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Theorem List for Metamath Proof Explorer - 18601-18700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremablinvadd 18601 The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑋) + (𝑁𝑌)))

Theoremablsub2inv 18602 Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   𝑁 = (invg𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑁𝑋) (𝑁𝑌)) = (𝑌 𝑋))

Theoremablsubadd 18603 Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑌 + 𝑍) = 𝑋))

Theoremablsub4 18604 Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑊)) = ((𝑋 𝑍) + (𝑌 𝑊)))

𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) = (𝑍 + 𝑊) ↔ (𝑋 𝑍) = (𝑊 𝑌)))

Theoremabladdsub 18606 Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = ((𝑋 𝑍) + 𝑌))

Theoremablpncan2 18607 Cancellation law for subtraction. (Contributed by NM, 2-Oct-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) 𝑋) = 𝑌)

Theoremablpncan3 18608 A cancellation law for commutative groups. (Contributed by NM, 23-Mar-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 + (𝑌 𝑋)) = 𝑌)

Theoremablsubsub 18609 Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) + 𝑍))

Theoremablsubsub4 18610 Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 + 𝑍)))

Theoremablpnpcan 18611 Cancellation law for mixed addition and subtraction. (pnpcan 10662 analog.) (Contributed by NM, 29-May-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = (𝑌 𝑍))

Theoremablnncan 18612 Cancellation law for group subtraction. (nncan 10652 analog.) (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 (𝑋 𝑌)) = 𝑌)

Theoremablsub32 18613 Swap the second and third terms in a double group subtraction. (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) 𝑌))

Theoremablnnncan 18614 Cancellation law for group subtraction. (nnncan 10658 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 (𝑌 𝑍)) 𝑍) = (𝑋 𝑌))

Theoremablnnncan1 18615 Cancellation law for group subtraction. (nnncan1 10659 analog.) (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 𝑌) (𝑋 𝑍)) = (𝑍 𝑌))

Theoremablsubsub23 18616 Swap subtrahend and result of group subtraction. (Contributed by NM, 14-Dec-2007.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐴 𝐶) = 𝐵))

Theoremmulgnn0di 18617 Group multiple of a sum, for nonnegative multiples. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ CMnd ∧ (𝑀 ∈ ℕ0𝑋𝐵𝑌𝐵)) → (𝑀 · (𝑋 + 𝑌)) = ((𝑀 · 𝑋) + (𝑀 · 𝑌)))

Theoremmulgdi 18618 Group multiple of a sum. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Abel ∧ (𝑀 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑀 · (𝑋 + 𝑌)) = ((𝑀 · 𝑋) + (𝑀 · 𝑌)))

Theoremmulgmhm 18619* The map from 𝑥 to 𝑛𝑥 for a fixed positive integer 𝑛 is a monoid homomorphism if the monoid is commutative. (Contributed by Mario Carneiro, 4-May-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0) → (𝑥𝐵 ↦ (𝑀 · 𝑥)) ∈ (𝐺 MndHom 𝐺))

Theoremmulgghm 18620* The map from 𝑥 to 𝑛𝑥 for a fixed integer 𝑛 is a group homomorphism if the group is commutative. (Contributed by Mario Carneiro, 4-May-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ) → (𝑥𝐵 ↦ (𝑀 · 𝑥)) ∈ (𝐺 GrpHom 𝐺))

Theoremmulgsubdi 18621 Group multiple of a difference. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑀 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑀 · (𝑋 𝑌)) = ((𝑀 · 𝑋) (𝑀 · 𝑌)))

Theoremghmfghm 18622* The function fulfilling the conditions of ghmgrp 17926 is a group homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.)
𝑋 = (Base‘𝐺)    &   𝑌 = (Base‘𝐻)    &    + = (+g𝐺)    &    = (+g𝐻)    &   ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ Grp)       (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))

Theoremghmcmn 18623* The image of a commutative monoid 𝐺 under a group homomorphism 𝐹 is a commutative monoid. (Contributed by Thierry Arnoux, 26-Jan-2020.)
𝑋 = (Base‘𝐺)    &   𝑌 = (Base‘𝐻)    &    + = (+g𝐺)    &    = (+g𝐻)    &   ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ CMnd)       (𝜑𝐻 ∈ CMnd)

Theoremghmabl 18624* The image of an abelian group 𝐺 under a group homomorphism 𝐹 is an abelian group. (Contributed by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)
𝑋 = (Base‘𝐺)    &   𝑌 = (Base‘𝐻)    &    + = (+g𝐺)    &    = (+g𝐻)    &   ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ Abel)       (𝜑𝐻 ∈ Abel)

Theoreminvghm 18625 The inversion map is a group automorphism if and only if the group is abelian. (In general it is only a group homomorphism into the opposite group, but in an abelian group the opposite group coincides with the group itself.) (Contributed by Mario Carneiro, 4-May-2015.)
𝐵 = (Base‘𝐺)    &   𝐼 = (invg𝐺)       (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺))

Theoremeqgabl 18626 Value of the subgroup coset equivalence relation on an abelian group. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &    = (𝐺 ~QG 𝑆)       ((𝐺 ∈ Abel ∧ 𝑆𝑋) → (𝐴 𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ (𝐵 𝐴) ∈ 𝑆)))

Theoremsubgabl 18627 A subgroup of an abelian group is also abelian. (Contributed by Mario Carneiro, 3-Dec-2014.)
𝐻 = (𝐺s 𝑆)       ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)

Theoremsubcmn 18628 A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐻 = (𝐺s 𝑆)       ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐻 ∈ CMnd)

Theoremsubmcmn 18629 A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝐻 = (𝐺s 𝑆)       ((𝐺 ∈ CMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ CMnd)

Theoremsubmcmn2 18630 A submonoid is commutative iff it is a subset of its own centralizer. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝐻 = (𝐺s 𝑆)    &   𝑍 = (Cntz‘𝐺)       (𝑆 ∈ (SubMnd‘𝐺) → (𝐻 ∈ CMnd ↔ 𝑆 ⊆ (𝑍𝑆)))

Theoremcntzcmn 18631 The centralizer of any subset in a commutative monoid is the whole monoid. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐵 = (Base‘𝐺)    &   𝑍 = (Cntz‘𝐺)       ((𝐺 ∈ CMnd ∧ 𝑆𝐵) → (𝑍𝑆) = 𝐵)

Theoremcntzcmnss 18632 Any subset in a commutative monoid is a subset of its centralizer. (Contributed by AV, 12-Jan-2019.)
𝐵 = (Base‘𝐺)    &   𝑍 = (Cntz‘𝐺)       ((𝐺 ∈ CMnd ∧ 𝑆𝐵) → 𝑆 ⊆ (𝑍𝑆))

Theoremcntzspan 18633 If the generators commute, the generated monoid is commutative. (Contributed by Mario Carneiro, 25-Apr-2016.)
𝑍 = (Cntz‘𝐺)    &   𝐾 = (mrCls‘(SubMnd‘𝐺))    &   𝐻 = (𝐺s (𝐾𝑆))       ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍𝑆)) → 𝐻 ∈ CMnd)

Theoremcntzcmnf 18634 Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))

Theoremghmplusg 18635 The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
+ = (+g𝑁)       ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹𝑓 + 𝐺) ∈ (𝑀 GrpHom 𝑁))

Theoremablnsg 18636 Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
(𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))

Theoremodadd1 18637 The order of a product in an abelian group divides the LCM of the orders of the factors. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑂 = (od‘𝐺)    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))) ∥ ((𝑂𝐴) · (𝑂𝐵)))

Theoremodadd2 18638 The order of a product in an abelian group is divisible by the LCM of the orders of the factors divided by the GCD. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑂 = (od‘𝐺)    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → ((𝑂𝐴) · (𝑂𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)))

Theoremodadd 18639 The order of a product is the product of the orders, if the factors have coprime order. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑂 = (od‘𝐺)    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 1) → (𝑂‘(𝐴 + 𝐵)) = ((𝑂𝐴) · (𝑂𝐵)))

Theoremgex2abl 18640 A group with exponent 2 (or 1) is abelian. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) → 𝐺 ∈ Abel)

Theoremgexexlem 18641* Lemma for gexex 18642. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝑂 = (od‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐸 ∈ ℕ)    &   (𝜑𝐴𝑋)    &   ((𝜑𝑦𝑋) → (𝑂𝑦) ≤ (𝑂𝐴))       (𝜑 → (𝑂𝐴) = 𝐸)

Theoremgexex 18642* In an abelian group with finite exponent, there is an element in the group with order equal to the exponent. In other words, all orders of elements divide the largest order of an element of the group. This fails if 𝐸 = 0, for example in an infinite p-group, where there are elements of arbitrarily large orders (so 𝐸 is zero) but no elements of infinite order. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ∃𝑥𝑋 (𝑂𝑥) = 𝐸)

Theoremtorsubg 18643 The set of all elements of finite order forms a subgroup of any abelian group, called the torsion subgroup. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑂 = (od‘𝐺)       (𝐺 ∈ Abel → (𝑂 “ ℕ) ∈ (SubGrp‘𝐺))

Theoremoddvdssubg 18644* The set of all elements whose order divides a fixed integer is a subgroup of any abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑂 = (od‘𝐺)    &   𝐵 = (Base‘𝐺)       ((𝐺 ∈ Abel ∧ 𝑁 ∈ ℤ) → {𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑁} ∈ (SubGrp‘𝐺))

Theoremlsmcomx 18645 Subgroup sum commutes (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       ((𝐺 ∈ Abel ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = (𝑈 𝑇))

Theoremablcntzd 18646 All subgroups in an abelian group commute. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))       (𝜑𝑇 ⊆ (𝑍𝑈))

Theoremlsmcom 18647 Subgroup sum commutes. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
= (LSSum‘𝐺)       ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 𝑈) = (𝑈 𝑇))

Theoremlsmsubg2 18648 The sum of two subgroups is a subgroup. (Contributed by NM, 4-Feb-2014.) (Proof shortened by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝐺 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 𝑈) ∈ (SubGrp‘𝐺))

Theoremlsm4 18649 Commutative/associative law for subgroup sum. (Contributed by NM, 26-Sep-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝐺 ∈ Abel ∧ (𝑄 ∈ (SubGrp‘𝐺) ∧ 𝑅 ∈ (SubGrp‘𝐺)) ∧ (𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺))) → ((𝑄 𝑅) (𝑇 𝑈)) = ((𝑄 𝑇) (𝑅 𝑈)))

Theoremprdscmnd 18650 The product of a family of commutative monoids is commutative. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶CMnd)       (𝜑𝑌 ∈ CMnd)

Theoremprdsabld 18651 The product of a family of Abelian groups is an Abelian group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Abel)       (𝜑𝑌 ∈ Abel)

Theorempwscmn 18652 The structure power on a commutative monoid is commutative. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ CMnd ∧ 𝐼𝑉) → 𝑌 ∈ CMnd)

Theorempwsabl 18653 The structure power on an Abelian group is Abelian. (Contributed by Mario Carneiro, 21-Jan-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ Abel ∧ 𝐼𝑉) → 𝑌 ∈ Abel)

Theoremqusabl 18654 If 𝑌 is a subgroup of the abelian group 𝐺, then 𝐻 = 𝐺 / 𝑌 is an abelian group. (Contributed by Mario Carneiro, 26-Apr-2016.)
𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))       ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)

Theoremabl1 18655 The (smallest) structure representing a trivial abelian group. (Contributed by AV, 28-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}       (𝐼𝑉𝑀 ∈ Abel)

Theoremabln0 18656 Abelian groups (and therefore also groups and monoids) exist. (Contributed by AV, 29-Apr-2019.)
Abel ≠ ∅

Theoremcnaddablx 18657 The complex numbers are an Abelian group under addition. This version of cnaddabl 18658 shows the explicit structure "scaffold" we chose for the definition for Abelian groups. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use cnaddabl 18658 instead. (New usage is discouraged.) (Contributed by NM, 18-Oct-2012.)
𝐺 = {⟨1, ℂ⟩, ⟨2, + ⟩}       𝐺 ∈ Abel

Theoremcnaddabl 18658 The complex numbers are an Abelian group under addition. This version of cnaddablx 18657 hides the explicit structure indices i.e. is "scaffold-independent". Note that the proof also does not reference explicit structure indices. The actual structure is dependent on how Base and +g is defined. This theorem should not be referenced in any proof. For the group/ring properties of the complex numbers, see cnring 20164. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.)
𝐺 = {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩}       𝐺 ∈ Abel

Theoremcnaddid 18659 The group identity element of complex number addition is zero. See also cnfld0 20166. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 26-Aug-2021.) (New usage is discouraged.)
𝐺 = {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩}       (0g𝐺) = 0

Theoremcnaddinv 18660 Value of the group inverse of complex number addition. See also cnfldneg 20168. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 26-Aug-2021.) (New usage is discouraged.)
𝐺 = {⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩}       (𝐴 ∈ ℂ → ((invg𝐺)‘𝐴) = -𝐴)

Theoremzaddablx 18661 The integers are an Abelian group under addition. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. Use zsubrg 20195 instead. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)
𝐺 = {⟨1, ℤ⟩, ⟨2, + ⟩}       𝐺 ∈ Abel

Theoremfrgpnabllem1 18662* Lemma for frgpnabl 18664. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐺 = (freeGrp‘𝐼)    &   𝑊 = ( I ‘Word (𝐼 × 2o))    &    = ( ~FG𝐼)    &    + = (+g𝐺)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑈 = (varFGrp𝐼)    &   (𝜑𝐼 ∈ V)    &   (𝜑𝐴𝐼)    &   (𝜑𝐵𝐼)       (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈𝐴) + (𝑈𝐵))))

Theoremfrgpnabllem2 18663* Lemma for frgpnabl 18664. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐺 = (freeGrp‘𝐼)    &   𝑊 = ( I ‘Word (𝐼 × 2o))    &    = ( ~FG𝐼)    &    + = (+g𝐺)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑈 = (varFGrp𝐼)    &   (𝜑𝐼 ∈ V)    &   (𝜑𝐴𝐼)    &   (𝜑𝐵𝐼)    &   (𝜑 → ((𝑈𝐴) + (𝑈𝐵)) = ((𝑈𝐵) + (𝑈𝐴)))       (𝜑𝐴 = 𝐵)

Theoremfrgpnabl 18664 The free group on two or more generators is not abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐺 = (freeGrp‘𝐼)       (1o𝐼 → ¬ 𝐺 ∈ Abel)

10.3.2  Cyclic groups

Syntaxccyg 18665 Cyclic group.
class CycGrp

Definitiondf-cyg 18666* Define a cyclic group, which is a group with an element 𝑥, called the generator of the group, such that all elements in the group are multiples of 𝑥. A generator is usually not unique. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔)}

Theoremiscyg 18667* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵))

Theoremiscyggen 18668* The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}       (𝑋𝐸 ↔ (𝑋𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))

Theoremiscyggen2 18669* The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}       (𝐺 ∈ Grp → (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋))))

Theoremiscyg2 18670* A cyclic group is a group which contains a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}       (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ 𝐸 ≠ ∅))

Theoremcyggeninv 18671* The inverse of a cyclic generator is a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐸) → (𝑁𝑋) ∈ 𝐸)

Theoremcyggenod 18672* An element is the generator of a finite group iff the order of the generator equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → (𝑋𝐸 ↔ (𝑋𝐵 ∧ (𝑂𝑋) = (♯‘𝐵))))

Theoremcyggenod2 18673* In an infinite cyclic group, the generator must have infinite order, but this property no longer characterizes the generators. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋𝐸) → (𝑂𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0))

Theoremiscyg3 18674* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥𝐵𝑦𝐵𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑥)))

Theoremiscygd 18675* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋𝐵)    &   ((𝜑𝑦𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋))       (𝜑𝐺 ∈ CycGrp)

Theoremiscygodd 18676 Show that a group with an element the same order as the group is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋𝐵)    &   (𝜑 → (𝑂𝑋) = (♯‘𝐵))       (𝜑𝐺 ∈ CycGrp)

Theoremcyggrp 18677 A cyclic group is a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝐺 ∈ CycGrp → 𝐺 ∈ Grp)

Theoremcygabl 18678 A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝐺 ∈ CycGrp → 𝐺 ∈ Abel)

Theoremcygctb 18679 A cyclic group is countable. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ CycGrp → 𝐵 ≼ ω)

Theorem0cyg 18680 The trivial group is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) → 𝐺 ∈ CycGrp)

Theoremprmcyg 18681 A group with prime order is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐵 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ (♯‘𝐵) ∈ ℙ) → 𝐺 ∈ CycGrp)

Theoremlt6abl 18682 A group with fewer than 6 elements is abelian. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐵 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ (♯‘𝐵) < 6) → 𝐺 ∈ Abel)

Theoremghmcyg 18683 The image of a cyclic group under a surjective group homomorphism is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝐶 = (Base‘𝐻)       ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝐹:𝐵onto𝐶) → (𝐺 ∈ CycGrp → 𝐻 ∈ CycGrp))

Theoremcyggex2 18684 The exponent of a cyclic group is 0 if the group is infinite, otherwise it equals the order of the group. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)       (𝐺 ∈ CycGrp → 𝐸 = if(𝐵 ∈ Fin, (♯‘𝐵), 0))

Theoremcyggex 18685 The exponent of a finite cyclic group is the order of the group. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)       ((𝐺 ∈ CycGrp ∧ 𝐵 ∈ Fin) → 𝐸 = (♯‘𝐵))

Theoremcyggexb 18686 A finite abelian group is cyclic iff the exponent equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)       ((𝐺 ∈ Abel ∧ 𝐵 ∈ Fin) → (𝐺 ∈ CycGrp ↔ 𝐸 = (♯‘𝐵)))

Theoremgiccyg 18687 Cyclicity is a group property, i.e. it is preserved under isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝐺𝑔 𝐻 → (𝐺 ∈ CycGrp → 𝐻 ∈ CycGrp))

Theoremcycsubgcyg 18688* The cyclic subgroup generated by 𝐴 is a cyclic group. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝑆 = ran (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐺s 𝑆) ∈ CycGrp)

Theoremcycsubgcyg2 18689 The cyclic subgroup generated by 𝐴 is a cyclic group. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐺 ∈ Grp ∧ 𝐴𝐵) → (𝐺s (𝐾‘{𝐴})) ∈ CycGrp)

10.3.3  Group sum operation

Theoremgsumval3a 18690* Value of the group sum operation over an index set with finite support. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by AV, 29-May-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝑊 ∈ Fin)    &   (𝜑𝑊 ≠ ∅)    &   𝑊 = (𝐹 supp 0 )    &   (𝜑 → ¬ 𝐴 ∈ ran ...)       (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))))

Theoremgsumval3eu 18691* The group sum as defined in gsumval3a 18690 is uniquely defined. (Contributed by Mario Carneiro, 8-Dec-2014.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝑊 ∈ Fin)    &   (𝜑𝑊 ≠ ∅)    &   (𝜑𝑊𝐴)       (𝜑 → ∃!𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))

Theoremgsumval3lem1 18692* Lemma 1 for gsumval3 18694. (Contributed by AV, 31-May-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐻:(1...𝑀)–1-1𝐴)    &   (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)    &   𝑊 = ((𝐹𝐻) supp 0 )       (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻𝑓):(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))

Theoremgsumval3lem2 18693* Lemma 2 for gsumval3 18694. (Contributed by AV, 31-May-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐻:(1...𝑀)–1-1𝐴)    &   (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)    &   𝑊 = ((𝐹𝐻) supp 0 )       (((𝜑𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ (𝐻𝑓)))‘(♯‘𝑊)))

Theoremgsumval3 18694 Value of the group sum operation over an arbitrary finite set. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 31-May-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐻:(1...𝑀)–1-1𝐴)    &   (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻)    &   𝑊 = ((𝐹𝐻) supp 0 )       (𝜑 → (𝐺 Σg 𝐹) = (seq1( + , (𝐹𝐻))‘𝑀))

Theoremgsumcllem 18695* Lemma for gsumcl 18702 and related theorems. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 31-May-2019.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝑍𝑈)    &   (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)       ((𝜑𝑊 = ∅) → 𝐹 = (𝑘𝐴𝑍))

Theoremgsumzres 18696 Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 31-May-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹))

Theoremgsumzcl2 18697 Closure of a finite group sum. This theorem has a weaker hypothesis than gsumzcl 18698, because it is not required that 𝐹 is a function (actually, the hypothesis always holds for any proper class 𝐹). (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 1-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑 → (𝐹 supp 0 ) ∈ Fin)       (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵)

Theoremgsumzcl 18698 Closure of a finite group sum. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 1-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵)

Theoremgsumzf1o 18699 Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 2-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))    &   (𝜑𝐹 finSupp 0 )    &   (𝜑𝐻:𝐶1-1-onto𝐴)       (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹𝐻)))

Theoremgsumres 18700 Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹))

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