Home | Metamath
Proof Explorer Theorem List (p. 228 of 450) | < 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: | Metamath Proof Explorer
(1-28695) |
Hilbert Space Explorer
(28696-30218) |
Users' Mathboxes
(30219-44926) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | tgpmulg 22701* | 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 22702 | 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 23424 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 22703* | 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 22704* | 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 22705 | 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 22706 | 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 22707 | 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 22708 | 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 22709 | The monoid of endofunctions on a set 𝐴 is a topological monoid. Formerly part of proof for symgtgp 22714. (Contributed by AV, 23-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ TopMnd) | ||
Theorem | tmdlactcn 22710* | 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 22711* | 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 22712 | 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 22713 | 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 22714 | 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 22715 | A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 22717, 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 22716 | 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 22717 | 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 22718 | 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 22719 | 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 22720* | 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 22721* | The identity component, the connected component containing the identity element, is a closed (conncompcld 22042) 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 22722* | 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 22723 | 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 22724 | 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 22725 | 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 22726 | 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 22727 | 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 22728* | 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 22729* | Lemma for qustgp 22730. (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 22730 | 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 22731 | 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 22732 | 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 22733 | 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 22734 | Extend class notation to include infinite group sums in a topological group. |
class tsums | ||
Definition | df-tsms 22735* | 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 15043) and Σg (df-gsum 16716), 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 22736* | 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 22737 | 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 22738* | 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 22739* | 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 22740 | 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 17936 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 22741* | 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 22742* | 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 22743 | 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 22744* | 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 22745 | 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 22746 | One half of tgptsmscls 22758, 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 22747 | 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 22748 | 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 22749 | 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 22750* | 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 22751 | 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 22752 | 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 22753 | 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 22754 | 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 22755 | 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 22756 | 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 22757 | 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 22758 | A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 22718, 0nsg 18321. (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 22759 | 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 22760 | 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 22761* | Lemma for tsmsxp 22763. (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 22762* | Lemma for tsmsxp 22763. (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 22763* | Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 19096 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 22761 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 22764 | The class of all topological division rings. |
class TopRing | ||
Syntax | ctdrg 22765 | The class of all topological division rings. |
class TopDRing | ||
Syntax | ctlm 22766 | The class of all topological modules. |
class TopMod | ||
Syntax | ctvc 22767 | The class of all topological vector spaces. |
class TopVec | ||
Definition | df-trg 22768 | 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 22769 | 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 22770 | 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 22771 | 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 22772 | 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 22773 | 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 22774 | 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 22775 | 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 22776 | A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp) | ||
Theorem | trgtmd2 22777 | A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd) | ||
Theorem | trgtps 22778 | A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopSp) | ||
Theorem | trgring 22779 | A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Ring) | ||
Theorem | trggrp 22780 | A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Grp) | ||
Theorem | tdrgtrg 22781 | A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing) | ||
Theorem | tdrgdrng 22782 | A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ DivRing) | ||
Theorem | tdrgring 22783 | A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ Ring) | ||
Theorem | tdrgtmd 22784 | A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopMnd) | ||
Theorem | tdrgtps 22785 | A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopSp) | ||
Theorem | istdrg2 22786 | 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 22787 | 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 22788 | 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 22789 | 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 22790* | Continuity of ring multiplication; analogue of cnmpt12f 22274 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 22791* | Continuity of ring multiplication; analogue of cnmpt22f 22283 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 22792 | 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 22793 | 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 22794 | 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 22795 | A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd) | ||
Theorem | tlmtps 22796 | A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp) | ||
Theorem | tlmlmod 22797 | A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod) | ||
Theorem | tlmtrg 22798 | 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 22799 | 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 22800 | 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 > |