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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | zartopon 33901 | The points of the Zariski topology are the prime ideals. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑆) & ⊢ 𝑃 = (PrmIdeal‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘𝑃)) | ||
| Theorem | zar0ring 33902 | The Zariski Topology of the trivial ring. (Contributed by Thierry Arnoux, 1-Jul-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑆) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐽 = {∅}) | ||
| Theorem | zart0 33903 | The Zariski topology is T0 . Corollary 1.1.8 of [EGA] p. 81. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑆) ⇒ ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Kol2) | ||
| Theorem | zarmxt1 33904 | The Zariski topology restricted to maximal ideals is T1 . (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑆) & ⊢ 𝑀 = (MaxIdeal‘𝑅) & ⊢ 𝑇 = (𝐽 ↾t 𝑀) ⇒ ⊢ (𝑅 ∈ CRing → 𝑇 ∈ Fre) | ||
| Theorem | zarcmplem 33905* | Lemma for zarcmp 33906. (Contributed by Thierry Arnoux, 2-Jul-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑆) & ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) ⇒ ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Comp) | ||
| Theorem | zarcmp 33906 | The Zariski topology is compact. Proposition 1.1.10(ii) of [EGA], p. 82. (Contributed by Thierry Arnoux, 2-Jul-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑆) ⇒ ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Comp) | ||
| Theorem | rspectps 33907 | The spectrum of a ring 𝑅 is a topological space. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝑆 ∈ TopSp) | ||
| Theorem | rhmpreimacnlem 33908* | Lemma for rhmpreimacn 33909. (Contributed by Thierry Arnoux, 7-Jul-2024.) |
| ⊢ 𝑇 = (Spec‘𝑅) & ⊢ 𝑈 = (Spec‘𝑆) & ⊢ 𝐴 = (PrmIdeal‘𝑅) & ⊢ 𝐵 = (PrmIdeal‘𝑆) & ⊢ 𝐽 = (TopOpen‘𝑇) & ⊢ 𝐾 = (TopOpen‘𝑈) & ⊢ 𝐺 = (𝑖 ∈ 𝐵 ↦ (◡𝐹 “ 𝑖)) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) & ⊢ (𝜑 → ran 𝐹 = (Base‘𝑆)) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ 𝑉 = (𝑗 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑗 ⊆ 𝑘}) & ⊢ 𝑊 = (𝑗 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑗 ⊆ 𝑘}) ⇒ ⊢ (𝜑 → (𝑊‘(𝐹 “ 𝐼)) = (◡𝐺 “ (𝑉‘𝐼))) | ||
| Theorem | rhmpreimacn 33909* | The function mapping a prime ideal to its preimage by a surjective ring homomorphism is continuous, when considering the Zariski topology. Corollary 1.2.3 of [EGA], p. 83. Notice that the direction of the continuous map 𝐺 is reverse: the original ring homomorphism 𝐹 goes from 𝑅 to 𝑆, but the continuous map 𝐺 goes from 𝐵 to 𝐴. This mapping is also called "induced map on prime spectra" or "pullback on primes". (Contributed by Thierry Arnoux, 8-Jul-2024.) |
| ⊢ 𝑇 = (Spec‘𝑅) & ⊢ 𝑈 = (Spec‘𝑆) & ⊢ 𝐴 = (PrmIdeal‘𝑅) & ⊢ 𝐵 = (PrmIdeal‘𝑆) & ⊢ 𝐽 = (TopOpen‘𝑇) & ⊢ 𝐾 = (TopOpen‘𝑈) & ⊢ 𝐺 = (𝑖 ∈ 𝐵 ↦ (◡𝐹 “ 𝑖)) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) & ⊢ (𝜑 → ran 𝐹 = (Base‘𝑆)) ⇒ ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽)) | ||
| Syntax | cmetid 33910 | Extend class notation with the class of metric identifications. |
| class ~Met | ||
| Syntax | cpstm 33911 | Extend class notation with the metric induced by a pseudometric. |
| class pstoMet | ||
| Definition | df-metid 33912* | Define the metric identification relation for a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ ~Met = (𝑑 ∈ ∪ ran PsMet ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)}) | ||
| Definition | df-pstm 33913* | Define the metric induced by a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ pstoMet = (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)})) | ||
| Theorem | metidval 33914* | Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑥𝐷𝑦) = 0)}) | ||
| Theorem | metidss 33915 | As a relation, the metric identification is a subset of a Cartesian product. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) ⊆ (𝑋 × 𝑋)) | ||
| Theorem | metidv 33916 | 𝐴 and 𝐵 identify by the metric 𝐷 if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0)) | ||
| Theorem | metideq 33917 | Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐵𝐷𝐹)) | ||
| Theorem | metider 33918 | The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
| ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) Er 𝑋) | ||
| Theorem | pstmval 33919* | Value of the metric induced by a pseudometric 𝐷. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ ∼ = (~Met‘𝐷) ⇒ ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)})) | ||
| Theorem | pstmfval 33920 | Function value of the metric induced by a pseudometric 𝐷 (Contributed by Thierry Arnoux, 11-Feb-2018.) |
| ⊢ ∼ = (~Met‘𝐷) ⇒ ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ([𝐴] ∼ (pstoMet‘𝐷)[𝐵] ∼ ) = (𝐴𝐷𝐵)) | ||
| Theorem | pstmxmet 33921 | The metric induced by a pseudometric is a full-fledged metric on the equivalence classes of the metric identification. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
| ⊢ ∼ = (~Met‘𝐷) ⇒ ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / ∼ ))) | ||
| Theorem | hauseqcn 33922 | In a Hausdorff topology, two continuous functions which agree on a dense set agree everywhere. (Contributed by Thierry Arnoux, 28-Dec-2017.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐾 ∈ Haus) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴)) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
| Theorem | elunitge0 33923 | An element of the closed unit interval is positive. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 20-Dec-2016.) |
| ⊢ (𝐴 ∈ (0[,]1) → 0 ≤ 𝐴) | ||
| Theorem | unitssxrge0 33924 | The closed unit interval is a subset of the set of the extended nonnegative reals. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 12-Dec-2016.) |
| ⊢ (0[,]1) ⊆ (0[,]+∞) | ||
| Theorem | unitdivcld 33925 | Necessary conditions for a quotient to be in the closed unit interval. (somewhat too strong, it would be sufficient that A and B are in RR+) (Contributed by Thierry Arnoux, 20-Dec-2016.) |
| ⊢ ((𝐴 ∈ (0[,]1) ∧ 𝐵 ∈ (0[,]1) ∧ 𝐵 ≠ 0) → (𝐴 ≤ 𝐵 ↔ (𝐴 / 𝐵) ∈ (0[,]1))) | ||
| Theorem | iistmd 33926 | 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 33927 | 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 33928 | 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 33929 | 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 33930 | 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 33931 | 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 33932 | 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 33933 | 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 33934* | Lemma for cnre2csqima 33935. (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 33935* | 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 33936* | 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 33937* | 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 33938 | 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 33939 | 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 33940 | 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 33941 | 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 33942* | 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 33943* | 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 33944 | 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 33945 | 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 33946* | Lemma for ordtrest2NEW 33947. (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 33947* | 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 33948* | Connectedness in the order topology of a toset. This is the "easy" direction of ordtconn 33949. See also reconnlem1 24752. (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 33949 | 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 33950* | 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 33951 | 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 33952* | Multiplication by a real constant is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.) Avoid ax-mulf 11096. (Revised by GG, 16-Mar-2025.) |
| ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ (𝑥 · 𝐶)) ∈ (𝐽 Cn 𝐽)) | ||
| Theorem | raddcn 33953* | 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 33954* | 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 33955 | 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 33956 | The extended nonnegative reals are homeomorphic to the closed unit interval. (Contributed by Thierry Arnoux, 24-Mar-2017.) |
| ⊢ II ≃ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | ||
| Theorem | xrge0iifcnv 33957* | 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 33958* | 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 33959* | 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 33960* | 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 33961* | 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 33962* | 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 33963* | 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 33964* | 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 33965* | 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 33966 | The extended nonnegative real numbers monoid forms a topological space. (Contributed by Thierry Arnoux, 19-Jun-2017.) |
| ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | ||
| Theorem | xrge0topn 33967 | 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 33968 | 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 33969 | 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 33970 | Alternate proof of xrge0tmd 33969. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopMnd | ||
| Theorem | lmlim 33971 | 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 33972 | 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 33973 | Implication of convergence for a nonnegative series. This could be used to shorten prmreclem6 16843. (Contributed by Thierry Arnoux, 28-Jul-2017.) |
| ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → sup(ran seq1( + , 𝐹), ℝ, < ) ∈ ℝ) | ||
| Theorem | fsumcvg4 33974 | 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 33975* | A neighborhood of +∞ contains an unbounded interval based at a real number. See pnfnei 23145. (Contributed by Thierry Arnoux, 31-Jul-2017.) |
| ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) ⇒ ⊢ ((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴) | ||
| Theorem | lmxrge0 33976* | 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 33977* | 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 33978* | 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 33979 | 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 33980 | Extend class notation with the Hausdorff uniform completion relation. |
| class HCmp | ||
| Definition | df-hcmp 33981* | 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 33982 | 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 33983 | 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 33984 | Zero of a ℤ-module. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
| ⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ 0 = (0g‘𝑊) | ||
| Theorem | zlm1 33985 | 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 33986 | 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 | zlmtset 33987 | 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 | zlmnm 33988 | 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 33989 | 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 33990 | 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 33991 | 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 33992 | The ℤ-module of ℂ is a normed module. (Contributed by Thierry Arnoux, 25-Feb-2018.) |
| ⊢ (ℤMod‘ℂfld) ∈ NrmMod | ||
| Theorem | rezh 33993 | The ℤ-module of ℝ is a normed module. (Contributed by Thierry Arnoux, 14-Feb-2018.) |
| ⊢ (ℤMod‘ℝfld) ∈ NrmMod | ||
| Syntax | cqqh 33994 | Map the rationals into a field. |
| class ℚHom | ||
| Definition | df-qqh 33995* | 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 33996* | 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 33997 | 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 33998 | 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 33999 | 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 34000 | 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})) | ||
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