𝐻)) = ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻)))
Theorem	gsummptfssub 19908*	The difference of two group sums expressed as mappings. (Contributed by AV, 7-Nov-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶))    &   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷))    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    &   ⊢ (𝜑 → 𝐻 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 − 𝐷))) = ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻)))
Theorem	gsummptfidmsub 19909*	The difference of two group sums expressed as mappings with finite domain. (Contributed by AV, 7-Nov-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Abel)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 − 𝐷))) = ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻)))
Theorem	gsumsnfd 19910*	Group sum of a singleton, deduction form, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Thierry Arnoux, 28-Mar-2018.) (Revised by AV, 11-Dec-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐶)    &   ⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐶    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)
Theorem	gsumsnd 19911*	Group sum of a singleton, deduction form. (Contributed by Thierry Arnoux, 30-Jan-2017.) (Proof shortened by AV, 11-Dec-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐶)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)
Theorem	gsumsnf 19912*	Group sum of a singleton, 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.)
⊢ Ⅎ𝑘𝐶    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐶)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ 𝑉 ∧ 𝐶 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶)
Theorem	gsumsn 19913*	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 19914*	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 19915*	Append an element to a finite group sum, more general version of gsumunsnd 19917. (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 19916*	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 19917*	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 19918*	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 19919*	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 19920*	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 19921	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 19922*	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 19923*	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 19924*	If only one summand in a finite group sum is not zero, the whole sum equals this summand. More general version of gsummptif1n0 19925. (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 19925*	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 19926*	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 19927*	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–1-1-onto→𝑁)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))
Theorem	gsummptfzcl 19928*	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 19929*	Lemma 1 for gsum2d 19931. (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 19930*	Lemma for gsum2d 19931. (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 19931*	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 19932*	Lemma for gsum2d2 19933: 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 19933*	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 19934*	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 19935*	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 19936*	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 19937*	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 19938*	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 19939*	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 19940*	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 19941*	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 19942*	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 19943*	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 19944*	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 19945*	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 19946*	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 19947*	Lemma for telgsumfzs 19948 (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 19948*	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 19949*	Telescoping group sum ranging over a finite set of sequential integers, using implicit substitution, analogous to telfsum 15782. (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 19950*	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 19951*	Telescoping finite group sum ranging over nonnegative integers, using implicit substitution, analogous to telfsum 15782. (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 19952*	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 19953*	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 19954	Internal direct product of a family of subgroups.
class DProd
Syntax	cdpj 19955	Projection operator for a direct product.
class dProj
Definition	df-dprd 19956*	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 19957*	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 19958	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 19959*	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 19960*	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 19961	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 19962	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 19963	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 19964*	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 19965*	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 19966	Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
Theorem	dprdf 19967	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 19968	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 19969	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 19970	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 19971*	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 19972*	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 19973*	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 19974*	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 19975*	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 19976*	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 19977	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 19978*	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 19979*	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 19980*	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 19981*	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 19982*	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 19983*	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 19984*	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 19985	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 19986	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 19987*	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 19988	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 19989	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 19990*	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 19991*	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 19992	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 19993	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 19994	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 19995	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 19996	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 19997	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 19998*	Lemma for dmdprdsplit 20008. (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 19999	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 20000	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 })