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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | oppgtmd 21901 | 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 21902 | 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 21903 | 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 21904 | 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 | symgtgp 21905 | The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.) |
⊢ 𝐺 = (SymGrp‘𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ TopGrp) | ||
Theorem | tmdlactcn 21906* | 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 21907* | 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 21908 | 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 21909 | 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 | subgntr 21910 | A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 21912, 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 21911 | 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 21912 | 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 21913 | 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 21914 | 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 21915* | 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 21916* | The identity component, the connected component containing the identity element, is a closed (conncompcld 21237) 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 21917* | 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 21918 | 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 21919 | 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 21920 | 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 21921 | 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 21922 | 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 21923* | 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 21924* | Lemma for qustgp 21925. (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 21925 | 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 21926 | 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 21927 | 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 21928 | 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 21929 | Extend class notation to include infinite group sums in a topological group. |
class tsums | ||
Definition | df-tsms 21930* | 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 14417) and Σg (df-gsum 16103), 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 21931* | 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 21932 | 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 21933* | 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 21934* | 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 21935 | 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 17316 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 21936* | 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 21937* | 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 21938 | 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 21939* | 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 21940 | 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 21941 | One half of tgptsmscls 21953, 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 21942 | 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 21943 | 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 21944 | 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 21945* | 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 21946 | 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 21947 | 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 21948 | 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 21949 | 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 21950 | 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 21951 | 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 21952 | 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 21953 | A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 21913, 0nsg 17639. (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 21954 | 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 21955 | 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 21956* | Lemma for tsmsxp 21958. (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 21957* | Lemma for tsmsxp 21958. (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 21958* | Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 18375 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 21956 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 21959 | The class of all topological division rings. |
class TopRing | ||
Syntax | ctdrg 21960 | The class of all topological division rings. |
class TopDRing | ||
Syntax | ctlm 21961 | The class of all topological modules. |
class TopMod | ||
Syntax | ctvc 21962 | The class of all topological vector spaces. |
class TopVec | ||
Definition | df-trg 21963 | 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 21964 | 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 21965 | 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 21966 | 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 21967 | 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 21968 | 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 21969 | 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 21970 | 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 21971 | A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp) | ||
Theorem | trgtmd2 21972 | A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd) | ||
Theorem | trgtps 21973 | A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopSp) | ||
Theorem | trgring 21974 | A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Ring) | ||
Theorem | trggrp 21975 | A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Grp) | ||
Theorem | tdrgtrg 21976 | A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing) | ||
Theorem | tdrgdrng 21977 | A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ DivRing) | ||
Theorem | tdrgring 21978 | A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ Ring) | ||
Theorem | tdrgtmd 21979 | A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopMnd) | ||
Theorem | tdrgtps 21980 | A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopSp) | ||
Theorem | istdrg2 21981 | 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 21982 | 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 21983 | 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 21984 | 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 21985* | Continuity of ring multiplication; analogue of cnmpt12f 21469 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 21986* | Continuity of ring multiplication; analogue of cnmpt22f 21478 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 21987 | 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 21988 | 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 21989 | 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 21990 | A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd) | ||
Theorem | tlmtps 21991 | A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp) | ||
Theorem | tlmlmod 21992 | A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod) | ||
Theorem | tlmtrg 21993 | 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 21994 | 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 21995 | 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)) | ||
Theorem | tvctdrg 21996 | The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ TopVec → 𝐹 ∈ TopDRing) | ||
Theorem | cnmpt1vsca 21997* | Continuity of scalar multiplication; analogue of cnmpt12f 21469 which cannot be used directly because ·𝑠 is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝐾 = (TopOpen‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ TopMod) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐿 Cn 𝐽)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽)) | ||
Theorem | cnmpt2vsca 21998* | Continuity of scalar multiplication; analogue of cnmpt22f 21478 which cannot be used directly because ·𝑠 is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝐾 = (TopOpen‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ TopMod) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐿 ×t 𝑀) Cn 𝐾)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐿 ×t 𝑀) Cn 𝐽)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽)) | ||
Theorem | tlmtgp 21999 | A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopGrp) | ||
Theorem | tvctlm 22000 | A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑊 ∈ TopVec → 𝑊 ∈ TopMod) |
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