HomeTheorem List (p. 245 of 506)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | ressust 24401 | The uniform structure of a restricted space. (Contributed by Thierry Arnoux, 22-Jan-2018.) |
| ⊢ 𝑋 = (Base‘𝑊) & ⊢ 𝑇 = (UnifSt‘(𝑊 ↾s 𝐴)) ⇒ ⊢ ((𝑊 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋) → 𝑇 ∈ (UnifOn‘𝐴)) | ||
| Theorem | ressusp 24402 | The restriction of a uniform topological space to an open set is a uniform space. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) ⇒ ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽) → (𝑊 ↾s 𝐴) ∈ UnifSp) | ||
| Theorem | tusval 24403 | The value of the uniform space mapping function. (Contributed by Thierry Arnoux, 5-Dec-2017.) |
| ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)) | ||
| Theorem | tuslem 24404 | Lemma for tusbas 24405, tusunif 24406, and tustopn 24408. (Contributed by Thierry Arnoux, 5-Dec-2017.) (Proof shortened by AV, 28-Oct-2024.) |
| ⊢ 𝐾 = (toUnifSp‘𝑈) ⇒ ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾))) | ||
| Theorem | tusbas 24405 | The base set of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.) |
| ⊢ 𝐾 = (toUnifSp‘𝑈) ⇒ ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾)) | ||
| Theorem | tusunif 24406 | The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.) |
| ⊢ 𝐾 = (toUnifSp‘𝑈) ⇒ ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾)) | ||
| Theorem | tususs 24407 | The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.) |
| ⊢ 𝐾 = (toUnifSp‘𝑈) ⇒ ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSt‘𝐾)) | ||
| Theorem | tustopn 24408 | The topology induced by a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.) |
| ⊢ 𝐾 = (toUnifSp‘𝑈) & ⊢ 𝐽 = (unifTop‘𝑈) ⇒ ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = (TopOpen‘𝐾)) | ||
| Theorem | tususp 24409 | A constructed uniform space is an uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.) |
| ⊢ 𝐾 = (toUnifSp‘𝑈) ⇒ ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 ∈ UnifSp) | ||
| Theorem | tustps 24410 | A constructed uniform space is a topological space. (Contributed by Thierry Arnoux, 25-Jan-2018.) |
| ⊢ 𝐾 = (toUnifSp‘𝑈) ⇒ ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 ∈ TopSp) | ||
| Theorem | uspreg 24411 | If a uniform space is Hausdorff, it is regular. Proposition 3 of [BourbakiTop1] p. II.5. (Contributed by Thierry Arnoux, 4-Jan-2018.) |
| ⊢ 𝐽 = (TopOpen‘𝑊) ⇒ ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg) | ||
| Syntax | cucn 24412 | Extend class notation with the uniform continuity operation. |
| class Cnu | ||
| Definition | df-ucn 24413* | Define a function on two uniform structures which value is the set of uniformly continuous functions from the first uniform structure to the second. A function 𝑓 is uniformly continuous if, roughly speaking, it is possible to guarantee that (𝑓‘𝑥) and (𝑓‘𝑦) be as close to each other as we please by requiring only that 𝑥 and 𝑦 are sufficiently close to each other; unlike ordinary continuity, the maximum distance between (𝑓‘𝑥) and (𝑓‘𝑦) cannot depend on 𝑥 and 𝑦 themselves. This formulation is the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.) |
| ⊢ Cnu = (𝑢 ∈ ∪ ran UnifOn, 𝑣 ∈ ∪ ran UnifOn ↦ {𝑓 ∈ (dom ∪ 𝑣 ↑m dom ∪ 𝑢) ∣ ∀𝑠 ∈ 𝑣 ∃𝑟 ∈ 𝑢 ∀𝑥 ∈ dom ∪ 𝑢∀𝑦 ∈ dom ∪ 𝑢(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))}) | ||
| Theorem | ucnval 24414* | The set of all uniformly continuous function from uniform space 𝑈 to uniform space 𝑉. (Contributed by Thierry Arnoux, 16-Nov-2017.) |
| ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑈 Cnu𝑉) = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))}) | ||
| Theorem | isucn 24415* | The predicate "𝐹 is a uniformly continuous function from uniform space 𝑈 to uniform space 𝑉". (Contributed by Thierry Arnoux, 16-Nov-2017.) |
| ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))) | ||
| Theorem | isucn2 24416* | The predicate "𝐹 is a uniformly continuous function from uniform space 𝑈 to uniform space 𝑉", expressed with filter bases for the entourages. (Contributed by Thierry Arnoux, 26-Jan-2018.) |
| ⊢ 𝑈 = ((𝑋 × 𝑋)filGen𝑅) & ⊢ 𝑉 = ((𝑌 × 𝑌)filGen𝑆) & ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋)) & ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌)) & ⊢ (𝜑 → 𝑅 ∈ (fBas‘(𝑋 × 𝑋))) & ⊢ (𝜑 → 𝑆 ∈ (fBas‘(𝑌 × 𝑌))) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑠 ∈ 𝑆 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))) | ||
| Theorem | ucnimalem 24417* | Reformulate the 𝐺 function as a mapping with one variable. (Contributed by Thierry Arnoux, 19-Nov-2017.) |
| ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋)) & ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌)) & ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ⇒ ⊢ 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩) | ||
| Theorem | ucnima 24418* | An equivalent statement of the definition of uniformly continuous function. (Contributed by Thierry Arnoux, 19-Nov-2017.) |
| ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋)) & ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌)) & ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ⇒ ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊) | ||
| Theorem | ucnprima 24419* | The preimage by a uniformly continuous function 𝐹 of an entourage 𝑊 of 𝑌 is an entourage of 𝑋. Note of the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 19-Nov-2017.) |
| ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋)) & ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌)) & ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ⇒ ⊢ (𝜑 → (◡𝐺 “ 𝑊) ∈ 𝑈) | ||
| Theorem | iducn 24420 | The identity is uniformly continuous from a uniform structure to itself. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.) |
| ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈)) | ||
| Theorem | cstucnd 24421 | A constant function is uniformly continuous. Deduction form. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.) |
| ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋)) & ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌)) & ⊢ (𝜑 → 𝐴 ∈ 𝑌) ⇒ ⊢ (𝜑 → (𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉)) | ||
| Theorem | ucncn 24422 | Uniform continuity implies continuity. Deduction form. Proposition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 30-Nov-2017.) |
| ⊢ 𝐽 = (TopOpen‘𝑅) & ⊢ 𝐾 = (TopOpen‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ UnifSp) & ⊢ (𝜑 → 𝑆 ∈ UnifSp) & ⊢ (𝜑 → 𝑅 ∈ TopSp) & ⊢ (𝜑 → 𝑆 ∈ TopSp) & ⊢ (𝜑 → 𝐹 ∈ ((UnifSt‘𝑅) Cnu(UnifSt‘𝑆))) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | ||
| Syntax | ccfilu 24423 | Extend class notation with the set of Cauchy filter bases. |
| class CauFilu | ||
| Definition | df-cfilu 24424* | Define the set of Cauchy filter bases on a uniform space. A Cauchy filter base is a filter base on the set such that for every entourage 𝑣, there is an element 𝑎 of the filter "small enough in 𝑣 " i.e. such that every pair {𝑥, 𝑦} of points in 𝑎 is related by 𝑣". Definition 2 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 16-Nov-2017.) |
| ⊢ CauFilu = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑓 ∈ (fBas‘dom ∪ 𝑢) ∣ ∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣}) | ||
| Theorem | iscfilu 24425* | The predicate "𝐹 is a Cauchy filter base on uniform space 𝑈". (Contributed by Thierry Arnoux, 18-Nov-2017.) |
| ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))) | ||
| Theorem | cfilufbas 24426 | A Cauchy filter base is a filter base. (Contributed by Thierry Arnoux, 19-Nov-2017.) |
| ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → 𝐹 ∈ (fBas‘𝑋)) | ||
| Theorem | cfiluexsm 24427* | For a Cauchy filter base and any entourage 𝑉, there is an element of the filter small in 𝑉. (Contributed by Thierry Arnoux, 19-Nov-2017.) |
| ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈) ∧ 𝑉 ∈ 𝑈) → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑉) | ||
| Theorem | fmucndlem 24428* | Lemma for fmucnd 24429. (Contributed by Thierry Arnoux, 19-Nov-2017.) |
| ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝐴 × 𝐴)) = ((𝐹 “ 𝐴) × (𝐹 “ 𝐴))) | ||
| Theorem | fmucnd 24429* | The image of a Cauchy filter base by an uniformly continuous function is a Cauchy filter base. Deduction form. Proposition 3 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 18-Nov-2017.) |
| ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋)) & ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌)) & ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉)) & ⊢ (𝜑 → 𝐶 ∈ (CauFilu‘𝑈)) & ⊢ 𝐷 = ran (𝑎 ∈ 𝐶 ↦ (𝐹 “ 𝑎)) ⇒ ⊢ (𝜑 → 𝐷 ∈ (CauFilu‘𝑉)) | ||
| Theorem | cfilufg 24430 | The filter generated by a Cauchy filter base is still a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.) |
| ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → (𝑋filGen𝐹) ∈ (CauFilu‘𝑈)) | ||
| Theorem | trcfilu 24431 | Condition for the trace of a Cauchy filter base to be a Cauchy filter base for the restricted uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.) |
| ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾t 𝐴) ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) | ||
| Theorem | cfiluweak 24432 | A Cauchy filter base is also a Cauchy filter base on any coarser uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.) |
| ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝐹 ∈ (CauFilu‘𝑈)) | ||
| Theorem | neipcfilu 24433 | In an uniform space, a neighboring filter is a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.) |
| ⊢ 𝑋 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑈 = (UnifSt‘𝑊) ⇒ ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFilu‘𝑈)) | ||
| Syntax | ccusp 24434 | Extend class notation with the class of all complete uniform spaces. |
| class CUnifSp | ||
| Definition | df-cusp 24435* | Define the class of all complete uniform spaces. Definition 3 of [BourbakiTop1] p. II.15. (Contributed by Thierry Arnoux, 1-Dec-2017.) |
| ⊢ CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)} | ||
| Theorem | iscusp 24436* | The predicate "𝑊 is a complete uniform space." (Contributed by Thierry Arnoux, 3-Dec-2017.) |
| ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))) | ||
| Theorem | cuspusp 24437 | A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.) |
| ⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp) | ||
| Theorem | cuspcvg 24438 | In a complete uniform space, any Cauchy filter 𝐶 has a limit. (Contributed by Thierry Arnoux, 3-Dec-2017.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) ⇒ ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐽 fLim 𝐶) ≠ ∅) | ||
| Theorem | iscusp2 24439* | The predicate "𝑊 is a complete uniform space." (Contributed by Thierry Arnoux, 15-Dec-2017.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝑈 = (UnifSt‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) ⇒ ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅))) | ||
| Theorem | cnextucn 24440* | Extension by continuity. Proposition 11 of [BourbakiTop1] p. II.20. Given a topology 𝐽 on 𝑋, a subset 𝐴 dense in 𝑋, this states a condition for 𝐹 from 𝐴 to a space 𝑌 Hausdorff and complete to be extensible by continuity. (Contributed by Thierry Arnoux, 4-Dec-2017.) |
| ⊢ 𝑋 = (Base‘𝑉) & ⊢ 𝑌 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑉) & ⊢ 𝐾 = (TopOpen‘𝑊) & ⊢ 𝑈 = (UnifSt‘𝑊) & ⊢ (𝜑 → 𝑉 ∈ TopSp) & ⊢ (𝜑 → 𝑊 ∈ TopSp) & ⊢ (𝜑 → 𝑊 ∈ CUnifSp) & ⊢ (𝜑 → 𝐾 ∈ Haus) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑌) & ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘𝑈)) ⇒ ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | ucnextcn 24441 | Extension by continuity. Theorem 2 of [BourbakiTop1] p. II.20. Given an uniform space on a set 𝑋, a subset 𝐴 dense in 𝑋, and a function 𝐹 uniformly continuous from 𝐴 to 𝑌, that function can be extended by continuity to the whole 𝑋, and its extension is uniformly continuous. (Contributed by Thierry Arnoux, 25-Jan-2018.) |
| ⊢ 𝑋 = (Base‘𝑉) & ⊢ 𝑌 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑉) & ⊢ 𝐾 = (TopOpen‘𝑊) & ⊢ 𝑆 = (UnifSt‘𝑉) & ⊢ 𝑇 = (UnifSt‘(𝑉 ↾s 𝐴)) & ⊢ 𝑈 = (UnifSt‘𝑊) & ⊢ (𝜑 → 𝑉 ∈ TopSp) & ⊢ (𝜑 → 𝑉 ∈ UnifSp) & ⊢ (𝜑 → 𝑊 ∈ TopSp) & ⊢ (𝜑 → 𝑊 ∈ CUnifSp) & ⊢ (𝜑 → 𝐾 ∈ Haus) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → 𝐹 ∈ (𝑇 Cnu𝑈)) & ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋) ⇒ ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | ispsmet 24442* | Express the predicate "𝐷 is a pseudometric." (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) | ||
| Theorem | psmetdmdm 24443 | Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷) | ||
| Theorem | psmetf 24444 | The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | ||
| Theorem | psmetcl 24445 | Closure of the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | ||
| Theorem | psmet0 24446 | The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0) | ||
| Theorem | psmettri2 24447 | Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
| ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))) | ||
| Theorem | psmetsym 24448 | The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴)) | ||
| Theorem | psmettri 24449 | Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
| ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵))) | ||
| Theorem | psmetge0 24450 | The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) | ||
| Theorem | psmetxrge0 24451 | The distance function of a pseudometric space is a function into the nonnegative extended real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.) |
| ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞)) | ||
| Theorem | psmetres2 24452 | Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
| ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅)) | ||
| Theorem | psmetlecl 24453 | Real closure of an extended metric value that is upper bounded by a real. (Contributed by Thierry Arnoux, 11-Mar-2018.) |
| ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ ℝ ∧ (𝐴𝐷𝐵) ≤ 𝐶)) → (𝐴𝐷𝐵) ∈ ℝ) | ||
| Theorem | distspace 24454 | A set 𝑋 together with a (distance) function 𝐷 which is a pseudometric is a distance space (according to E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006), i.e. a (base) set 𝑋 equipped with a distance 𝐷, which is a mapping of two elements of the base set to the (extended) reals and which is nonnegative, symmetric and equal to 0 if the two elements are equal. (Contributed by AV, 15-Oct-2021.) (Revised by AV, 5-Jul-2022.) |
| ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ (𝐴𝐷𝐴) = 0) ∧ (0 ≤ (𝐴𝐷𝐵) ∧ (𝐴𝐷𝐵) = (𝐵𝐷𝐴)))) | ||
| Syntax | cxms 24455 | Extend class notation with the class of extended metric spaces. |
| class ∞MetSp | ||
| Syntax | cms 24456 | Extend class notation with the class of metric spaces. |
| class MetSp | ||
| Syntax | ctms 24457 | Extend class notation with the function mapping a metric to the metric space it defines. |
| class toMetSp | ||
| Definition | df-xms 24458 | Define the (proper) class of extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))} | ||
| Definition | df-ms 24459 | Define the (proper) class of metric spaces. (Contributed by NM, 27-Aug-2006.) |
| ⊢ MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))} | ||
| Definition | df-tms 24460 | Define the function mapping a metric to the metric space which it defines. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ toMetSp = (𝑑 ∈ ∪ ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩)) | ||
| Theorem | ismet 24461* | Express the predicate "𝐷 is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.) |
| ⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))))) | ||
| Theorem | isxmet 24462* | Express the predicate "𝐷 is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) | ||
| Theorem | ismeti 24463* | Properties that determine a metric. (Contributed by NM, 17-Nov-2006.) (Revised by Mario Carneiro, 14-Aug-2015.) |
| ⊢ 𝑋 ∈ V & ⊢ 𝐷:(𝑋 × 𝑋)⟶ℝ & ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦)) & ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))) ⇒ ⊢ 𝐷 ∈ (Met‘𝑋) | ||
| Theorem | isxmetd 24464* | Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by AV, 9-Apr-2024.) |
| ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ*) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) ⇒ ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) | ||
| Theorem | isxmet2d 24465* | It is safe to only require the triangle inequality when the values are real (so that we can use the standard addition over the reals), but in this case the nonnegativity constraint cannot be deduced and must be provided separately. (Counterexample: 𝐷(𝑥, 𝑦) = if(𝑥 = 𝑦, 0, -∞) satisfies all hypotheses except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ*) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ≤ (𝑥𝐷𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) ≤ 0 ↔ 𝑥 = 𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))) ⇒ ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) | ||
| Theorem | metflem 24466* | Lemma for metf 24468 and others. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.) |
| ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))) | ||
| Theorem | xmetf 24467 | Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | ||
| Theorem | metf 24468 | Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) |
| ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) | ||
| Theorem | xmetcl 24469 | Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.) |
| ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | ||
| Theorem | metcl 24470 | Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.) |
| ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ) | ||
| Theorem | ismet2 24471 | An extended metric is a metric exactly when it takes real values for all values of the arguments. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐷 ∈ (Met‘𝑋) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐷:(𝑋 × 𝑋)⟶ℝ)) | ||
| Theorem | metxmet 24472 | A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | ||
| Theorem | xmetdmdm 24473 | Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.) |
| ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷) | ||
| Theorem | metdmdm 24474 | Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.) |
| ⊢ (𝐷 ∈ (Met‘𝑋) → 𝑋 = dom dom 𝐷) | ||
| Theorem | xmetunirn 24475 | Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
| ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷)) | ||
| Theorem | xmeteq0 24476 | The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)) | ||
| Theorem | meteq0 24477 | The value of a metric is zero iff its arguments are equal. Property M2 of [Kreyszig] p. 4. (Contributed by NM, 30-Aug-2006.) |
| ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵)) | ||
| Theorem | xmettri2 24478 | Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))) | ||
| Theorem | mettri2 24479 | Triangle inequality for the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) + (𝐶𝐷𝐵))) | ||
| Theorem | xmet0 24480 | The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0) | ||
| Theorem | met0 24481 | The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by NM, 30-Aug-2006.) |
| ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0) | ||
| Theorem | xmetge0 24482 | The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) | ||
| Theorem | metge0 24483 | The distance function of a metric space is nonnegative. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.) |
| ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) | ||
| Theorem | xmetlecl 24484 | Real closure of an extended metric value that is upper bounded by a real. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ ℝ ∧ (𝐴𝐷𝐵) ≤ 𝐶)) → (𝐴𝐷𝐵) ∈ ℝ) | ||
| Theorem | xmetsym 24485 | The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴)) | ||
| Theorem | xmetpsmet 24486 | An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋)) | ||
| Theorem | xmettpos 24487 | The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐷 ∈ (∞Met‘𝑋) → tpos 𝐷 = 𝐷) | ||
| Theorem | metsym 24488 | The distance function of a metric space is symmetric. Definition 14-1.1(c) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴)) | ||
| Theorem | xmettri 24489 | Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵))) | ||
| Theorem | mettri 24490 | Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.) |
| ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐶𝐷𝐵))) | ||
| Theorem | xmettri3 24491 | Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐵𝐷𝐶))) | ||
| Theorem | mettri3 24492 | Triangle inequality for the distance function of a metric space. (Contributed by NM, 13-Mar-2007.) |
| ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐵𝐷𝐶))) | ||
| Theorem | xmetrtri 24493 | One half of the reverse triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.) |
| ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶)) ≤ (𝐴𝐷𝐵)) | ||
| Theorem | xmetrtri2 24494 | The reverse triangle inequality for the distance function of an extended metric. In order to express the "extended absolute value function", we use the distance function xrsdsval 21542 defined on the extended real structure. (Contributed by Mario Carneiro, 4-Sep-2015.) |
| ⊢ 𝐾 = (dist‘ℝ*𝑠) ⇒ ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐶)𝐾(𝐵𝐷𝐶)) ≤ (𝐴𝐷𝐵)) | ||
| Theorem | metrtri 24495 | Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.) |
| ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (abs‘((𝐴𝐷𝐶) − (𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵)) | ||
| Theorem | xmetgt0 24496 | The distance function of an extended metric space is positive for unequal points. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≠ 𝐵 ↔ 0 < (𝐴𝐷𝐵))) | ||
| Theorem | metgt0 24497 | The distance function of a metric space is positive for unequal points. Definition 14-1.1(b) of [Gleason] p. 223 and its converse. (Contributed by NM, 27-Aug-2006.) |
| ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≠ 𝐵 ↔ 0 < (𝐴𝐷𝐵))) | ||
| Theorem | metn0 24498 | A metric space is nonempty iff its base set is nonempty. (Contributed by NM, 4-Oct-2007.) (Revised by Mario Carneiro, 14-Aug-2015.) |
| ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅)) | ||
| Theorem | xmetres2 24499 | Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘𝑅)) | ||
| Theorem | metreslem 24500 | Lemma for metres 24503. (Contributed by Mario Carneiro, 24-Aug-2015.) |
| ⊢ (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅)))) | ||
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