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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cntzcmnss 19701 | 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 | cntrcmnd 19702 | The center of a monoid is a commutative submonoid. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
⊢ 𝑍 = (𝑀 ↾s (Cntr‘𝑀)) ⇒ ⊢ (𝑀 ∈ Mnd → 𝑍 ∈ CMnd) | ||
Theorem | cntrabl 19703 | The center of a group is an abelian group. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
⊢ 𝑍 = (𝑀 ↾s (Cntr‘𝑀)) ⇒ ⊢ (𝑀 ∈ Grp → 𝑍 ∈ Abel) | ||
Theorem | cntzspan 19704 | 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 19705 | 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 19706 | 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 19707 | Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺)) | ||
Theorem | odadd1 19708 | 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 19709 | 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 19710 | 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 19711 | 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 19712* | Lemma for gexex 19713. (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 19713* | 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 19714 | 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 19715* | 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 19716 | 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 19717 | 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 19718 | 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 19719 | 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 19720 | 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 19721 | 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 19722 | 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 19723 | 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 19724 | 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 19725 | 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 19726 | 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 19727 | Abelian groups (and therefore also groups and monoids) exist. (Contributed by AV, 29-Apr-2019.) |
⊢ Abel ≠ ∅ | ||
Theorem | cnaddablx 19728 | The complex numbers are an Abelian group under addition. This version of cnaddabl 19729 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 19729 instead. (New usage is discouraged.) (Contributed by NM, 18-Oct-2012.) |
⊢ 𝐺 = {〈1, ℂ〉, 〈2, + 〉} ⇒ ⊢ 𝐺 ∈ Abel | ||
Theorem | cnaddabl 19729 | The complex numbers are an Abelian group under addition. This version of cnaddablx 19728 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 20952. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.) |
⊢ 𝐺 = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ⇒ ⊢ 𝐺 ∈ Abel | ||
Theorem | cnaddid 19730 | The group identity element of complex number addition is zero. See also cnfld0 20954. (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 19731 | Value of the group inverse of complex number addition. See also cnfldneg 20956. (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 19732 | 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 20983 instead. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.) |
⊢ 𝐺 = {〈1, ℤ〉, 〈2, + 〉} ⇒ ⊢ 𝐺 ∈ Abel | ||
Theorem | frgpnabllem1 19733* | Lemma for frgpnabl 19735. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 25-Apr-2024.) |
⊢ 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 19734* | Lemma for frgpnabl 19735. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 25-Apr-2024.) |
⊢ 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 19735 | 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) | ||
Theorem | imasabl 19736* | The image structure of an abelian group is an abelian group (imasgrp 18935 analog). (Contributed by AV, 22-Feb-2025.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ (𝜑 → + = (+g‘𝑅)) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ (𝜑 → 𝑅 ∈ Abel) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘ 0 ) = (0g‘𝑈))) | ||
Syntax | ccyg 19737 | Cyclic group. |
class CycGrp | ||
Definition | df-cyg 19738* | 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 19739* | 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 19740* | 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 19741* | 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 19742* | 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 19743* | 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 19744* | 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 19745* | 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 19746* | 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 19747* | 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 19748 | 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 | cycsubmcmn 19749* | The set of nonnegative integer powers of an element 𝐴 of a monoid forms a commutative monoid. (Contributed by AV, 20-Jan-2024.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) & ⊢ 𝐶 = ran 𝐹 ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → (𝐺 ↾s 𝐶) ∈ CMnd) | ||
Theorem | cyggrp 19750 | A cyclic group is a group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp) | ||
Theorem | cygabl 19751 | A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 20-Jan-2024.) |
⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Abel) | ||
Theorem | cygctb 19752 | A cyclic group is countable. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ CycGrp → 𝐵 ≼ ω) | ||
Theorem | 0cyg 19753 | The trivial group is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) → 𝐺 ∈ CycGrp) | ||
Theorem | prmcyg 19754 | 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 19755 | 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 19756 | 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 19757 | 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 19758 | 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 19759 | 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 19760 | 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 19761* | 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 19762 | 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 19763* | 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 19764* | The group sum as defined in gsumval3a 19763 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 19765* | Lemma 1 for gsumval3 19767. (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 19766* | Lemma 2 for gsumval3 19767. (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 19767 | 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 19768* | Lemma for gsumcl 19775 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 19769 | 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 19770 | Closure of a finite group sum. This theorem has a weaker hypothesis than gsumzcl 19771, 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 19771 | 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 19772 | 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 19773 | 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)) | ||
Theorem | gsumcl2 19774 | Closure of a finite group sum. This theorem has a weaker hypothesis than gsumcl 19775, because it is not required that 𝐹 is a function (actually, the hypothesis always holds for any proper class 𝐹). (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 ) ∈ Fin) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) | ||
Theorem | gsumcl 19775 | Closure of a finite group sum. (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 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) | ||
Theorem | gsumf1o 19776 | Re-index a finite group sum using a bijection. (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 finSupp 0 ) & ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻))) | ||
Theorem | gsumreidx 19777 | Re-index a finite group sum using a bijection. Corresponds to the first equation in [Lang] p. 5 with 𝑀 = 1. (Contributed by AV, 26-Dec-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵) & ⊢ (𝜑 → 𝐻:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻))) | ||
Theorem | gsumzsubmcl 19778 | Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.) |
⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆) | ||
Theorem | gsumsubmcl 19779 | Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.) |
⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆) | ||
Theorem | gsumsubgcl 19780 | Closure of a group sum in a subgroup. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 3-Jun-2019.) |
⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝑆) | ||
Theorem | gsumzaddlem 19781* | The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 finSupp 0 ) & ⊢ (𝜑 → 𝐻 finSupp 0 ) & ⊢ 𝑊 = ((𝐹 ∪ 𝐻) supp 0 ) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) & ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) & ⊢ (𝜑 → ran 𝐻 ⊆ (𝑍‘ran 𝐻)) & ⊢ (𝜑 → ran (𝐹 ∘f + 𝐻) ⊆ (𝑍‘ran (𝐹 ∘f + 𝐻))) & ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑘 ∈ (𝐴 ∖ 𝑥))) → (𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))})) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) | ||
Theorem | gsumzadd 19782 | The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 finSupp 0 ) & ⊢ (𝜑 → 𝐻 finSupp 0 ) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) & ⊢ (𝜑 → 𝑆 ⊆ (𝑍‘𝑆)) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐻:𝐴⟶𝑆) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) | ||
Theorem | gsumadd 19783 | The sum of two group sums. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (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 | gsummptfsadd 19784* | The sum of two group sums expressed as mappings. (Contributed by AV, 4-Apr-2019.) (Revised by AV, 9-Jul-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (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 | gsummptfidmadd 19785* | The sum of two group sums expressed as mappings with finite domain. (Contributed by AV, 23-Jul-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (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 | gsummptfidmadd2 19786* | The sum of two group sums expressed as mappings with finite domain, using a function operation. (Contributed by AV, 23-Jul-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) | ||
Theorem | gsumzsplit 19787 | Split a group sum into two parts. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-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 → 𝐹 finSupp 0 ) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷)))) | ||
Theorem | gsumsplit 19788 | Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 5-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐹 finSupp 0 ) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷)))) | ||
Theorem | gsumsplit2 19789* | Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 5-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋)))) | ||
Theorem | gsummptfidmsplit 19790* | Split a group sum expressed as mapping with a finite domain into two parts. (Contributed by AV, 23-Jul-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑌)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑌)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑌)))) | ||
Theorem | gsummptfidmsplitres 19791* | Split a group sum expressed as mapping with a finite domain into two parts using restrictions. (Contributed by AV, 23-Jul-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) & ⊢ 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑌) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷)))) | ||
Theorem | gsummptfzsplit 19792* | Split a group sum expressed as mapping with a finite set of sequential integers as domain into two parts, extracting a singleton from the right. (Contributed by AV, 25-Oct-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ 𝑌)) = ((𝐺 Σg (𝑘 ∈ (0...𝑁) ↦ 𝑌)) + (𝐺 Σg (𝑘 ∈ {(𝑁 + 1)} ↦ 𝑌)))) | ||
Theorem | gsummptfzsplitl 19793* | Split a group sum expressed as mapping with a finite set of sequential integers as domain into two parts, , extracting a singleton from the left. (Contributed by AV, 7-Nov-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (0...𝑁) ↦ 𝑌)) = ((𝐺 Σg (𝑘 ∈ (1...𝑁) ↦ 𝑌)) + (𝐺 Σg (𝑘 ∈ {0} ↦ 𝑌)))) | ||
Theorem | gsumconst 19794* | Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋)) | ||
Theorem | gsumconstf 19795* | Sum of a constant series. (Contributed by Thierry Arnoux, 5-Jul-2017.) |
⊢ Ⅎ𝑘𝑋 & ⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋)) | ||
Theorem | gsummptshft 19796* | Index shift of a finite group sum over a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ 𝐵) & ⊢ (𝑗 = (𝑘 − 𝐾) → 𝐴 = 𝐶) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ (𝑀...𝑁) ↦ 𝐴)) = (𝐺 Σg (𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ 𝐶))) | ||
Theorem | gsumzmhm 19797 | Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 6-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐻 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻)) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹))) | ||
Theorem | gsummhm 19798 | Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 6-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐻 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻)) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹))) | ||
Theorem | gsummhm2 19799* | Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.) (Revised by AV, 6-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐻 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ (𝐺 MndHom 𝐻)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) & ⊢ (𝑥 = 𝑋 → 𝐶 = 𝐷) & ⊢ (𝑥 = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) → 𝐶 = 𝐸) ⇒ ⊢ (𝜑 → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ 𝐷)) = 𝐸) | ||
Theorem | gsummptmhm 19800* | Apply a group homomorphism to a group sum expressed with a mapping. (Contributed by Thierry Arnoux, 7-Sep-2018.) (Revised by AV, 8-Sep-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐻 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐻 Σg (𝑥 ∈ 𝐴 ↦ (𝐾‘𝐶))) = (𝐾‘(𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)))) |
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