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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | odadd 18901 | 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 18902 | 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 18903* | Lemma for gexex 18904. (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 18904* | 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 18905 | 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 18906* | 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 18907 | 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 18908 | 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 18909 | 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 18910 | 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 18911 | 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 18912 | 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 18913 | 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 18914 | 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 18915 | 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 18916 | 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 18917 | 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 18918 | Abelian groups (and therefore also groups and monoids) exist. (Contributed by AV, 29-Apr-2019.) |
⊢ Abel ≠ ∅ | ||
Theorem | cnaddablx 18919 | The complex numbers are an Abelian group under addition. This version of cnaddabl 18920 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 18920 instead. (New usage is discouraged.) (Contributed by NM, 18-Oct-2012.) |
⊢ 𝐺 = {〈1, ℂ〉, 〈2, + 〉} ⇒ ⊢ 𝐺 ∈ Abel | ||
Theorem | cnaddabl 18920 | The complex numbers are an Abelian group under addition. This version of cnaddablx 18919 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 20497. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.) |
⊢ 𝐺 = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ⇒ ⊢ 𝐺 ∈ Abel | ||
Theorem | cnaddid 18921 | The group identity element of complex number addition is zero. See also cnfld0 20499. (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 18922 | Value of the group inverse of complex number addition. See also cnfldneg 20501. (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 18923 | 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 20528 instead. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.) |
⊢ 𝐺 = {〈1, ℤ〉, 〈2, + 〉} ⇒ ⊢ 𝐺 ∈ Abel | ||
Theorem | frgpnabllem1 18924* | Lemma for frgpnabl 18926. (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 18925* | Lemma for frgpnabl 18926. (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 18926 | 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 18927 | Cyclic group. |
class CycGrp | ||
Definition | df-cyg 18928* | 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 18929* | 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 18930* | 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 18931* | 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 18932* | 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 18933* | 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 18934* | 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 18935* | 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 18936* | 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 18937* | 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 18938 | 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 18939* | 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 18940 | A cyclic group is a group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp) | ||
Theorem | cygabl 18941 | A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 20-Jan-2024.) |
⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Abel) | ||
Theorem | cygablOLD 18942 | Obsolete proof of cygabl 18941 as of 20-Jan-2024. A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Abel) | ||
Theorem | cygctb 18943 | A cyclic group is countable. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ CycGrp → 𝐵 ≼ ω) | ||
Theorem | 0cyg 18944 | The trivial group is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) → 𝐺 ∈ CycGrp) | ||
Theorem | prmcyg 18945 | 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 18946 | 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 18947 | 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 18948 | 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 18949 | 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 18950 | 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 18951 | 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 18952* | 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 18953 | 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 18954* | 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 18955* | The group sum as defined in gsumval3a 18954 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 18956* | Lemma 1 for gsumval3 18958. (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 18957* | Lemma 2 for gsumval3 18958. (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 18958 | 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 18959* | Lemma for gsumcl 18966 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 18960 | 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 18961 | Closure of a finite group sum. This theorem has a weaker hypothesis than gsumzcl 18962, 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 18962 | 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 18963 | 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 18964 | 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 18965 | Closure of a finite group sum. This theorem has a weaker hypothesis than gsumcl 18966, 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 18966 | 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 18967 | 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 18968 | 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 18969 | 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 18970 | 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 18971 | 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 18972* | 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 18973 | 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 18974 | 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 18975* | 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 18976* | 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 18977* | 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 18978 | 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 18979 | 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 18980* | 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 18981* | 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 18982* | 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 18983* | 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 18984* | 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 18985* | 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 18986* | 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 18987* | 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 18988 | 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 18989 | 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 18990* | 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 18991* | 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)))) | ||
Theorem | gsummulglem 18992* | Lemma for gsummulg 18993 and gsummulgz 18994. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 6-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → (𝐺 ∈ Abel ∨ 𝑁 ∈ ℕ0)) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ (𝑁 · 𝑋))) = (𝑁 · (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)))) | ||
Theorem | gsummulg 18993* | Nonnegative multiple of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 6-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ (𝑁 · 𝑋))) = (𝑁 · (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)))) | ||
Theorem | gsummulgz 18994* | Integer multiple of a group sum. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 6-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ (𝑁 · 𝑋))) = (𝑁 · (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)))) | ||
Theorem | gsumzoppg 18995 | The opposite of a group sum is the same as the original. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ 𝑂 = (oppg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)) | ||
Theorem | gsumzinv 18996 | Inverse of a group sum. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝐼 ∘ 𝐹)) = (𝐼‘(𝐺 Σg 𝐹))) | ||
Theorem | gsuminv 18997 | Inverse of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 4-May-2015.) (Revised by AV, 6-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝐼 ∘ 𝐹)) = (𝐼‘(𝐺 Σg 𝐹))) | ||
Theorem | gsummptfidminv 18998* | Inverse of a group sum expressed as mapping with a finite domain. (Contributed by AV, 23-Jul-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝐼 ∘ 𝐹)) = (𝐼‘(𝐺 Σg 𝐹))) | ||
Theorem | gsumsub 18999 | The difference of two group sums. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 6-Jun-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐹 finSupp 0 ) & ⊢ (𝜑 → 𝐻 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f − 𝐻)) = ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻))) | ||
Theorem | gsummptfssub 19000* | The difference of two group sums expressed as mappings. (Contributed by AV, 7-Nov-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)) & ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷)) & ⊢ (𝜑 → 𝐹 finSupp 0 ) & ⊢ (𝜑 → 𝐻 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 − 𝐷))) = ((𝐺 Σg 𝐹) − (𝐺 Σg 𝐻))) |
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