| Metamath
Proof Explorer Theorem List (p. 242 of 494) | < Previous Next > | |
| Bad symbols? Try the
GIF version. |
||
|
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
||
| Color key: | (1-30937) |
(30938-32460) |
(32461-49324) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | tgpmulg 24101* | In a topological group, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽)) | ||
| Theorem | tgpmulg2 24102 | In a topological monoid, the group multiple function is jointly continuous (although this is not saying much as one of the factors is discrete). Use zdis 24838 to write the left topology as a subset of the complex numbers. (Contributed by Mario Carneiro, 19-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝐺 ∈ TopGrp → · ∈ ((𝒫 ℤ ×t 𝐽) Cn 𝐽)) | ||
| Theorem | tmdgsum 24103* | In a topological monoid, the group sum operation is a continuous function from the function space to the base topology. This theorem is not true when 𝐴 is infinite, because in this case for any basic open set of the domain one of the factors will be the whole space, so by varying the value of the functions to sum at this index, one can achieve any desired sum. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.) |
| ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((𝐽 ↑ko 𝒫 𝐴) Cn 𝐽)) | ||
| Theorem | tmdgsum2 24104* | For any neighborhood 𝑈 of 𝑛𝑋, there is a neighborhood 𝑢 of 𝑋 such that any sum of 𝑛 elements in 𝑢 sums to an element of 𝑈. (Contributed by Mario Carneiro, 19-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑈 ∈ 𝐽) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → ((♯‘𝐴) · 𝑋) ∈ 𝑈) ⇒ ⊢ (𝜑 → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈)) | ||
| Theorem | oppgtmd 24105 | The opposite of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.) |
| ⊢ 𝑂 = (oppg‘𝐺) ⇒ ⊢ (𝐺 ∈ TopMnd → 𝑂 ∈ TopMnd) | ||
| Theorem | oppgtgp 24106 | The opposite of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.) |
| ⊢ 𝑂 = (oppg‘𝐺) ⇒ ⊢ (𝐺 ∈ TopGrp → 𝑂 ∈ TopGrp) | ||
| Theorem | distgp 24107 | Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp) | ||
| Theorem | indistgp 24108 | Any group equipped with the indiscrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopGrp) | ||
| Theorem | efmndtmd 24109 | The monoid of endofunctions on a set 𝐴 is a topological monoid. Formerly part of proof for symgtgp 24114. (Contributed by AV, 23-Feb-2024.) |
| ⊢ 𝑀 = (EndoFMnd‘𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ TopMnd) | ||
| Theorem | tmdlactcn 24110* | The left group action of element 𝐴 in a topological monoid 𝐺 is a continuous function. (Contributed by FL, 18-Mar-2008.) (Revised by Mario Carneiro, 14-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) ⇒ ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽)) | ||
| Theorem | tgplacthmeo 24111* | The left group action of element 𝐴 in a topological group 𝐺 is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) ⇒ ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽Homeo𝐽)) | ||
| Theorem | submtmd 24112 | A submonoid of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 6-Oct-2015.) |
| ⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ ((𝐺 ∈ TopMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ TopMnd) | ||
| Theorem | subgtgp 24113 | A subgroup of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.) |
| ⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopGrp) | ||
| Theorem | symgtgp 24114 | The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.) (Proof shortened by AV, 30-Mar-2024.) |
| ⊢ 𝐺 = (SymGrp‘𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ TopGrp) | ||
| Theorem | subgntr 24115 | A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 24117, the interior of a subgroup is either a subgroup, or empty. (Contributed by Mario Carneiro, 19-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝐺) ⇒ ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ∈ 𝐽) | ||
| Theorem | opnsubg 24116 | An open subgroup of a topological group is also closed. (Contributed by Mario Carneiro, 17-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝐺) ⇒ ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ (Clsd‘𝐽)) | ||
| Theorem | clssubg 24117 | The closure of a subgroup in a topological group is a subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝐺) ⇒ ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺)) | ||
| Theorem | clsnsg 24118 | The closure of a normal subgroup is a normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝐺) ⇒ ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺)) | ||
| Theorem | cldsubg 24119 | A subgroup of finite index is closed iff it is open. (Contributed by Mario Carneiro, 20-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝑅 = (𝐺 ~QG 𝑆) & ⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ (Clsd‘𝐽) ↔ 𝑆 ∈ 𝐽)) | ||
| Theorem | tgpconncompeqg 24120* | The connected component containing 𝐴 is the left coset of the identity component containing 𝐴. (Contributed by Mario Carneiro, 17-Sep-2015.) |
| ⊢ 𝑋 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} & ⊢ ∼ = (𝐺 ~QG 𝑆) ⇒ ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}) | ||
| Theorem | tgpconncomp 24121* | The identity component, the connected component containing the identity element, is a closed (conncompcld 23442) normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.) |
| ⊢ 𝑋 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ⇒ ⊢ (𝐺 ∈ TopGrp → 𝑆 ∈ (NrmSGrp‘𝐺)) | ||
| Theorem | tgpconncompss 24122* | The identity component is a subset of any open subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.) |
| ⊢ 𝑋 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ⇒ ⊢ ((𝐺 ∈ TopGrp ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ 𝐽) → 𝑆 ⊆ 𝑇) | ||
| Theorem | ghmcnp 24123 | A group homomorphism on topological groups is continuous everywhere if it is continuous at any point. (Contributed by Mario Carneiro, 21-Oct-2015.) |
| ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝐾 = (TopOpen‘𝐻) ⇒ ⊢ ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐴 ∈ 𝑋 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)))) | ||
| Theorem | snclseqg 24124 | The coset of the closure of the identity is the closure of a point. (Contributed by Mario Carneiro, 22-Sep-2015.) |
| ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ ∼ = (𝐺 ~QG 𝑆) & ⊢ 𝑆 = ((cls‘𝐽)‘{ 0 }) ⇒ ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((cls‘𝐽)‘{𝐴})) | ||
| Theorem | tgphaus 24125 | A topological group is Hausdorff iff the identity subgroup is closed. (Contributed by Mario Carneiro, 18-Sep-2015.) |
| ⊢ 0 = (0g‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) ⇒ ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ { 0 } ∈ (Clsd‘𝐽))) | ||
| Theorem | tgpt1 24126 | Hausdorff and T1 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝐺) ⇒ ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre)) | ||
| Theorem | tgpt0 24127 | Hausdorff and T0 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝐺) ⇒ ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2)) | ||
| Theorem | qustgpopn 24128* | A quotient map in a topological group is an open map. (Contributed by Mario Carneiro, 18-Sep-2015.) |
| ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌)) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝐾 = (TopOpen‘𝐻) & ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) ⇒ ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ∈ 𝐾) | ||
| Theorem | qustgplem 24129* | Lemma for qustgp 24130. (Contributed by Mario Carneiro, 18-Sep-2015.) |
| ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌)) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝐾 = (TopOpen‘𝐻) & ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) & ⊢ − = (𝑧 ∈ 𝑋, 𝑤 ∈ 𝑋 ↦ [(𝑧(-g‘𝐺)𝑤)](𝐺 ~QG 𝑌)) ⇒ ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → 𝐻 ∈ TopGrp) | ||
| Theorem | qustgp 24130 | The quotient of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.) |
| ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌)) ⇒ ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → 𝐻 ∈ TopGrp) | ||
| Theorem | qustgphaus 24131 | The quotient of a topological group by a closed normal subgroup is a Hausdorff topological group. In particular, the quotient by the closure of the identity is a Hausdorff topological group, isomorphic to both the Kolmogorov quotient and the Hausdorff quotient operations on topological spaces (because T0 and Hausdorff coincide for topological groups). (Contributed by Mario Carneiro, 22-Sep-2015.) |
| ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌)) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝐾 = (TopOpen‘𝐻) ⇒ ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 ∈ Haus) | ||
| Theorem | prdstmdd 24132 | The product of a family of topological monoids is a topological monoid. (Contributed by Mario Carneiro, 22-Sep-2015.) |
| ⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶TopMnd) ⇒ ⊢ (𝜑 → 𝑌 ∈ TopMnd) | ||
| Theorem | prdstgpd 24133 | The product of a family of topological groups is a topological group. (Contributed by Mario Carneiro, 22-Sep-2015.) |
| ⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶TopGrp) ⇒ ⊢ (𝜑 → 𝑌 ∈ TopGrp) | ||
| Syntax | ctsu 24134 | Extend class notation to include infinite group sums in a topological group. |
| class tsums | ||
| Definition | df-tsms 24135* | Define the set of limit points of an infinite group sum for the topological group 𝐺. If 𝐺 is Hausdorff, then there will be at most one element in this set and ∪ (𝑊 tsums 𝐹) selects this unique element if it exists. (𝑊 tsums 𝐹) ≈ 1o is a way to say that the sum exists and is unique. Note that unlike Σ (df-sum 15723) and Σg (df-gsum 17487), this does not return the sum itself, but rather the set of all such sums, which is usually either empty or a singleton. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ tsums = (𝑤 ∈ V, 𝑓 ∈ V ↦ ⦋(𝒫 dom 𝑓 ∩ Fin) / 𝑠⦌(((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦})))‘(𝑦 ∈ 𝑠 ↦ (𝑤 Σg (𝑓 ↾ 𝑦))))) | ||
| Theorem | tsmsfbas 24136* | The collection of all sets of the form 𝐹(𝑧) = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}, which can be read as the set of all finite subsets of 𝐴 which contain 𝑧 as a subset, for each finite subset 𝑧 of 𝐴, form a filter base. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin) & ⊢ 𝐹 = (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) & ⊢ 𝐿 = ran 𝐹 & ⊢ (𝜑 → 𝐴 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐿 ∈ (fBas‘𝑆)) | ||
| Theorem | tsmslem1 24137 | The finite partial sums of a function 𝐹 are defined in a commutative monoid. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑊) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐺 Σg (𝐹 ↾ 𝑋)) ∈ 𝐵) | ||
| Theorem | tsmsval2 24138* | Definition of the topological group sum(s) of a collection 𝐹(𝑥) of values in the group with index set 𝐴. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin) & ⊢ 𝐿 = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝑊) & ⊢ (𝜑 → dom 𝐹 = 𝐴) ⇒ ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦))))) | ||
| Theorem | tsmsval 24139* | Definition of the topological group sum(s) of a collection 𝐹(𝑥) of values in the group with index set 𝐴. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin) & ⊢ 𝐿 = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑊) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦))))) | ||
| Theorem | tsmspropd 24140 | The group sum depends only on the base set, additive operation, and topology components. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 18772 etc. (Contributed by Mario Carneiro, 18-Sep-2015.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) & ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻)) & ⊢ (𝜑 → (TopOpen‘𝐺) = (TopOpen‘𝐻)) ⇒ ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹)) | ||
| Theorem | eltsms 24141* | The property of being a sum of the sequence 𝐹 in the topological commutative monoid 𝐺. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐺 tsums 𝐹) ↔ (𝐶 ∈ 𝐵 ∧ ∀𝑢 ∈ 𝐽 (𝐶 ∈ 𝑢 → ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑢))))) | ||
| Theorem | tsmsi 24142* | The property of being a sum of the sequence 𝐹 in the topological commutative monoid 𝐺. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐶 ∈ (𝐺 tsums 𝐹)) & ⊢ (𝜑 → 𝑈 ∈ 𝐽) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑈)) | ||
| Theorem | tsmscl 24143 | A sum in a topological group is an element of the group. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵) | ||
| Theorem | haustsms 24144* | In a Hausdorff topological group, a sum has at most one limit point. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ (𝜑 → 𝐽 ∈ Haus) ⇒ ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ (𝐺 tsums 𝐹)) | ||
| Theorem | haustsms2 24145 | In a Hausdorff topological group, a sum has at most one limit point. (Contributed by Mario Carneiro, 13-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ (𝜑 → 𝐽 ∈ Haus) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐺 tsums 𝐹) → (𝐺 tsums 𝐹) = {𝑋})) | ||
| Theorem | tsmscls 24146 | One half of tgptsmscls 24158, true in any commutative monoid topological space. (Contributed by Mario Carneiro, 21-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹)) ⇒ ⊢ (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹)) | ||
| Theorem | tsmsgsum 24147 | The convergent points of a finite topological group sum are the closure of the finite group sum operation. (Contributed by Mario Carneiro, 19-Sep-2015.) (Revised by AV, 24-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐹 finSupp 0 ) & ⊢ 𝐽 = (TopOpen‘𝐺) ⇒ ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{(𝐺 Σg 𝐹)})) | ||
| Theorem | tsmsid 24148 | If a sum is finite, the usual sum is always a limit point of the topological sum (although it may not be the only limit point). (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by AV, 24-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 tsums 𝐹)) | ||
| Theorem | haustsmsid 24149 | In a Hausdorff topological group, a finite sum sums to exactly the usual number with no extraneous limit points. By setting the topology to the discrete topology (which is Hausdorff), this theorem can be used to turn any tsums theorem into a Σg theorem, so that the infinite group sum operation can be viewed as a generalization of the finite group sum. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by AV, 24-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐹 finSupp 0 ) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ (𝜑 → 𝐽 ∈ Haus) ⇒ ⊢ (𝜑 → (𝐺 tsums 𝐹) = {(𝐺 Σg 𝐹)}) | ||
| Theorem | tsms0 24150* | The sum of zero is zero. (Contributed by Mario Carneiro, 18-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.) |
| ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 0 ∈ (𝐺 tsums (𝑥 ∈ 𝐴 ↦ 0 ))) | ||
| Theorem | tsmssubm 24151 | Evaluate an infinite group sum in a submonoid. (Contributed by Mario Carneiro, 18-Sep-2015.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopSp) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ (𝜑 → (𝐻 tsums 𝐹) = ((𝐺 tsums 𝐹) ∩ 𝑆)) | ||
| Theorem | tsmsres 24152 | Extend an infinite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 18-Sep-2015.) (Revised by AV, 25-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊) ⇒ ⊢ (𝜑 → (𝐺 tsums (𝐹 ↾ 𝑊)) = (𝐺 tsums 𝐹)) | ||
| Theorem | tsmsf1o 24153 | Re-index an infinite group sum using a bijection. (Contributed by Mario Carneiro, 18-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopSp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴) ⇒ ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐺 tsums (𝐹 ∘ 𝐻))) | ||
| Theorem | tsmsmhm 24154 | Apply a continuous group homomorphism to an infinite group sum. (Contributed by Mario Carneiro, 18-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝐾 = (TopOpen‘𝐻) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopSp) & ⊢ (𝜑 → 𝐻 ∈ CMnd) & ⊢ (𝜑 → 𝐻 ∈ TopSp) & ⊢ (𝜑 → 𝐶 ∈ (𝐺 MndHom 𝐻)) & ⊢ (𝜑 → 𝐶 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹)) ⇒ ⊢ (𝜑 → (𝐶‘𝑋) ∈ (𝐻 tsums (𝐶 ∘ 𝐹))) | ||
| Theorem | tsmsadd 24155 | The sum of two infinite group sums. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) & ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹)) & ⊢ (𝜑 → 𝑌 ∈ (𝐺 tsums 𝐻)) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝐺 tsums (𝐹 ∘f + 𝐻))) | ||
| Theorem | tsmsinv 24156 | Inverse of an infinite group sum. (Contributed by Mario Carneiro, 20-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopGrp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹)) ⇒ ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝐺 tsums (𝐼 ∘ 𝐹))) | ||
| Theorem | tsmssub 24157 | The difference of two infinite group sums. (Contributed by Mario Carneiro, 20-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopGrp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) & ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹)) & ⊢ (𝜑 → 𝑌 ∈ (𝐺 tsums 𝐻)) ⇒ ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝐺 tsums (𝐹 ∘f − 𝐻))) | ||
| Theorem | tgptsmscls 24158 | A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 24118, 0nsg 19187. (Contributed by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopGrp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹)) ⇒ ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋})) | ||
| Theorem | tgptsmscld 24159 | The set of limit points to an infinite sum in a topological group is closed. (Contributed by Mario Carneiro, 22-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopGrp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽)) | ||
| Theorem | tsmssplit 24160 | Split a topological group sum into two parts. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums (𝐹 ↾ 𝐶))) & ⊢ (𝜑 → 𝑌 ∈ (𝐺 tsums (𝐹 ↾ 𝐷))) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝐺 tsums 𝐹)) | ||
| Theorem | tsmsxplem1 24161* | Lemma for tsmsxp 24163. (Contributed by Mario Carneiro, 21-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopGrp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵) & ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)))) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐿 ∈ 𝐽) & ⊢ (𝜑 → 0 ∈ 𝐿) & ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐴 ∩ Fin)) & ⊢ (𝜑 → dom 𝐷 ⊆ 𝐾) & ⊢ (𝜑 → 𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin)) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ (𝒫 𝐶 ∩ Fin)(ran 𝐷 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿)) | ||
| Theorem | tsmsxplem2 24162* | Lemma for tsmsxp 24163. (Contributed by Mario Carneiro, 21-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopGrp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵) & ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)))) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐿 ∈ 𝐽) & ⊢ (𝜑 → 0 ∈ 𝐿) & ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐴 ∩ Fin)) & ⊢ (𝜑 → ∀𝑐 ∈ 𝑆 ∀𝑑 ∈ 𝑇 (𝑐 + 𝑑) ∈ 𝑈) & ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝐶 ∩ Fin)) & ⊢ (𝜑 → 𝐷 ⊆ (𝐾 × 𝑁)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑁)))) ∈ 𝐿) & ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐾 × 𝑁))) ∈ 𝑆) & ⊢ (𝜑 → ∀𝑔 ∈ (𝐿 ↑m 𝐾)(𝐺 Σg 𝑔) ∈ 𝑇) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝐻 ↾ 𝐾)) ∈ 𝑈) | ||
| Theorem | tsmsxp 24163* | Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 19994 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 24161 for the main proof; this part mostly sets up the local assumptions. (Contributed by Mario Carneiro, 21-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐺 ∈ TopGrp) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵) & ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)))) ⇒ ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ (𝐺 tsums 𝐻)) | ||
| Syntax | ctrg 24164 | The class of all topological division rings. |
| class TopRing | ||
| Syntax | ctdrg 24165 | The class of all topological division rings. |
| class TopDRing | ||
| Syntax | ctlm 24166 | The class of all topological modules. |
| class TopMod | ||
| Syntax | ctvc 24167 | The class of all topological vector spaces. |
| class TopVec | ||
| Definition | df-trg 24168 | Define a topological ring, which is a ring such that the addition is a topological group operation and the multiplication is continuous. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd} | ||
| Definition | df-tdrg 24169 | Define a topological division ring (which differs from a topological field only in being potentially noncommutative), which is a division ring and topological ring such that the unit group of the division ring (which is the set of nonzero elements) is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp} | ||
| Definition | df-tlm 24170 | Define a topological left module, which is just what its name suggests: instead of a group over a ring with a scalar product connecting them, it is a topological group over a topological ring with a continuous scalar product. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))} | ||
| Definition | df-tvc 24171 | Define a topological left vector space, which is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ TopVec = {𝑤 ∈ TopMod ∣ (Scalar‘𝑤) ∈ TopDRing} | ||
| Theorem | istrg 24172 | Express the predicate "𝑅 is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd)) | ||
| Theorem | trgtmd 24173 | The multiplicative monoid of a topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ (𝑅 ∈ TopRing → 𝑀 ∈ TopMnd) | ||
| Theorem | istdrg 24174 | Express the predicate "𝑅 is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp)) | ||
| Theorem | tdrgunit 24175 | The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝑅 ∈ TopDRing → (𝑀 ↾s 𝑈) ∈ TopGrp) | ||
| Theorem | trgtgp 24176 | A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp) | ||
| Theorem | trgtmd2 24177 | A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd) | ||
| Theorem | trgtps 24178 | A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopSp) | ||
| Theorem | trgring 24179 | A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Ring) | ||
| Theorem | trggrp 24180 | A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Grp) | ||
| Theorem | tdrgtrg 24181 | A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing) | ||
| Theorem | tdrgdrng 24182 | A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ DivRing) | ||
| Theorem | tdrgring 24183 | A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ Ring) | ||
| Theorem | tdrgtmd 24184 | A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopMnd) | ||
| Theorem | tdrgtps 24185 | A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopSp) | ||
| Theorem | istdrg2 24186 | A topological-ring division ring is a topological division ring iff the group of nonzero elements is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp)) | ||
| Theorem | mulrcn 24187 | The functionalization of the ring multiplication operation is a continuous function in a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝑅) & ⊢ 𝑇 = (+𝑓‘(mulGrp‘𝑅)) ⇒ ⊢ (𝑅 ∈ TopRing → 𝑇 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) | ||
| Theorem | invrcn2 24188 | The multiplicative inverse function is a continuous function from the unit group (that is, the nonzero numbers) to itself. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝑅 ∈ TopDRing → 𝐼 ∈ ((𝐽 ↾t 𝑈) Cn (𝐽 ↾t 𝑈))) | ||
| Theorem | invrcn 24189 | The multiplicative inverse function is a continuous function from the unit group (that is, the nonzero numbers) to the field. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝑅 ∈ TopDRing → 𝐼 ∈ ((𝐽 ↾t 𝑈) Cn 𝐽)) | ||
| Theorem | cnmpt1mulr 24190* | Continuity of ring multiplication; analogue of cnmpt12f 23674 which cannot be used directly because .r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ TopRing) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐾 Cn 𝐽)) | ||
| Theorem | cnmpt2mulr 24191* | Continuity of ring multiplication; analogue of cnmpt22f 23683 which cannot be used directly because .r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ TopRing) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) | ||
| Theorem | dvrcn 24192 | The division function is continuous in a topological field. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ 𝐽 = (TopOpen‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝑅 ∈ TopDRing → / ∈ ((𝐽 ×t (𝐽 ↾t 𝑈)) Cn 𝐽)) | ||
| Theorem | istlm 24193 | The predicate "𝑊 is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ · = ( ·sf ‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (TopOpen‘𝐹) ⇒ ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))) | ||
| Theorem | vscacn 24194 | The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ · = ( ·sf ‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (TopOpen‘𝐹) ⇒ ⊢ (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) | ||
| Theorem | tlmtmd 24195 | A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd) | ||
| Theorem | tlmtps 24196 | A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp) | ||
| Theorem | tlmlmod 24197 | A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod) | ||
| Theorem | tlmtrg 24198 | The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing) | ||
| Theorem | tlmscatps 24199 | The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp) | ||
| Theorem | istvc 24200 | A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing)) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |