HomeTheorem List (p. 340 of 494)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | iistmd 33901 | The closed unit interval forms a topological monoid under multiplication. (Contributed by Thierry Arnoux, 25-Mar-2017.) |
| ⊢ 𝐼 = ((mulGrp‘ℂfld) ↾s (0[,]1)) ⇒ ⊢ 𝐼 ∈ TopMnd | ||
| Theorem | unicls 33902 | The union of the closed set is the underlying set of the topology. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
| ⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ∪ (Clsd‘𝐽) = 𝑋 | ||
| Theorem | tpr2tp 33903 | The usual topology on (ℝ × ℝ) is the product topology of the usual topology on ℝ. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
| ⊢ 𝐽 = (topGen‘ran (,)) ⇒ ⊢ (𝐽 ×t 𝐽) ∈ (TopOn‘(ℝ × ℝ)) | ||
| Theorem | tpr2uni 33904 | The usual topology on (ℝ × ℝ) is the product topology of the usual topology on ℝ. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
| ⊢ 𝐽 = (topGen‘ran (,)) ⇒ ⊢ ∪ (𝐽 ×t 𝐽) = (ℝ × ℝ) | ||
| Theorem | xpinpreima 33905 | Rewrite the cartesian product of two sets as the intersection of their preimage by 1st and 2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.) |
| ⊢ (𝐴 × 𝐵) = ((◡(1st ↾ (V × V)) “ 𝐴) ∩ (◡(2nd ↾ (V × V)) “ 𝐵)) | ||
| Theorem | xpinpreima2 33906 | Rewrite the cartesian product of two sets as the intersection of their preimage by 1st and 2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.) |
| ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ((◡(1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ (◡(2nd ↾ (𝐸 × 𝐹)) “ 𝐵))) | ||
| Theorem | sqsscirc1 33907 | The complex square of side 𝐷 is a subset of the complex circle of radius 𝐷. (Contributed by Thierry Arnoux, 25-Sep-2017.) |
| ⊢ ((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ+) → ((𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2)) → (√‘((𝑋↑2) + (𝑌↑2))) < 𝐷)) | ||
| Theorem | sqsscirc2 33908 | The complex square of side 𝐷 is a subset of the complex disc of radius 𝐷. (Contributed by Thierry Arnoux, 25-Sep-2017.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐷 ∈ ℝ+) → (((abs‘(ℜ‘(𝐵 − 𝐴))) < (𝐷 / 2) ∧ (abs‘(ℑ‘(𝐵 − 𝐴))) < (𝐷 / 2)) → (abs‘(𝐵 − 𝐴)) < 𝐷)) | ||
| Theorem | cnre2csqlem 33909* | Lemma for cnre2csqima 33910. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
| ⊢ (𝐺 ↾ (ℝ × ℝ)) = (𝐻 ∘ 𝐹) & ⊢ 𝐹 Fn (ℝ × ℝ) & ⊢ 𝐺 Fn V & ⊢ (𝑥 ∈ (ℝ × ℝ) → (𝐺‘𝑥) ∈ ℝ) & ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹) → (𝐻‘(𝑥 − 𝑦)) = ((𝐻‘𝑥) − (𝐻‘𝑦))) ⇒ ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ (◡(𝐺 ↾ (ℝ × ℝ)) “ (((𝐺‘𝑋) − 𝐷)(,)((𝐺‘𝑋) + 𝐷))) → (abs‘(𝐻‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷)) | ||
| Theorem | cnre2csqima 33910* | Image of a centered square by the canonical bijection from (ℝ × ℝ) to ℂ. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ⇒ ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) × (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷))) → ((abs‘(ℜ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷 ∧ (abs‘(ℑ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷))) | ||
| Theorem | tpr2rico 33911* | For any point of an open set of the usual topology on (ℝ × ℝ) there is an open square which contains that point and is entirely in the open set. This is square is actually a ball by the (𝑙↑+∞) norm 𝑋. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
| ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐺 = (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣))) & ⊢ 𝐵 = ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)) ⇒ ⊢ ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑟 ∈ 𝐵 (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴)) | ||
| Theorem | cnvordtrestixx 33912* | The restriction of the 'greater than' order to an interval gives the same topology as the subspace topology. (Contributed by Thierry Arnoux, 1-Apr-2017.) |
| ⊢ 𝐴 ⊆ ℝ* & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴) ⇒ ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) | ||
| Theorem | prsdm 33913 | Domain of the relation of a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ (𝐾 ∈ Proset → dom ≤ = 𝐵) | ||
| Theorem | prsrn 33914 | Range of the relation of a proset. (Contributed by Thierry Arnoux, 11-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ (𝐾 ∈ Proset → ran ≤ = 𝐵) | ||
| Theorem | prsss 33915 | Relation of a subproset. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) | ||
| Theorem | prsssdm 33916 | Domain of a subproset relation. (Contributed by Thierry Arnoux, 12-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴) | ||
| Theorem | ordtprsval 33917* | Value of the order topology for a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ 𝐸 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) & ⊢ 𝐹 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}) ⇒ ⊢ (𝐾 ∈ Proset → (ordTop‘ ≤ ) = (topGen‘(fi‘({𝐵} ∪ (𝐸 ∪ 𝐹))))) | ||
| Theorem | ordtprsuni 33918* | Value of the order topology. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ 𝐸 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) & ⊢ 𝐹 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}) ⇒ ⊢ (𝐾 ∈ Proset → 𝐵 = ∪ ({𝐵} ∪ (𝐸 ∪ 𝐹))) | ||
| Theorem | ordtcnvNEW 33919 | The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.) (Revised by Thierry Arnoux, 13-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ (𝐾 ∈ Proset → (ordTop‘◡ ≤ ) = (ordTop‘ ≤ )) | ||
| Theorem | ordtrestNEW 33920 | The subspace topology of an order topology is in general finer than the topology generated by the restricted order, but we do have inclusion in one direction. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴)) | ||
| Theorem | ordtrest2NEWlem 33921* | Lemma for ordtrest2NEW 33922. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ (𝜑 → 𝐾 ∈ Toset) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴) ⇒ ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))) | ||
| Theorem | ordtrest2NEW 33922* | An interval-closed set 𝐴 in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in ℝ, but in other sets like ℚ there are interval-closed sets like (π, +∞) ∩ ℚ that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ (𝜑 → 𝐾 ∈ Toset) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴) ⇒ ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘ ≤ ) ↾t 𝐴)) | ||
| Theorem | ordtconnlem1 33923* | Connectedness in the order topology of a toset. This is the "easy" direction of ordtconn 33924. See also reconnlem1 24848. (Contributed by Thierry Arnoux, 14-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ 𝐽 = (ordTop‘ ≤ ) ⇒ ⊢ ((𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵) → ((𝐽 ↾t 𝐴) ∈ Conn → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑟 ∈ 𝐵 ((𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦) → 𝑟 ∈ 𝐴))) | ||
| Theorem | ordtconn 33924 | Connectedness in the order topology of a complete uniform totally ordered space. (Contributed by Thierry Arnoux, 15-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ 𝐽 = (ordTop‘ ≤ ) ⇒ ⊢ ⊤ | ||
| Theorem | mndpluscn 33925* | A mapping that is both a homeomorphism and a monoid homomorphism preserves the "continuousness" of the operation. (Contributed by Thierry Arnoux, 25-Mar-2017.) |
| ⊢ 𝐹 ∈ (𝐽Homeo𝐾) & ⊢ + :(𝐵 × 𝐵)⟶𝐵 & ⊢ ∗ :(𝐶 × 𝐶)⟶𝐶 & ⊢ 𝐽 ∈ (TopOn‘𝐵) & ⊢ 𝐾 ∈ (TopOn‘𝐶) & ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))) & ⊢ + ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ⇒ ⊢ ∗ ∈ ((𝐾 ×t 𝐾) Cn 𝐾) | ||
| Theorem | mhmhmeotmd 33926 | Deduce a Topological Monoid using mapping that is both a homeomorphism and a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.) |
| ⊢ 𝐹 ∈ (𝑆 MndHom 𝑇) & ⊢ 𝐹 ∈ ((TopOpen‘𝑆)Homeo(TopOpen‘𝑇)) & ⊢ 𝑆 ∈ TopMnd & ⊢ 𝑇 ∈ TopSp ⇒ ⊢ 𝑇 ∈ TopMnd | ||
| Theorem | rmulccn 33927* | Multiplication by a real constant is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.) Avoid ax-mulf 11235. (Revised by GG, 16-Mar-2025.) |
| ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ (𝑥 · 𝐶)) ∈ (𝐽 Cn 𝐽)) | ||
| Theorem | raddcn 33928* | Addition in the real numbers is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.) |
| ⊢ 𝐽 = (topGen‘ran (,)) ⇒ ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) | ||
| Theorem | xrmulc1cn 33929* | The operation multiplying an extended real number by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 5-Jul-2017.) |
| ⊢ 𝐽 = (ordTop‘ ≤ ) & ⊢ 𝐹 = (𝑥 ∈ ℝ* ↦ (𝑥 ·e 𝐶)) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐽)) | ||
| Theorem | fmcncfil 33930 | The image of a Cauchy filter by a continuous filter map is a Cauchy filter. (Contributed by Thierry Arnoux, 12-Nov-2017.) |
| ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐾 = (MetOpen‘𝐸) ⇒ ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝐸 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝐵 ∈ (CauFil‘𝐷)) → ((𝑌 FilMap 𝐹)‘𝐵) ∈ (CauFil‘𝐸)) | ||
| Theorem | xrge0hmph 33931 | The extended nonnegative reals are homeomorphic to the closed unit interval. (Contributed by Thierry Arnoux, 24-Mar-2017.) |
| ⊢ II ≃ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | ||
| Theorem | xrge0iifcnv 33932* | Define a bijection from [0, 1] onto [0, +∞]. (Contributed by Thierry Arnoux, 29-Mar-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) ⇒ ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦)))) | ||
| Theorem | xrge0iifcv 33933* | The defined function's value in the real. (Contributed by Thierry Arnoux, 1-Apr-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) ⇒ ⊢ (𝑋 ∈ (0(,]1) → (𝐹‘𝑋) = -(log‘𝑋)) | ||
| Theorem | xrge0iifiso 33934* | The defined bijection from the closed unit interval onto the extended nonnegative reals is an order isomorphism. (Contributed by Thierry Arnoux, 31-Mar-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) ⇒ ⊢ 𝐹 Isom < , ◡ < ((0[,]1), (0[,]+∞)) | ||
| Theorem | xrge0iifhmeo 33935* | Expose a homeomorphism from the closed unit interval to the extended nonnegative reals. (Contributed by Thierry Arnoux, 1-Apr-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) & ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ⇒ ⊢ 𝐹 ∈ (IIHomeo𝐽) | ||
| Theorem | xrge0iifhom 33936* | The defined function from the closed unit interval to the extended nonnegative reals is a monoid homomorphism. (Contributed by Thierry Arnoux, 5-Apr-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) & ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ⇒ ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +𝑒 (𝐹‘𝑌))) | ||
| Theorem | xrge0iif1 33937* | Condition for the defined function, -(log‘𝑥) to be a monoid homomorphism. (Contributed by Thierry Arnoux, 20-Jun-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) & ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ⇒ ⊢ (𝐹‘1) = 0 | ||
| Theorem | xrge0iifmhm 33938* | The defined function from the closed unit interval to the extended nonnegative reals is a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) & ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ⇒ ⊢ 𝐹 ∈ (((mulGrp‘ℂfld) ↾s (0[,]1)) MndHom (ℝ*𝑠 ↾s (0[,]+∞))) | ||
| Theorem | xrge0pluscn 33939* | The addition operation of the extended nonnegative real numbers monoid is continuous. (Contributed by Thierry Arnoux, 24-Mar-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥))) & ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) & ⊢ + = ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) ⇒ ⊢ + ∈ ((𝐽 ×t 𝐽) Cn 𝐽) | ||
| Theorem | xrge0mulc1cn 33940* | The operation multiplying a nonnegative real numbers by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 6-Jul-2017.) |
| ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) & ⊢ 𝐹 = (𝑥 ∈ (0[,]+∞) ↦ (𝑥 ·e 𝐶)) & ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐽)) | ||
| Theorem | xrge0tps 33941 | The extended nonnegative real numbers monoid forms a topological space. (Contributed by Thierry Arnoux, 19-Jun-2017.) |
| ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | ||
| Theorem | xrge0topn 33942 | The topology of the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 20-Jun-2017.) |
| ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | ||
| Theorem | xrge0haus 33943 | The topology of the extended nonnegative real numbers is Hausdorff. (Contributed by Thierry Arnoux, 26-Jul-2017.) |
| ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) ∈ Haus | ||
| Theorem | xrge0tmd 33944 | The extended nonnegative real numbers monoid is a topological monoid. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof Shortened by Thierry Arnoux, 21-Jun-2017.) |
| ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd | ||
| Theorem | xrge0tmdALT 33945 | Alternate proof of xrge0tmd 33944. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd | ||
| Theorem | lmlim 33946 | Relate a limit in a given topology to a complex number limit, provided that topology agrees with the common topology on ℂ on the required subset. (Contributed by Thierry Arnoux, 11-Jul-2017.) |
| ⊢ 𝐽 ∈ (TopOn‘𝑌) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) & ⊢ (𝐽 ↾t 𝑋) = ((TopOpen‘ℂfld) ↾t 𝑋) & ⊢ 𝑋 ⊆ ℂ ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) | ||
| Theorem | lmlimxrge0 33947 | Relate a limit in the nonnegative extended reals to a complex limit, provided the considered function is a real function. (Contributed by Thierry Arnoux, 11-Jul-2017.) |
| ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) & ⊢ 𝑋 ⊆ (0[,)+∞) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) | ||
| Theorem | rge0scvg 33948 | Implication of convergence for a nonnegative series. This could be used to shorten prmreclem6 16959. (Contributed by Thierry Arnoux, 28-Jul-2017.) |
| ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → sup(ran seq1( + , 𝐹), ℝ, < ) ∈ ℝ) | ||
| Theorem | fsumcvg4 33949 | A serie with finite support is a finite sum, and therefore converges. (Contributed by Thierry Arnoux, 6-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
| ⊢ 𝑆 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑆⟶ℂ) & ⊢ (𝜑 → (◡𝐹 “ (ℂ ∖ {0})) ∈ Fin) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | ||
| Theorem | pnfneige0 33950* | A neighborhood of +∞ contains an unbounded interval based at a real number. See pnfnei 23228. (Contributed by Thierry Arnoux, 31-Jul-2017.) |
| ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) ⇒ ⊢ ((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴) | ||
| Theorem | lmxrge0 33951* | Express "sequence 𝐹 converges to plus infinity" (i.e. diverges), for a sequence of nonnegative extended real numbers. (Contributed by Thierry Arnoux, 2-Aug-2017.) |
| ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) & ⊢ (𝜑 → 𝐹:ℕ⟶(0[,]+∞)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = 𝐴) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴)) | ||
| Theorem | lmdvg 33952* | If a monotonic sequence of real numbers diverges, it is unbounded. (Contributed by Thierry Arnoux, 4-Aug-2017.) |
| ⊢ (𝜑 → 𝐹:ℕ⟶(0[,)+∞)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) & ⊢ (𝜑 → ¬ 𝐹 ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < (𝐹‘𝑘)) | ||
| Theorem | lmdvglim 33953* | If a monotonic real number sequence 𝐹 diverges, it converges in the extended real numbers and its limit is plus infinity. (Contributed by Thierry Arnoux, 3-Aug-2017.) |
| ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) & ⊢ (𝜑 → 𝐹:ℕ⟶(0[,)+∞)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) & ⊢ (𝜑 → ¬ 𝐹 ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)+∞) | ||
| Theorem | pl1cn 33954 | A univariate polynomial is continuous. (Contributed by Thierry Arnoux, 17-Sep-2018.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐸 = (eval1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ TopRing) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐸‘𝐹) ∈ (𝐽 Cn 𝐽)) | ||
| Syntax | chcmp 33955 | Extend class notation with the Hausdorff uniform completion relation. |
| class HCmp | ||
| Definition | df-hcmp 33956* | Definition of the Hausdorff completion. In this definition, a structure 𝑤 is a Hausdorff completion of a uniform structure 𝑢 if 𝑤 is a complete uniform space, in which 𝑢 is dense, and which admits the same uniform structure. Theorem 3 of [BourbakiTop1] p. II.21. states the existence and uniqueness of such a completion. (Contributed by Thierry Arnoux, 5-Mar-2018.) |
| ⊢ HCmp = {⟨𝑢, 𝑤⟩ ∣ ((𝑢 ∈ ∪ ran UnifOn ∧ 𝑤 ∈ CUnifSp) ∧ ((UnifSt‘𝑤) ↾t dom ∪ 𝑢) = 𝑢 ∧ ((cls‘(TopOpen‘𝑤))‘dom ∪ 𝑢) = (Base‘𝑤))} | ||
| Theorem | zringnm 33957 | The norm (function) for a ring of integers is the absolute value function (restricted to the integers). (Contributed by AV, 13-Jun-2019.) |
| ⊢ (norm‘ℤring) = (abs ↾ ℤ) | ||
| Theorem | zzsnm 33958 | The norm of the ring of the integers. (Contributed by Thierry Arnoux, 8-Nov-2017.) (Revised by AV, 13-Jun-2019.) |
| ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) = ((norm‘ℤring)‘𝑀)) | ||
| Theorem | zlm0 33959 | Zero of a ℤ-module. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
| ⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ 0 = (0g‘𝑊) | ||
| Theorem | zlm1 33960 | Unity element of a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) |
| ⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 1 = (1r‘𝐺) ⇒ ⊢ 1 = (1r‘𝑊) | ||
| Theorem | zlmds 33961 | Distance in a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof shortened by AV, 11-Nov-2024.) |
| ⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → 𝐷 = (dist‘𝑊)) | ||
| Theorem | zlmdsOLD 33962 | Obsolete proof of zlmds 33961 as of 11-Nov-2024. Distance in a ℤ -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → 𝐷 = (dist‘𝑊)) | ||
| Theorem | zlmtset 33963 | Topology in a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof shortened by AV, 12-Nov-2024.) |
| ⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 𝐽 = (TopSet‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → 𝐽 = (TopSet‘𝑊)) | ||
| Theorem | zlmtsetOLD 33964 | Obsolete proof of zlmtset 33963 as of 11-Nov-2024. Topology in a ℤ -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 𝐽 = (TopSet‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → 𝐽 = (TopSet‘𝑊)) | ||
| Theorem | zlmnm 33965 | Norm of a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) |
| ⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 𝑁 = (norm‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → 𝑁 = (norm‘𝑊)) | ||
| Theorem | zhmnrg 33966 | The ℤ-module built from a normed ring is also a normed ring. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
| ⊢ 𝑊 = (ℤMod‘𝐺) ⇒ ⊢ (𝐺 ∈ NrmRing → 𝑊 ∈ NrmRing) | ||
| Theorem | nmmulg 33967 | The norm of a group product, provided the ℤ-module is normed. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑁 = (norm‘𝑅) & ⊢ 𝑍 = (ℤMod‘𝑅) & ⊢ · = (.g‘𝑅) ⇒ ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑀 · 𝑋)) = ((abs‘𝑀) · (𝑁‘𝑋))) | ||
| Theorem | zrhnm 33968 | The norm of the image by ℤRHom of an integer in a normed ring. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑁 = (norm‘𝑅) & ⊢ 𝑍 = (ℤMod‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = (abs‘𝑀)) | ||
| Theorem | cnzh 33969 | The ℤ-module of ℂ is a normed module. (Contributed by Thierry Arnoux, 25-Feb-2018.) |
| ⊢ (ℤMod‘ℂfld) ∈ NrmMod | ||
| Theorem | rezh 33970 | The ℤ-module of ℝ is a normed module. (Contributed by Thierry Arnoux, 14-Feb-2018.) |
| ⊢ (ℤMod‘ℝfld) ∈ NrmMod | ||
| Syntax | cqqh 33971 | Map the rationals into a field. |
| class ℚHom | ||
| Definition | df-qqh 33972* | Define the canonical homomorphism from the rationals into any field. (Contributed by Mario Carneiro, 22-Oct-2017.) (Revised by Thierry Arnoux, 23-Oct-2017.) |
| ⊢ ℚHom = (𝑟 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡(ℤRHom‘𝑟) “ (Unit‘𝑟)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑟)‘𝑥)(/r‘𝑟)((ℤRHom‘𝑟)‘𝑦))⟩)) | ||
| Theorem | qqhval 33973* | Value of the canonical homormorphism from the rational number to a field. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
| ⊢ / = (/r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ (𝑅 ∈ V → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)) | ||
| Theorem | zrhf1ker 33974 | The kernel of the homomorphism from the integers to a ring, if it is injective. (Contributed by Thierry Arnoux, 26-Oct-2017.) (Revised by Thierry Arnoux, 23-May-2023.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐿:ℤ–1-1→𝐵 ↔ (◡𝐿 “ { 0 }) = {0})) | ||
| Theorem | zrhchr 33975 | The kernel of the homomorphism from the integers to a ring is injective if and only if the ring has characteristic 0 . (Contributed by Thierry Arnoux, 8-Nov-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) = 0 ↔ 𝐿:ℤ–1-1→𝐵)) | ||
| Theorem | zrhker 33976 | The kernel of the homomorphism from the integers to a ring with characteristic 0. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) = 0 ↔ (◡𝐿 “ { 0 }) = {0})) | ||
| Theorem | zrhunitpreima 33977 | The preimage by ℤRHom of the units of a division ring is (ℤ ∖ {0}). (Contributed by Thierry Arnoux, 22-Oct-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (◡𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0})) | ||
| Theorem | elzrhunit 33978 | Condition for the image by ℤRHom to be a unit. (Contributed by Thierry Arnoux, 30-Oct-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → (𝐿‘𝑀) ∈ (Unit‘𝑅)) | ||
| Theorem | zrhneg 33979 | The canonical homomorphism from the integers to a ring 𝑅 maps additive inverses to additive inverses. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| ⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ 𝐼 = (invg‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐿‘-𝑁) = (𝐼‘(𝐿‘𝑁))) | ||
| Theorem | zrhcntr 33980 | The canonical representation of an integer 𝑁 in a ring 𝑅 is in the centralizer of the ring's multiplicative monoid. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝐶 = (Cntr‘𝑀) & ⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐿‘𝑁) ∈ 𝐶) | ||
| Theorem | elzdif0 33981 | Lemma for qqhval2 33983. (Contributed by Thierry Arnoux, 29-Oct-2017.) |
| ⊢ (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ)) | ||
| Theorem | qqhval2lem 33982 | Lemma for qqhval2 33983. (Contributed by Thierry Arnoux, 29-Oct-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0)) → ((𝐿‘(numer‘(𝑋 / 𝑌))) / (𝐿‘(denom‘(𝑋 / 𝑌)))) = ((𝐿‘𝑋) / (𝐿‘𝑌))) | ||
| Theorem | qqhval2 33983* | Value of the canonical homormorphism from the rational number when the target ring is a division ring. (Contributed by Thierry Arnoux, 26-Oct-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) = (𝑞 ∈ ℚ ↦ ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))) | ||
| Theorem | qqhvval 33984 | Value of the canonical homormorphism from the rational number when the target ring is a division ring. (Contributed by Thierry Arnoux, 30-Oct-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ 𝑄 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑄) = ((𝐿‘(numer‘𝑄)) / (𝐿‘(denom‘𝑄)))) | ||
| Theorem | qqh0 33985 | The image of 0 by the ℚHom homomorphism is the ring's zero. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = (0g‘𝑅)) | ||
| Theorem | qqh1 33986 | The image of 1 by the ℚHom homomorphism is the ring unity. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = (1r‘𝑅)) | ||
| Theorem | qqhf 33987 | ℚHom as a function. (Contributed by Thierry Arnoux, 28-Oct-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵) | ||
| Theorem | qqhvq 33988 | The image of a quotient by the ℚHom homomorphism. (Contributed by Thierry Arnoux, 28-Oct-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0)) → ((ℚHom‘𝑅)‘(𝑋 / 𝑌)) = ((𝐿‘𝑋) / (𝐿‘𝑌))) | ||
| Theorem | qqhghm 33989 | The ℚHom homomorphism is a group homomorphism if the target structure is a division ring. (Contributed by Thierry Arnoux, 9-Nov-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ 𝑄 = (ℂfld ↾s ℚ) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅)) | ||
| Theorem | qqhrhm 33990 | The ℚHom homomorphism is a ring homomorphism if the target structure is a field. If the target structure is a division ring, it is a group homomorphism, but not a ring homomorphism, because it does not preserve the ring multiplication operation. (Contributed by Thierry Arnoux, 29-Oct-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ 𝑄 = (ℂfld ↾s ℚ) ⇒ ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 RingHom 𝑅)) | ||
| Theorem | qqhnm 33991 | The norm of the image by ℚHom of a rational number in a topological division ring. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
| ⊢ 𝑁 = (norm‘𝑅) & ⊢ 𝑍 = (ℤMod‘𝑅) ⇒ ⊢ (((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑄 ∈ ℚ) → (𝑁‘((ℚHom‘𝑅)‘𝑄)) = (abs‘𝑄)) | ||
| Theorem | qqhcn 33992 | The ℚHom homomorphism is a continuous function. (Contributed by Thierry Arnoux, 9-Nov-2017.) |
| ⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝐽 = (TopOpen‘𝑄) & ⊢ 𝑍 = (ℤMod‘𝑅) & ⊢ 𝐾 = (TopOpen‘𝑅) ⇒ ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | qqhucn 33993 | The ℚHom homomorphism is uniformly continuous. (Contributed by Thierry Arnoux, 28-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑄 = (ℂfld ↾s ℚ) & ⊢ 𝑈 = (UnifSt‘𝑄) & ⊢ 𝑉 = (metUnif‘((dist‘𝑅) ↾ (𝐵 × 𝐵))) & ⊢ 𝑍 = (ℤMod‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ NrmRing) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑍 ∈ NrmMod) & ⊢ (𝜑 → (chr‘𝑅) = 0) ⇒ ⊢ (𝜑 → (ℚHom‘𝑅) ∈ (𝑈 Cnu𝑉)) | ||
| Syntax | crrh 33994 | Map the real numbers into a complete field. |
| class ℝHom | ||
| Syntax | crrext 33995 | Extend class notation with the class of extension fields of ℝ. |
| class ℝExt | ||
| Definition | df-rrh 33996 | Define the canonical homomorphism from the real numbers to any complete field, as the extension by continuity of the canonical homomorphism from the rational numbers. (Contributed by Mario Carneiro, 22-Oct-2017.) (Revised by Thierry Arnoux, 23-Oct-2017.) |
| ⊢ ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟))) | ||
| Theorem | rrhval 33997 | Value of the canonical homormorphism from the real numbers to a complete space. (Contributed by Thierry Arnoux, 2-Nov-2017.) |
| ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐾 = (TopOpen‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅))) | ||
| Theorem | rrhcn 33998 | If the topology of 𝑅 is Hausdorff, and 𝑅 is a complete uniform space, then the canonical homomorphism from the real numbers to 𝑅 is continuous. (Contributed by Thierry Arnoux, 17-Jan-2018.) |
| ⊢ 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵)) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐾 = (TopOpen‘𝑅) & ⊢ 𝑍 = (ℤMod‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑅 ∈ NrmRing) & ⊢ (𝜑 → 𝑍 ∈ NrmMod) & ⊢ (𝜑 → (chr‘𝑅) = 0) & ⊢ (𝜑 → 𝑅 ∈ CUnifSp) & ⊢ (𝜑 → (UnifSt‘𝑅) = (metUnif‘𝐷)) ⇒ ⊢ (𝜑 → (ℝHom‘𝑅) ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | rrhf 33999 | If the topology of 𝑅 is Hausdorff, Cauchy sequences have at most one limit, i.e. the canonical homomorphism of ℝ into 𝑅 is a function. (Contributed by Thierry Arnoux, 2-Nov-2017.) |
| ⊢ 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵)) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐾 = (TopOpen‘𝑅) & ⊢ 𝑍 = (ℤMod‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑅 ∈ NrmRing) & ⊢ (𝜑 → 𝑍 ∈ NrmMod) & ⊢ (𝜑 → (chr‘𝑅) = 0) & ⊢ (𝜑 → 𝑅 ∈ CUnifSp) & ⊢ (𝜑 → (UnifSt‘𝑅) = (metUnif‘𝐷)) ⇒ ⊢ (𝜑 → (ℝHom‘𝑅):ℝ⟶𝐵) | ||
| Definition | df-rrext 34000 | Define the class of extensions of ℝ. This is a shorthand for listing the necessary conditions for a structure to admit a canonical embedding of ℝ into it. Interestingly, this is not coming from a mathematical reference, but was from the necessary conditions to build the embedding at each step (ℤ, ℚ and ℝ). It would be interesting see if this is formally treated in the literature. See isrrext 34001 for a better readable version. (Contributed by Thierry Arnoux, 2-May-2018.) |
| ⊢ ℝExt = {𝑟 ∈ (NrmRing ∩ DivRing) ∣ (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ∧ (𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))))} | ||
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