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Theorem List for Metamath Proof Explorer - 18401-18500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgsumzunsnd 18401* Append an element to a finite group sum, more general version of gsumunsnd 18403. (Contributed by AV, 7-Oct-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   𝐹 = (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑 → ¬ 𝑀𝐴)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘 = 𝑀) → 𝑋 = 𝑌)       (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝑘𝐴𝑋)) + 𝑌))

Theoremgsumunsnfd 18402* 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)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑 → ¬ 𝑀𝐴)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘 = 𝑀) → 𝑋 = 𝑌)    &   𝑘𝑌       (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘𝐴𝑋)) + 𝑌))

Theoremgsumunsnd 18403* 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)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑 → ¬ 𝑀𝐴)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘 = 𝑀) → 𝑋 = 𝑌)       (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘𝐴𝑋)) + 𝑌))

Theoremgsumunsnf 18404* 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.)
𝑘𝑌    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑 → ¬ 𝑀𝐴)    &   (𝜑𝑌𝐵)    &   (𝑘 = 𝑀𝑋 = 𝑌)       (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘𝐴𝑋)) + 𝑌))

Theoremgsumunsn 18405* 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)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑 → ¬ 𝑀𝐴)    &   (𝜑𝑌𝐵)    &   (𝑘 = 𝑀𝑋 = 𝑌)       (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘𝐴𝑋)) + 𝑌))

Theoremgsumdifsnd 18406* Extract a summand from a finitely supported group sum. (Contributed by AV, 21-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑊)    &   (𝜑 → (𝑘𝐴𝑋) finSupp (0g𝐺))    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑𝑀𝐴)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘 = 𝑀) → 𝑋 = 𝑌)       (𝜑 → (𝐺 Σg (𝑘𝐴𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌))

Theoremgsumpt 18407 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 = (0g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → (𝐹 supp 0 ) ⊆ {𝑋})       (𝜑 → (𝐺 Σg 𝐹) = (𝐹𝑋))

Theoremgsummptf1o 18408* Re-index a finite group sum using a bijection. (Contributed by Thierry Arnoux, 29-Mar-2018.)
𝑥𝐻    &   𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝑥 = 𝐸𝐶 = 𝐻)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐹𝐵)    &   ((𝜑𝑥𝐴) → 𝐶𝐹)    &   ((𝜑𝑦𝐷) → 𝐸𝐴)    &   ((𝜑𝑥𝐴) → ∃!𝑦𝐷 𝑥 = 𝐸)       (𝜑 → (𝐺 Σg (𝑥𝐴𝐶)) = (𝐺 Σg (𝑦𝐷𝐻)))

Theoremgsummptun 18409* 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)    &    0 = (0g𝑊)    &    + = (+g𝑊)    &   (𝜑𝑊 ∈ CMnd)    &   (𝜑 → (𝐴𝐶) ∈ Fin)    &   (𝜑 → (𝐴𝐶) = ∅)    &   ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝐷𝐵)       (𝜑 → (𝑊 Σg (𝑥 ∈ (𝐴𝐶) ↦ 𝐷)) = ((𝑊 Σg (𝑥𝐴𝐷)) + (𝑊 Σg (𝑥𝐶𝐷))))

Theoremgsummpt1n0 18410* If only one summand in a finite group sum is not zero, the whole sum equals this summand. More general version of gsummptif1n0 18411. (Contributed by AV, 11-Oct-2019.)
0 = (0g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐼𝑊)    &   (𝜑𝑋𝐼)    &   𝐹 = (𝑛𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 ))    &   (𝜑 → ∀𝑛𝐼 𝐴 ∈ (Base‘𝐺))       (𝜑 → (𝐺 Σg 𝐹) = 𝑋 / 𝑛𝐴)

Theoremgsummptif1n0 18411* 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 = (0g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐼𝑊)    &   (𝜑𝑋𝐼)    &   𝐹 = (𝑛𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 ))    &   (𝜑𝐴 ∈ (Base‘𝐺))       (𝜑 → (𝐺 Σg 𝐹) = 𝐴)

Theoremgsummptcl 18412* Closure of a finite group sum over a finite set as map. (Contributed by AV, 29-Dec-2018.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑 → ∀𝑖𝑁 𝑋𝐵)       (𝜑 → (𝐺 Σg (𝑖𝑁𝑋)) ∈ 𝐵)

Theoremgsummptfif1o 18413* Re-index a finite group sum as map, using a bijection. (Contributed by by AV, 23-Jul-2019.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑 → ∀𝑖𝑁 𝑋𝐵)    &   𝐹 = (𝑖𝑁𝑋)    &   (𝜑𝐻:𝐶1-1-onto𝑁)       (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹𝐻)))

Theoremgsummptfzcl 18414* Closure of a finite group sum over a finite set of sequential integers as map. (Contributed by AV, 14-Dec-2018.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐼 = (𝑀...𝑁))    &   (𝜑 → ∀𝑖𝐼 𝑋𝐵)       (𝜑 → (𝐺 Σg (𝑖𝐼𝑋)) ∈ 𝐵)

Theoremgsum2dlem1 18415* Lemma 1 for gsum2d 18417. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑 → Rel 𝐴)    &   (𝜑𝐷𝑊)    &   (𝜑 → dom 𝐴𝐷)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)

Theoremgsum2dlem2 18416* Lemma for gsum2d 18417. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑 → Rel 𝐴)    &   (𝜑𝐷𝑊)    &   (𝜑 → dom 𝐴𝐷)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))

Theoremgsum2d 18417* 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 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑 → Rel 𝐴)    &   (𝜑𝐷𝑊)    &   (𝜑 → dom 𝐴𝐷)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑗𝐷 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))

Theoremgsum2d2lem 18418* Lemma for gsum2d2 18419: show the function is finitely supported. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 9-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐶𝑊)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )       (𝜑 → (𝑗𝐴, 𝑘𝐶𝑋) finSupp 0 )

Theoremgsum2d2 18419* 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 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐶𝑊)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )       (𝜑 → (𝐺 Σg (𝑗𝐴, 𝑘𝐶𝑋)) = (𝐺 Σg (𝑗𝐴 ↦ (𝐺 Σg (𝑘𝐶𝑋)))))

Theoremgsumcom2 18420* 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 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐶𝑊)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )    &   (𝜑𝐷𝑌)    &   (𝜑 → ((𝑗𝐴𝑘𝐶) ↔ (𝑘𝐷𝑗𝐸)))       (𝜑 → (𝐺 Σg (𝑗𝐴, 𝑘𝐶𝑋)) = (𝐺 Σg (𝑘𝐷, 𝑗𝐸𝑋)))

Theoremgsumxp 18421* 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 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑𝐹:(𝐴 × 𝐶)⟶𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑗𝐴 ↦ (𝐺 Σg (𝑘𝐶 ↦ (𝑗𝐹𝑘))))))

Theoremgsumcom 18422* Commute the arguments of a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )       (𝜑 → (𝐺 Σg (𝑗𝐴, 𝑘𝐶𝑋)) = (𝐺 Σg (𝑘𝐶, 𝑗𝐴𝑋)))

Theoremprdsgsum 18423* 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(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑌)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑆𝑋)    &   ((𝜑𝑥𝐼) → 𝑅 ∈ CMnd)    &   ((𝜑 ∧ (𝑥𝐼𝑦𝐽)) → 𝑈𝐵)    &   (𝜑 → (𝑦𝐽 ↦ (𝑥𝐼𝑈)) finSupp 0 )       (𝜑 → (𝑌 Σg (𝑦𝐽 ↦ (𝑥𝐼𝑈))) = (𝑥𝐼 ↦ (𝑅 Σg (𝑦𝐽𝑈))))

Theorempwsgsum 18424* 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 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑌)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ CMnd)    &   ((𝜑 ∧ (𝑥𝐼𝑦𝐽)) → 𝑈𝐵)    &   (𝜑 → (𝑦𝐽 ↦ (𝑥𝐼𝑈)) finSupp 0 )       (𝜑 → (𝑌 Σg (𝑦𝐽 ↦ (𝑥𝐼𝑈))) = (𝑥𝐼 ↦ (𝑅 Σg (𝑦𝐽𝑈))))

10.3.4  Group sums over (ranges of) integers

Theoremfsfnn0gsumfsffz 18425* 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 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐹 ∈ (𝐵𝑚0))    &   (𝜑𝑆 ∈ ℕ0)    &   𝐻 = (𝐹 ↾ (0...𝑆))       (𝜑 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg 𝐻)))

Theoremnn0gsumfz 18426* 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 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐹 ∈ (𝐵𝑚0))    &   (𝜑𝐹 finSupp 0 )       (𝜑 → ∃𝑠 ∈ ℕ0𝑓 ∈ (𝐵𝑚 (0...𝑠))(𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)))

Theoremnn0gsumfz0 18427* 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 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐹 ∈ (𝐵𝑚0))    &   (𝜑𝐹 finSupp 0 )       (𝜑 → ∃𝑠 ∈ ℕ0𝑓 ∈ (𝐵𝑚 (0...𝑠))(𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))

Theoremgsummptnn0fz 18428* 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.)
𝑘𝜑    &   𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑 → ∀𝑘 ∈ ℕ0 𝐶𝐵)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘𝐶 = 0 ))       (𝜑 → (𝐺 Σg (𝑘 ∈ ℕ0𝐶)) = (𝐺 Σg (𝑘 ∈ (0...𝑆) ↦ 𝐶)))

Theoremgsummptnn0fzv 18429* 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.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑 → ∀𝑘 ∈ ℕ0 𝐶𝐵)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘𝐶 = 0 ))       (𝜑 → (𝐺 Σg (𝑘 ∈ ℕ0𝐶)) = (𝐺 Σg (𝑘 ∈ (0...𝑆) ↦ 𝐶)))

Theoremgsummptnn0fzfv 18430* 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 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐹 ∈ (𝐵𝑚0))    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 0 ))       (𝜑 → (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (𝐹𝑘))) = (𝐺 Σg (𝑘 ∈ (0...𝑆) ↦ (𝐹𝑘))))

Theoremtelgsumfzslem 18431* Lemma for telgsumfzs 18432 (induction step). (Contributed by AV, 23-Nov-2019.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &    = (-g𝐺)       ((𝑦 ∈ (ℤ𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵)) → ((𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑦 + 1) / 𝑘𝐶) → (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 ((𝑦 + 1) + 1) / 𝑘𝐶)))

Theoremtelgsumfzs 18432* Telescoping group sum ranging over a finite set of sequential integers, using explicit substitution. (Contributed by AV, 23-Nov-2019.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &    = (-g𝐺)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶𝐵)       (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶))

Theoremtelgsumfz 18433* Telescoping group sum ranging over a finite set of sequential integers, using implicit substitution, analogous to telfsum 14580. (Contributed by AV, 23-Nov-2019.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &    = (-g𝐺)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐴𝐵)    &   (𝑘 = 𝑖𝐴 = 𝐿)    &   (𝑘 = (𝑖 + 1) → 𝐴 = 𝐶)    &   (𝑘 = 𝑀𝐴 = 𝐷)    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸)       (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 𝐶))) = (𝐷 𝐸))

Theoremtelgsumfz0s 18434* 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)    &   (𝜑𝐺 ∈ Abel)    &    = (-g𝐺)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → ∀𝑘 ∈ (0...(𝑆 + 1))𝐶𝐵)       (𝜑 → (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (0 / 𝑘𝐶 (𝑆 + 1) / 𝑘𝐶))

Theoremtelgsumfz0 18435* Telescoping finite group sum ranging over nonnegative integers, using implicit substitution, analogous to telfsum 14580. (Contributed by AV, 23-Nov-2019.)
𝐾 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &    = (-g𝐺)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → ∀𝑘 ∈ (0...(𝑆 + 1))𝐴𝐾)    &   (𝑘 = 𝑖𝐴 = 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))

Theoremtelgsums 18436* Telescoping finitely supported group sum ranging over nonnegative integers, using explicit substitution. (Contributed by AV, 24-Oct-2019.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &    = (-g𝐺)    &    0 = (0g𝐺)    &   (𝜑 → ∀𝑘 ∈ ℕ0 𝐶𝐵)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘𝐶 = 0 ))       (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = 0 / 𝑘𝐶)

Theoremtelgsum 18437* Telescoping finitely supported group sum ranging over nonnegative integers, using implicit substitution. (Contributed by AV, 31-Dec-2019.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &    = (-g𝐺)    &    0 = (0g𝐺)    &   (𝜑 → ∀𝑘 ∈ ℕ0 𝐴𝐵)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘𝐴 = 0 ))    &   (𝑘 = 𝑖𝐴 = 𝐶)    &   (𝑘 = (𝑖 + 1) → 𝐴 = 𝐷)    &   (𝑘 = 0 → 𝐴 = 𝐸)       (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (𝐶 𝐷))) = 𝐸)

10.3.5  Internal direct products

Syntaxcdprd 18438 Internal direct product of a family of subgroups.
class DProd

Syntaxcdpj 18439 Projection operator for a direct product.
class dProj

Definitiondf-dprd 18440* 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 ∈ { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))} ↦ ran (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)))

Definitiondf-dpj 18441* 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 ↦ ((𝑠𝑖)(proj1𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))))))

Theoremreldmdprd 18442 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

Theoremdmdprd 18443* 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 = (0g𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐼𝑉 ∧ dom 𝑆 = 𝐼) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))

Theoremdmdprdd 18444* 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 = (0g𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))    &   ((𝜑 ∧ (𝑥𝐼𝑦𝐼𝑥𝑦)) → (𝑆𝑥) ⊆ (𝑍‘(𝑆𝑦)))    &   ((𝜑𝑥𝐼) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑥})))) ⊆ { 0 })       (𝜑𝐺dom DProd 𝑆)

Theoremdprddomprc 18445 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)

Theoremdprddomcld 18446 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.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)       (𝜑𝐼 ∈ V)

Theoremdprdval0prc 18447 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) = ∅)

Theoremdprdval 18448* 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 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }       ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))

Theoremeldprd 18449* 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 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }       (dom 𝑆 = 𝐼 → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓))))

Theoremdprdgrp 18450 Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝐺dom DProd 𝑆𝐺 ∈ Grp)

Theoremdprdf 18451 The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝐺dom DProd 𝑆𝑆:dom 𝑆⟶(SubGrp‘𝐺))

Theoremdprdf2 18452 The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)       (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))

Theoremdprdcntz 18453 The function 𝑆 is a family having pairwise commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)    &   (𝜑𝑌𝐼)    &   (𝜑𝑋𝑌)    &   𝑍 = (Cntz‘𝐺)       (𝜑 → (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑌)))

Theoremdprddisj 18454 The function 𝑆 is a family having trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)    &    0 = (0g𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       (𝜑 → ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))) = { 0 })

Theoremdprdw 18455* 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 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)       (𝜑 → (𝐹𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥) ∧ 𝐹 finSupp 0 )))

Theoremdprdwd 18456* 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 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   ((𝜑𝑥𝐼) → 𝐴 ∈ (𝑆𝑥))    &   (𝜑 → (𝑥𝐼𝐴) finSupp 0 )       (𝜑 → (𝑥𝐼𝐴) ∈ 𝑊)

Theoremdprdff 18457* 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 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)    &   𝐵 = (Base‘𝐺)       (𝜑𝐹:𝐼𝐵)

Theoremdprdfcl 18458* 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 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)       ((𝜑𝑋𝐼) → (𝐹𝑋) ∈ (𝑆𝑋))

Theoremdprdffsupp 18459* 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 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)       (𝜑𝐹 finSupp 0 )

Theoremdprdfcntz 18460* 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 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)    &   𝑍 = (Cntz‘𝐺)       (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))

Theoremdprdssv 18461 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

Theoremdprdfid 18462* 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 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)    &   (𝜑𝐴 ∈ (𝑆𝑋))    &   𝐹 = (𝑛𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 ))       (𝜑 → (𝐹𝑊 ∧ (𝐺 Σg 𝐹) = 𝐴))

Theoremeldprdi 18463* 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 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)       (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆))

Theoremdprdfinv 18464* 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 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)    &   𝑁 = (invg𝐺)       (𝜑 → ((𝑁𝐹) ∈ 𝑊 ∧ (𝐺 Σg (𝑁𝐹)) = (𝑁‘(𝐺 Σg 𝐹))))

Theoremdprdfadd 18465* 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 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)    &   (𝜑𝐻𝑊)    &    + = (+g𝐺)       (𝜑 → ((𝐹𝑓 + 𝐻) ∈ 𝑊 ∧ (𝐺 Σg (𝐹𝑓 + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))))

Theoremdprdfsub 18466* 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 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)    &   (𝜑𝐻𝑊)    &    = (-g𝐺)       (𝜑 → ((𝐹𝑓 𝐻) ∈ 𝑊 ∧ (𝐺 Σg (𝐹𝑓 𝐻)) = ((𝐺 Σg 𝐹) (𝐺 Σg 𝐻))))

Theoremdprdfeq0 18467* 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 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)       (𝜑 → ((𝐺 Σg 𝐹) = 0𝐹 = (𝑥𝐼0 )))

Theoremdprdf11 18468* 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 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹𝑊)    &   (𝜑𝐻𝑊)       (𝜑 → ((𝐺 Σg 𝐹) = (𝐺 Σg 𝐻) ↔ 𝐹 = 𝐻))

Theoremdprdsubg 18469 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))

Theoremdprdub 18470 Each factor is a subset of the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑆𝑋) ⊆ (𝐺 DProd 𝑆))

Theoremdprdlub 18471* The direct product is smaller than any subgroup which contains the factors. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   ((𝜑𝑘𝐼) → (𝑆𝑘) ⊆ 𝑇)       (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝑇)

Theoremdprdspan 18472 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))

Theoremdprdres 18473 Restriction of a direct product (dropping factors). (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐴𝐼)       (𝜑 → (𝐺dom DProd (𝑆𝐴) ∧ (𝐺 DProd (𝑆𝐴)) ⊆ (𝐺 DProd 𝑆)))

Theoremdprdss 18474* Create a direct product by finding subgroups inside each factor of another direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑇)    &   (𝜑 → dom 𝑇 = 𝐼)    &   (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))    &   ((𝜑𝑘𝐼) → (𝑆𝑘) ⊆ (𝑇𝑘))       (𝜑 → (𝐺dom DProd 𝑆 ∧ (𝐺 DProd 𝑆) ⊆ (𝐺 DProd 𝑇)))

Theoremdprdz 18475* 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 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐼𝑉) → (𝐺dom DProd (𝑥𝐼 ↦ { 0 }) ∧ (𝐺 DProd (𝑥𝐼 ↦ { 0 })) = { 0 }))

Theoremdprd0 18476 The empty family is an internal direct product, the product of which is the trivial subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
0 = (0g𝐺)       (𝐺 ∈ Grp → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = { 0 }))

Theoremdprdf1o 18477 Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹:𝐽1-1-onto𝐼)       (𝜑 → (𝐺dom DProd (𝑆𝐹) ∧ (𝐺 DProd (𝑆𝐹)) = (𝐺 DProd 𝑆)))

Theoremdprdf1 18478 Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐹:𝐽1-1𝐼)       (𝜑 → (𝐺dom DProd (𝑆𝐹) ∧ (𝐺 DProd (𝑆𝐹)) ⊆ (𝐺 DProd 𝑆)))

Theoremsubgdmdprd 18479 A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐻 = (𝐺s 𝐴)       (𝐴 ∈ (SubGrp‘𝐺) → (𝐻dom DProd 𝑆 ↔ (𝐺dom DProd 𝑆 ∧ ran 𝑆 ⊆ 𝒫 𝐴)))

Theoremsubgdprd 18480 A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐻 = (𝐺s 𝐴)    &   (𝜑𝐴 ∈ (SubGrp‘𝐺))    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → ran 𝑆 ⊆ 𝒫 𝐴)       (𝜑 → (𝐻 DProd 𝑆) = (𝐺 DProd 𝑆))

Theoremdprdsn 18481 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
((𝐴𝑉𝑆 ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {⟨𝐴, 𝑆⟩} ∧ (𝐺 DProd {⟨𝐴, 𝑆⟩}) = 𝑆))

Theoremdmdprdsplitlem 18482* Lemma for dmdprdsplit 18492. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
0 = (0g𝐺)    &   𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }    &   (𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐴𝐼)    &   (𝜑𝐹𝑊)    &   (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd (𝑆𝐴)))       ((𝜑𝑋 ∈ (𝐼𝐴)) → (𝐹𝑋) = 0 )

Theoremdprdcntz2 18483 The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐶𝐼)    &   (𝜑𝐷𝐼)    &   (𝜑 → (𝐶𝐷) = ∅)    &   𝑍 = (Cntz‘𝐺)       (𝜑 → (𝐺 DProd (𝑆𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆𝐷))))

Theoremdprddisj2 18484 The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.) (Revised by AV, 14-Jul-2019.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝐶𝐼)    &   (𝜑𝐷𝐼)    &   (𝜑 → (𝐶𝐷) = ∅)    &    0 = (0g𝐺)       (𝜑 → ((𝐺 DProd (𝑆𝐶)) ∩ (𝐺 DProd (𝑆𝐷))) = { 0 })

Theoremdprd2dlem2 18485* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑 → Rel 𝐴)    &   (𝜑𝑆:𝐴⟶(SubGrp‘𝐺))    &   (𝜑 → dom 𝐴𝐼)    &   ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))    &   (𝜑𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝜑𝑋𝐴) → (𝑆𝑋) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑋)}) ↦ ((1st𝑋)𝑆𝑗))))

Theoremdprd2dlem1 18486* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑 → Rel 𝐴)    &   (𝜑𝑆:𝐴⟶(SubGrp‘𝐺))    &   (𝜑 → dom 𝐴𝐼)    &   ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))    &   (𝜑𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))    &   𝐾 = (mrCls‘(SubGrp‘𝐺))    &   (𝜑𝐶𝐼)       (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) = (𝐺 DProd (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))

Theoremdprd2da 18487* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑 → Rel 𝐴)    &   (𝜑𝑆:𝐴⟶(SubGrp‘𝐺))    &   (𝜑 → dom 𝐴𝐼)    &   ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))    &   (𝜑𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       (𝜑𝐺dom DProd 𝑆)

Theoremdprd2db 18488* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑 → Rel 𝐴)    &   (𝜑𝑆:𝐴⟶(SubGrp‘𝐺))    &   (𝜑 → dom 𝐴𝐼)    &   ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))    &   (𝜑𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       (𝜑 → (𝐺 DProd 𝑆) = (𝐺 DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))

Theoremdprd2d2 18489* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
((𝜑 ∧ (𝑖𝐼𝑗𝐽)) → 𝑆 ∈ (SubGrp‘𝐺))    &   ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗𝐽𝑆))    &   (𝜑𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗𝐽𝑆))))       (𝜑 → (𝐺dom DProd (𝑖𝐼, 𝑗𝐽𝑆) ∧ (𝐺 DProd (𝑖𝐼, 𝑗𝐽𝑆)) = (𝐺 DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗𝐽𝑆))))))

Theoremdmdprdsplit2lem 18490 Lemma for dmdprdsplit 18492. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝑆:𝐼⟶(SubGrp‘𝐺))    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐼 = (𝐶𝐷))    &   𝑍 = (Cntz‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺dom DProd (𝑆𝐶))    &   (𝜑𝐺dom DProd (𝑆𝐷))    &   (𝜑 → (𝐺 DProd (𝑆𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆𝐷))))    &   (𝜑 → ((𝐺 DProd (𝑆𝐶)) ∩ (𝐺 DProd (𝑆𝐷))) = { 0 })    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝜑𝑋𝐶) → ((𝑌𝐼 → (𝑋𝑌 → (𝑆𝑋) ⊆ (𝑍‘(𝑆𝑌)))) ∧ ((𝑆𝑋) ∩ (𝐾 (𝑆 “ (𝐼 ∖ {𝑋})))) ⊆ { 0 }))

Theoremdmdprdsplit2 18491 The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝑆:𝐼⟶(SubGrp‘𝐺))    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐼 = (𝐶𝐷))    &   𝑍 = (Cntz‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺dom DProd (𝑆𝐶))    &   (𝜑𝐺dom DProd (𝑆𝐷))    &   (𝜑 → (𝐺 DProd (𝑆𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆𝐷))))    &   (𝜑 → ((𝐺 DProd (𝑆𝐶)) ∩ (𝐺 DProd (𝑆𝐷))) = { 0 })       (𝜑𝐺dom DProd 𝑆)

Theoremdmdprdsplit 18492 The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝑆:𝐼⟶(SubGrp‘𝐺))    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐼 = (𝐶𝐷))    &   𝑍 = (Cntz‘𝐺)    &    0 = (0g𝐺)       (𝜑 → (𝐺dom DProd 𝑆 ↔ ((𝐺dom DProd (𝑆𝐶) ∧ 𝐺dom DProd (𝑆𝐷)) ∧ (𝐺 DProd (𝑆𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆𝐷))) ∧ ((𝐺 DProd (𝑆𝐶)) ∩ (𝐺 DProd (𝑆𝐷))) = { 0 })))

Theoremdprdsplit 18493 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 25-Apr-2016.)
(𝜑𝑆:𝐼⟶(SubGrp‘𝐺))    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐼 = (𝐶𝐷))    &    = (LSSum‘𝐺)    &   (𝜑𝐺dom DProd 𝑆)       (𝜑 → (𝐺 DProd 𝑆) = ((𝐺 DProd (𝑆𝐶)) (𝐺 DProd (𝑆𝐷))))

Theoremdmdprdpr 18494 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 = (0g𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))       (𝜑 → (𝐺dom DProd ({𝑆} +𝑐 {𝑇}) ↔ (𝑆 ⊆ (𝑍𝑇) ∧ (𝑆𝑇) = { 0 })))

Theoremdprdpr 18495 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 = (0g𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &    = (LSSum‘𝐺)    &   (𝜑𝑆 ⊆ (𝑍𝑇))    &   (𝜑 → (𝑆𝑇) = { 0 })       (𝜑 → (𝐺 DProd ({𝑆} +𝑐 {𝑇})) = (𝑆 𝑇))

Theoremdpjlem 18496 Lemma for theorems about direct product projection. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)       (𝜑 → (𝐺 DProd (𝑆 ↾ {𝑋})) = (𝑆𝑋))

Theoremdpjcntz 18497 The two subgroups that appear in dpjval 18501 commute. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)    &   𝑍 = (Cntz‘𝐺)       (𝜑 → (𝑆𝑋) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))

Theoremdpjdisj 18498 The two subgroups that appear in dpjval 18501 are disjoint. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)    &    0 = (0g𝐺)       (𝜑 → ((𝑆𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = { 0 })

Theoremdpjlsm 18499 The two subgroups that appear in dpjval 18501 add to the full direct product. (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   (𝜑𝑋𝐼)    &    = (LSSum‘𝐺)       (𝜑 → (𝐺 DProd 𝑆) = ((𝑆𝑋) (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))

Theoremdpjfval 18500* Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
(𝜑𝐺dom DProd 𝑆)    &   (𝜑 → dom 𝑆 = 𝐼)    &   𝑃 = (𝐺dProj𝑆)    &   𝑄 = (proj1𝐺)       (𝜑𝑃 = (𝑖𝐼 ↦ ((𝑆𝑖)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑖}))))))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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