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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ablinvadd 18601 | The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁‘𝑋) + (𝑁‘𝑌))) | ||
Theorem | ablsub2inv 18602 | Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑁‘𝑋) − (𝑁‘𝑌)) = (𝑌 − 𝑋)) | ||
Theorem | ablsubadd 18603 | Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑌) = 𝑍 ↔ (𝑌 + 𝑍) = 𝑋)) | ||
Theorem | ablsub4 18604 | Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 + 𝑌) − (𝑍 + 𝑊)) = ((𝑋 − 𝑍) + (𝑌 − 𝑊))) | ||
Theorem | abladdsub4 18605 | Abelian group addition/subtraction law. (Contributed by NM, 31-Mar-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 + 𝑌) = (𝑍 + 𝑊) ↔ (𝑋 − 𝑍) = (𝑊 − 𝑌))) | ||
Theorem | abladdsub 18606 | Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) − 𝑍) = ((𝑋 − 𝑍) + 𝑌)) | ||
Theorem | ablpncan2 18607 | Cancellation law for subtraction. (Contributed by NM, 2-Oct-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑌) − 𝑋) = 𝑌) | ||
Theorem | ablpncan3 18608 | A cancellation law for commutative groups. (Contributed by NM, 23-Mar-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 + (𝑌 − 𝑋)) = 𝑌) | ||
Theorem | ablsubsub 18609 | Law for double subtraction. (Contributed by NM, 7-Apr-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 − (𝑌 − 𝑍)) = ((𝑋 − 𝑌) + 𝑍)) | ||
Theorem | ablsubsub4 18610 | Law for double subtraction. (Contributed by NM, 7-Apr-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = (𝑋 − (𝑌 + 𝑍))) | ||
Theorem | ablpnpcan 18611 | Cancellation law for mixed addition and subtraction. (pnpcan 10662 analog.) (Contributed by NM, 29-May-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = (𝑌 − 𝑍)) | ||
Theorem | ablnncan 18612 | Cancellation law for group subtraction. (nncan 10652 analog.) (Contributed by NM, 7-Apr-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = 𝑌) | ||
Theorem | ablsub32 18613 | Swap the second and third terms in a double group subtraction. (Contributed by NM, 7-Apr-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = ((𝑋 − 𝑍) − 𝑌)) | ||
Theorem | ablnnncan 18614 | Cancellation law for group subtraction. (nnncan 10658 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 − (𝑌 − 𝑍)) − 𝑍) = (𝑋 − 𝑌)) | ||
Theorem | ablnnncan1 18615 | Cancellation law for group subtraction. (nnncan1 10659 analog.) (Contributed by NM, 7-Apr-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = (𝑍 − 𝑌)) | ||
Theorem | ablsubsub23 18616 | Swap subtrahend and result of group subtraction. (Contributed by NM, 14-Dec-2007.) (Revised by AV, 7-Oct-2021.) |
⊢ 𝑉 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐴 − 𝐶) = 𝐵)) | ||
Theorem | mulgnn0di 18617 | Group multiple of a sum, for nonnegative multiples. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ (𝑀 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑀 · (𝑋 + 𝑌)) = ((𝑀 · 𝑋) + (𝑀 · 𝑌))) | ||
Theorem | mulgdi 18618 | Group multiple of a sum. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑀 · (𝑋 + 𝑌)) = ((𝑀 · 𝑋) + (𝑀 · 𝑌))) | ||
Theorem | mulgmhm 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) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0) → (𝑥 ∈ 𝐵 ↦ (𝑀 · 𝑥)) ∈ (𝐺 MndHom 𝐺)) | ||
Theorem | mulgghm 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) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ) → (𝑥 ∈ 𝐵 ↦ (𝑀 · 𝑥)) ∈ (𝐺 GrpHom 𝐺)) | ||
Theorem | mulgsubdi 18621 | Group multiple of a difference. (Contributed by Mario Carneiro, 13-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑀 · (𝑋 − 𝑌)) = ((𝑀 · 𝑋) − (𝑀 · 𝑌))) | ||
Theorem | ghmfghm 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) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Grp) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) | ||
Theorem | ghmcmn 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) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ CMnd) ⇒ ⊢ (𝜑 → 𝐻 ∈ CMnd) | ||
Theorem | ghmabl 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) & ⊢ + = (+g‘𝐺) & ⊢ ⨣ = (+g‘𝐻) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ (𝜑 → 𝐺 ∈ Abel) ⇒ ⊢ (𝜑 → 𝐻 ∈ Abel) | ||
Theorem | invghm 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 = (invg‘𝐺) ⇒ ⊢ (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺)) | ||
Theorem | eqgabl 18626 | Value of the subgroup coset equivalence relation on an abelian group. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ ∼ = (𝐺 ~QG 𝑆) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐵 − 𝐴) ∈ 𝑆))) | ||
Theorem | subgabl 18627 | A subgroup of an abelian group is also abelian. (Contributed by Mario Carneiro, 3-Dec-2014.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel) | ||
Theorem | subcmn 18628 | A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐻 ∈ CMnd) | ||
Theorem | submcmn 18629 | A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ CMnd) | ||
Theorem | submcmn2 18630 | A submonoid is commutative iff it is a subset of its own centralizer. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) & ⊢ 𝑍 = (Cntz‘𝐺) ⇒ ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (𝐻 ∈ CMnd ↔ 𝑆 ⊆ (𝑍‘𝑆))) | ||
Theorem | cntzcmn 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 ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) = 𝐵) | ||
Theorem | cntzcmnss 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 ∧ 𝑆 ⊆ 𝐵) → 𝑆 ⊆ (𝑍‘𝑆)) | ||
Theorem | cntzspan 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 = (𝐺 ↾s (𝐾‘𝑆)) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍‘𝑆)) → 𝐻 ∈ CMnd) | ||
Theorem | cntzcmnf 18634 | Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) | ||
Theorem | ghmplusg 18635 | The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ + = (+g‘𝑁) ⇒ ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘𝑓 + 𝐺) ∈ (𝑀 GrpHom 𝑁)) | ||
Theorem | ablnsg 18636 | Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺)) | ||
Theorem | odadd1 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) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵))) | ||
Theorem | odadd2 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) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2))) | ||
Theorem | odadd 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) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (𝑂‘(𝐴 + 𝐵)) = ((𝑂‘𝐴) · (𝑂‘𝐵))) | ||
Theorem | gex2abl 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) | ||
Theorem | gexexlem 18641* | Lemma for gexex 18642. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐸 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ≤ (𝑂‘𝐴)) ⇒ ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸) | ||
Theorem | gexex 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) → ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = 𝐸) | ||
Theorem | torsubg 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)) | ||
Theorem | oddvdssubg 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) → {𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ 𝑁} ∈ (SubGrp‘𝐺)) | ||
Theorem | lsmcomx 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 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) | ||
Theorem | ablcntzd 18646 | All subgroups in an abelian group commute. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) ⇒ ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) | ||
Theorem | lsmcom 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)) | ||
Theorem | lsmsubg2 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)) | ||
Theorem | lsm4 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))) | ||
Theorem | prdscmnd 18650 | The product of a family of commutative monoids is commutative. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶CMnd) ⇒ ⊢ (𝜑 → 𝑌 ∈ CMnd) | ||
Theorem | prdsabld 18651 | The product of a family of Abelian groups is an Abelian group. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Abel) ⇒ ⊢ (𝜑 → 𝑌 ∈ Abel) | ||
Theorem | pwscmn 18652 | The structure power on a commutative monoid is commutative. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) ⇒ ⊢ ((𝑅 ∈ CMnd ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ CMnd) | ||
Theorem | pwsabl 18653 | The structure power on an Abelian group is Abelian. (Contributed by Mario Carneiro, 21-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) ⇒ ⊢ ((𝑅 ∈ Abel ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Abel) | ||
Theorem | qusabl 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) | ||
Theorem | abl1 18655 | The (smallest) structure representing a trivial abelian group. (Contributed by AV, 28-Apr-2019.) |
⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Abel) | ||
Theorem | abln0 18656 | Abelian groups (and therefore also groups and monoids) exist. (Contributed by AV, 29-Apr-2019.) |
⊢ Abel ≠ ∅ | ||
Theorem | cnaddablx 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 | ||
Theorem | cnaddabl 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 | ||
Theorem | cnaddid 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), + 〉} ⇒ ⊢ (0g‘𝐺) = 0 | ||
Theorem | cnaddinv 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 → ((invg‘𝐺)‘𝐴) = -𝐴) | ||
Theorem | zaddablx 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 | ||
Theorem | frgpnabllem1 18662* | Lemma for frgpnabl 18664. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ + = (+g‘𝐺) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑈 = (varFGrp‘𝐼) & ⊢ (𝜑 → 𝐼 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ (𝜑 → 𝐵 ∈ 𝐼) ⇒ ⊢ (𝜑 → 〈“〈𝐴, ∅〉〈𝐵, ∅〉”〉 ∈ (𝐷 ∩ ((𝑈‘𝐴) + (𝑈‘𝐵)))) | ||
Theorem | frgpnabllem2 18663* | Lemma for frgpnabl 18664. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ + = (+g‘𝐺) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑈 = (varFGrp‘𝐼) & ⊢ (𝜑 → 𝐼 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ (𝜑 → 𝐵 ∈ 𝐼) & ⊢ (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = ((𝑈‘𝐵) + (𝑈‘𝐴))) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | frgpnabl 18664 | The free group on two or more generators is not abelian. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐺 = (freeGrp‘𝐼) ⇒ ⊢ (1o ≺ 𝐼 → ¬ 𝐺 ∈ Abel) | ||
Syntax | ccyg 18665 | Cyclic group. |
class CycGrp | ||
Definition | df-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(.g‘𝑔)𝑥)) = (Base‘𝑔)} | ||
Theorem | iscyg 18667* | Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵)) | ||
Theorem | iscyggen 18668* | The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ⇒ ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) | ||
Theorem | iscyggen2 18669* | The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ⇒ ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋)))) | ||
Theorem | iscyg2 18670* | A cyclic group is a group which contains a generator. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ⇒ ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ 𝐸 ≠ ∅)) | ||
Theorem | cyggeninv 18671* | The inverse of a cyclic generator is a generator. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸) → (𝑁‘𝑋) ∈ 𝐸) | ||
Theorem | cyggenod 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) & ⊢ · = (.g‘𝐺) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} & ⊢ 𝑂 = (od‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin) → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ (𝑂‘𝑋) = (♯‘𝐵)))) | ||
Theorem | cyggenod2 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) & ⊢ · = (.g‘𝐺) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} & ⊢ 𝑂 = (od‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸) → (𝑂‘𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0)) | ||
Theorem | iscyg3 18674* | Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑥))) | ||
Theorem | iscygd 18675* | Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋)) ⇒ ⊢ (𝜑 → 𝐺 ∈ CycGrp) | ||
Theorem | iscygodd 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) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑂‘𝑋) = (♯‘𝐵)) ⇒ ⊢ (𝜑 → 𝐺 ∈ CycGrp) | ||
Theorem | cyggrp 18677 | A cyclic group is a group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp) | ||
Theorem | cygabl 18678 | A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Abel) | ||
Theorem | cygctb 18679 | A cyclic group is countable. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ CycGrp → 𝐵 ≼ ω) | ||
Theorem | 0cyg 18680 | The trivial group is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) → 𝐺 ∈ CycGrp) | ||
Theorem | prmcyg 18681 | A group with prime order is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (♯‘𝐵) ∈ ℙ) → 𝐺 ∈ CycGrp) | ||
Theorem | lt6abl 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) | ||
Theorem | ghmcyg 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:𝐵–onto→𝐶) → (𝐺 ∈ CycGrp → 𝐻 ∈ CycGrp)) | ||
Theorem | cyggex2 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)) | ||
Theorem | cyggex 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)) | ||
Theorem | cyggexb 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))) | ||
Theorem | giccyg 18687 | Cyclicity is a group property, i.e. it is preserved under isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ (𝐺 ≃𝑔 𝐻 → (𝐺 ∈ CycGrp → 𝐻 ∈ CycGrp)) | ||
Theorem | cycsubgcyg 18688* | The cyclic subgroup generated by 𝐴 is a cyclic group. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝑆 = ran (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐺 ↾s 𝑆) ∈ CycGrp) | ||
Theorem | cycsubgcyg2 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 ∧ 𝐴 ∈ 𝐵) → (𝐺 ↾s (𝐾‘{𝐴})) ∈ CycGrp) | ||
Theorem | gsumval3a 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 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) & ⊢ (𝜑 → 𝑊 ∈ Fin) & ⊢ (𝜑 → 𝑊 ≠ ∅) & ⊢ 𝑊 = (𝐹 supp 0 ) & ⊢ (𝜑 → ¬ 𝐴 ∈ ran ...) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))) | ||
Theorem | gsumval3eu 18691* | The group sum as defined in gsumval3a 18690 is uniquely defined. (Contributed by Mario Carneiro, 8-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) & ⊢ (𝜑 → 𝑊 ∈ Fin) & ⊢ (𝜑 → 𝑊 ≠ ∅) & ⊢ (𝜑 → 𝑊 ⊆ 𝐴) ⇒ ⊢ (𝜑 → ∃!𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))) | ||
Theorem | gsumval3lem1 18692* | Lemma 1 for gsumval3 18694. (Contributed by AV, 31-May-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1→𝐴) & ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻) & ⊢ 𝑊 = ((𝐹 ∘ 𝐻) supp 0 ) ⇒ ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐻 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )) | ||
Theorem | gsumval3lem2 18693* | Lemma 2 for gsumval3 18694. (Contributed by AV, 31-May-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1→𝐴) & ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻) & ⊢ 𝑊 = ((𝐹 ∘ 𝐻) supp 0 ) ⇒ ⊢ (((𝜑 ∧ 𝑊 ≠ ∅) ∧ (¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(♯‘𝑊)), 𝑊))) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ (𝐻 ∘ 𝑓)))‘(♯‘𝑊))) | ||
Theorem | gsumval3 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 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐻:(1...𝑀)–1-1→𝐴) & ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ ran 𝐻) & ⊢ 𝑊 = ((𝐹 ∘ 𝐻) supp 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝐻))‘𝑀)) | ||
Theorem | gsumcllem 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.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) ⇒ ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑍)) | ||
Theorem | gsumzres 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 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) & ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑊)) = (𝐺 Σg 𝐹)) | ||
Theorem | gsumzcl2 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 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) & ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) | ||
Theorem | gsumzcl 18698 | Closure of a finite group sum. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 1-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) | ||
Theorem | gsumzf1o 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 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) & ⊢ (𝜑 → 𝐹 finSupp 0 ) & ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻))) | ||
Theorem | gsumres 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 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑊)) = (𝐺 Σg 𝐹)) |
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