HomeTheorem List (p. 379 of 491)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | heiborlem5 37801* | Lemma for heibor 37807. The function 𝑀 is a set of point-and-radius pairs suitable for application to caubl 25355. (Contributed by Jeff Madsen, 23-Jan-2014.) |
| ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣} & ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} & ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) & ⊢ (𝜑 → 𝐶𝐺0) & ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))) & ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩) ⇒ ⊢ (𝜑 → 𝑀:ℕ⟶(𝑋 × ℝ+)) | ||
| Theorem | heiborlem6 37802* | Lemma for heibor 37807. Since the sequence of balls connected by the function 𝑇 ensures that each ball nontrivially intersects with the next (since the empty set has a finite subcover, the intersection of any two successive balls in the sequence is nonempty), and each ball is half the size of the previous one, the distance between the centers is at most 3 / 2 times the size of the larger, and so if we expand each ball by a factor of 3 we get a nested sequence of balls. (Contributed by Jeff Madsen, 23-Jan-2014.) |
| ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣} & ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} & ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) & ⊢ (𝜑 → 𝐶𝐺0) & ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))) & ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘))) | ||
| Theorem | heiborlem7 37803* | Lemma for heibor 37807. Since the sizes of the balls decrease exponentially, the sequence converges to zero. (Contributed by Jeff Madsen, 23-Jan-2014.) |
| ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣} & ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} & ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) & ⊢ (𝜑 → 𝐶𝐺0) & ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))) & ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩) ⇒ ⊢ ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟 | ||
| Theorem | heiborlem8 37804* | Lemma for heibor 37807. The previous lemmas establish that the sequence 𝑀 is Cauchy, so using completeness we now consider the convergent point 𝑌. By assumption, 𝑈 is an open cover, so 𝑌 is an element of some 𝑍 ∈ 𝑈, and some ball centered at 𝑌 is contained in 𝑍. But the sequence contains arbitrarily small balls close to 𝑌, so some element ball(𝑀‘𝑛) of the sequence is contained in 𝑍. And finally we arrive at a contradiction, because {𝑍} is a finite subcover of 𝑈 that covers ball(𝑀‘𝑛), yet ball(𝑀‘𝑛) ∈ 𝐾. For convenience, we write this contradiction as 𝜑 → 𝜓 where 𝜑 is all the accumulated hypotheses and 𝜓 is anything at all. (Contributed by Jeff Madsen, 22-Jan-2014.) |
| ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣} & ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} & ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) & ⊢ (𝜑 → 𝐶𝐺0) & ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))) & ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩) & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ 𝑌 ∈ V & ⊢ (𝜑 → 𝑌 ∈ 𝑍) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → (1st ∘ 𝑀)(⇝𝑡‘𝐽)𝑌) ⇒ ⊢ (𝜑 → 𝜓) | ||
| Theorem | heiborlem9 37805* | Lemma for heibor 37807. Discharge the hypotheses of heiborlem8 37804 by applying caubl 25355 to get a convergent point and adding the open cover assumption. (Contributed by Jeff Madsen, 20-Jan-2014.) |
| ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣} & ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} & ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) & ⊢ (𝜑 → 𝐶𝐺0) & ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))) & ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩) & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ (𝜑 → ∪ 𝑈 = 𝑋) ⇒ ⊢ (𝜑 → 𝜓) | ||
| Theorem | heiborlem10 37806* | Lemma for heibor 37807. The last remaining piece of the proof is to find an element 𝐶 such that 𝐶𝐺0, i.e. 𝐶 is an element of (𝐹‘0) that has no finite subcover, which is true by heiborlem1 37797, since (𝐹‘0) is a finite cover of 𝑋, which has no finite subcover. Thus, the rest of the proof follows to a contradiction, and thus there must be a finite subcover of 𝑈 that covers 𝑋, i.e. 𝑋 is compact. (Contributed by Jeff Madsen, 22-Jan-2014.) |
| ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣} & ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} & ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) ⇒ ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∪ 𝐽 = ∪ 𝑣) | ||
| Theorem | heibor 37807 | Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 37796 and heiborlem1 37797 for a description of the proof. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jan-2014.) |
| ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ↔ (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋))) | ||
| Theorem | bfplem1 37808* | Lemma for bfp 37810. The sequence 𝐺, which simply starts from any point in the space and iterates 𝐹, satisfies the property that the distance from 𝐺(𝑛) to 𝐺(𝑛 + 1) decreases by at least 𝐾 after each step. Thus, the total distance from any 𝐺(𝑖) to 𝐺(𝑗) is bounded by a geometric series, and the sequence is Cauchy. Therefore, it converges to a point ((⇝𝑡‘𝐽)‘𝐺) since the space is complete. (Contributed by Jeff Madsen, 17-Jun-2014.) |
| ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝐾 ∈ ℝ+) & ⊢ (𝜑 → 𝐾 < 1) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑋) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦))) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐺 = seq1((𝐹 ∘ 1st ), (ℕ × {𝐴})) ⇒ ⊢ (𝜑 → 𝐺(⇝𝑡‘𝐽)((⇝𝑡‘𝐽)‘𝐺)) | ||
| Theorem | bfplem2 37809* | Lemma for bfp 37810. Using the point found in bfplem1 37808, we show that this convergent point is a fixed point of 𝐹. Since for any positive 𝑥, the sequence 𝐺 is in 𝐵(𝑥 / 2, 𝑃) for all 𝑘 ∈ (ℤ≥‘𝑗) (where 𝑃 = ((⇝𝑡‘𝐽)‘𝐺)), we have 𝐷(𝐺(𝑗 + 1), 𝐹(𝑃)) ≤ 𝐷(𝐺(𝑗), 𝑃) < 𝑥 / 2 and 𝐷(𝐺(𝑗 + 1), 𝑃) < 𝑥 / 2, so 𝐹(𝑃) is in every neighborhood of 𝑃 and 𝑃 is a fixed point of 𝐹. (Contributed by Jeff Madsen, 5-Jun-2014.) |
| ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝐾 ∈ ℝ+) & ⊢ (𝜑 → 𝐾 < 1) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑋) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦))) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐺 = seq1((𝐹 ∘ 1st ), (ℕ × {𝐴})) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑧) | ||
| Theorem | bfp 37810* | Banach fixed point theorem, also known as contraction mapping theorem. A contraction on a complete metric space has a unique fixed point. We show existence in the lemmas, and uniqueness here - if 𝐹 has two fixed points, then the distance between them is less than 𝐾 times itself, a contradiction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.) |
| ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝐾 ∈ ℝ+) & ⊢ (𝜑 → 𝐾 < 1) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑋) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦))) ⇒ ⊢ (𝜑 → ∃!𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑧) | ||
| Syntax | crrn 37811 | Extend class notation with the n-dimensional Euclidean space. |
| class ℝn | ||
| Definition | df-rrn 37812* | Define n-dimensional Euclidean space as a metric space with the standard Euclidean norm given by the quadratic mean. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| ⊢ ℝn = (𝑖 ∈ Fin ↦ (𝑥 ∈ (ℝ ↑m 𝑖), 𝑦 ∈ (ℝ ↑m 𝑖) ↦ (√‘Σ𝑘 ∈ 𝑖 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))) | ||
| Theorem | rrnval 37813* | The n-dimensional Euclidean space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.) |
| ⊢ 𝑋 = (ℝ ↑m 𝐼) ⇒ ⊢ (𝐼 ∈ Fin → (ℝn‘𝐼) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))) | ||
| Theorem | rrnmval 37814* | The value of the Euclidean metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.) |
| ⊢ 𝑋 = (ℝ ↑m 𝐼) ⇒ ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹(ℝn‘𝐼)𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))) | ||
| Theorem | rrnmet 37815 | Euclidean space is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.) |
| ⊢ 𝑋 = (ℝ ↑m 𝐼) ⇒ ⊢ (𝐼 ∈ Fin → (ℝn‘𝐼) ∈ (Met‘𝑋)) | ||
| Theorem | rrndstprj1 37816 | The distance between two points in Euclidean space is greater than the distance between the projections onto one coordinate. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.) |
| ⊢ 𝑋 = (ℝ ↑m 𝐼) & ⊢ 𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ)) ⇒ ⊢ (((𝐼 ∈ Fin ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹(ℝn‘𝐼)𝐺)) | ||
| Theorem | rrndstprj2 37817* | Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 37816 can be used to show that the supremum norm and Euclidean norm are equivalent. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.) |
| ⊢ 𝑋 = (ℝ ↑m 𝐼) & ⊢ 𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ)) ⇒ ⊢ (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝐹(ℝn‘𝐼)𝐺) < (𝑅 · (√‘(♯‘𝐼)))) | ||
| Theorem | rrncmslem 37818* | Lemma for rrncms 37819. (Contributed by Jeff Madsen, 6-Jun-2014.) (Revised by Mario Carneiro, 13-Sep-2015.) |
| ⊢ 𝑋 = (ℝ ↑m 𝐼) & ⊢ 𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ)) & ⊢ 𝐽 = (MetOpen‘(ℝn‘𝐼)) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐹 ∈ (Cau‘(ℝn‘𝐼))) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) & ⊢ 𝑃 = (𝑚 ∈ 𝐼 ↦ ( ⇝ ‘(𝑡 ∈ ℕ ↦ ((𝐹‘𝑡)‘𝑚)))) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽)) | ||
| Theorem | rrncms 37819 | Euclidean space is complete. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.) |
| ⊢ 𝑋 = (ℝ ↑m 𝐼) ⇒ ⊢ (𝐼 ∈ Fin → (ℝn‘𝐼) ∈ (CMet‘𝑋)) | ||
| Theorem | repwsmet 37820 | The supremum metric on ℝ↑𝐼 is a metric. (Contributed by Jeff Madsen, 15-Sep-2015.) |
| ⊢ 𝑌 = ((ℂfld ↾s ℝ) ↑s 𝐼) & ⊢ 𝐷 = (dist‘𝑌) & ⊢ 𝑋 = (ℝ ↑m 𝐼) ⇒ ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘𝑋)) | ||
| Theorem | rrnequiv 37821 | The supremum metric on ℝ↑𝐼 is equivalent to the ℝn metric. (Contributed by Jeff Madsen, 15-Sep-2015.) |
| ⊢ 𝑌 = ((ℂfld ↾s ℝ) ↑s 𝐼) & ⊢ 𝐷 = (dist‘𝑌) & ⊢ 𝑋 = (ℝ ↑m 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) ⇒ ⊢ ((𝜑 ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹𝐷𝐺) ≤ (𝐹(ℝn‘𝐼)𝐺) ∧ (𝐹(ℝn‘𝐼)𝐺) ≤ ((√‘(♯‘𝐼)) · (𝐹𝐷𝐺)))) | ||
| Theorem | rrntotbnd 37822 | A set in Euclidean space is totally bounded iff its is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 16-Sep-2015.) |
| ⊢ 𝑋 = (ℝ ↑m 𝐼) & ⊢ 𝑀 = ((ℝn‘𝐼) ↾ (𝑌 × 𝑌)) ⇒ ⊢ (𝐼 ∈ Fin → (𝑀 ∈ (TotBnd‘𝑌) ↔ 𝑀 ∈ (Bnd‘𝑌))) | ||
| Theorem | rrnheibor 37823 | Heine-Borel theorem for Euclidean space. A subset of Euclidean space is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.) |
| ⊢ 𝑋 = (ℝ ↑m 𝐼) & ⊢ 𝑀 = ((ℝn‘𝐼) ↾ (𝑌 × 𝑌)) & ⊢ 𝑇 = (MetOpen‘𝑀) & ⊢ 𝑈 = (MetOpen‘(ℝn‘𝐼)) ⇒ ⊢ ((𝐼 ∈ Fin ∧ 𝑌 ⊆ 𝑋) → (𝑇 ∈ Comp ↔ (𝑌 ∈ (Clsd‘𝑈) ∧ 𝑀 ∈ (Bnd‘𝑌)))) | ||
| Theorem | ismrer1 37824* | An isometry between ℝ and ℝ↑1. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.) |
| ⊢ 𝑅 = ((abs ∘ − ) ↾ (ℝ × ℝ)) & ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ({𝐴} × {𝑥})) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ (𝑅 Ismty (ℝn‘{𝐴}))) | ||
| Theorem | reheibor 37825 | Heine-Borel theorem for real numbers. A subset of ℝ is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.) |
| ⊢ 𝑀 = ((abs ∘ − ) ↾ (𝑌 × 𝑌)) & ⊢ 𝑇 = (MetOpen‘𝑀) & ⊢ 𝑈 = (topGen‘ran (,)) ⇒ ⊢ (𝑌 ⊆ ℝ → (𝑇 ∈ Comp ↔ (𝑌 ∈ (Clsd‘𝑈) ∧ 𝑀 ∈ (Bnd‘𝑌)))) | ||
| Theorem | iccbnd 37826 | A closed interval in ℝ is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Sep-2015.) |
| ⊢ 𝐽 = (𝐴[,]𝐵) & ⊢ 𝑀 = ((abs ∘ − ) ↾ (𝐽 × 𝐽)) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑀 ∈ (Bnd‘𝐽)) | ||
| Theorem | icccmpALT 37827 | A closed interval in ℝ is compact. Alternate proof of icccmp 24860 using the Heine-Borel theorem heibor 37807. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Aug-2014.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝐽 = (𝐴[,]𝐵) & ⊢ 𝑀 = ((abs ∘ − ) ↾ (𝐽 × 𝐽)) & ⊢ 𝑇 = (MetOpen‘𝑀) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑇 ∈ Comp) | ||
| Syntax | cass 37828 | Extend class notation with a device to add associativity to internal operations. |
| class Ass | ||
| Definition | df-ass 37829* | A device to add associativity to various sorts of internal operations. The definition is meaningful when 𝑔 is a magma at least. (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.) |
| ⊢ Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔∀𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))} | ||
| Syntax | cexid 37830 | Extend class notation with the class of all the internal operations with an identity element. |
| class ExId | ||
| Definition | df-exid 37831* | A device to add an identity element to various sorts of internal operations. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) |
| ⊢ ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)} | ||
| Theorem | isass 37832* | The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.) |
| ⊢ 𝑋 = dom dom 𝐺 ⇒ ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Ass ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))) | ||
| Theorem | isexid 37833* | The predicate 𝐺 has a left and right identity element. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = dom dom 𝐺 ⇒ ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) | ||
| Syntax | cmagm 37834 | Extend class notation with the class of all magmas. |
| class Magma | ||
| Definition | df-mgmOLD 37835* | Obsolete version of df-mgm 18665 as of 3-Feb-2020. A magma is a binary internal operation. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) |
| ⊢ Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡} | ||
| Theorem | ismgmOLD 37836 | Obsolete version of ismgm 18666 as of 3-Feb-2020. The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑋 = dom dom 𝐺 ⇒ ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋)) | ||
| Theorem | clmgmOLD 37837 | Obsolete version of mgmcl 18668 as of 3-Feb-2020. Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑋 = dom dom 𝐺 ⇒ ⊢ ((𝐺 ∈ Magma ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) | ||
| Theorem | opidonOLD 37838 | Obsolete version of mndpfo 18782 as of 23-Jan-2020. An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑋 = dom dom 𝐺 ⇒ ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto→𝑋) | ||
| Theorem | rngopidOLD 37839 | Obsolete version of mndpfo 18782 as of 23-Jan-2020. Range of an operation with a left and right identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺) | ||
| Theorem | opidon2OLD 37840 | Obsolete version of mndpfo 18782 as of 23-Jan-2020. An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto→𝑋) | ||
| Theorem | isexid2 37841* | If 𝐺 ∈ (Magma ∩ ExId ), then it has a left and right identity element that belongs to the range of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) | ||
| Theorem | exidu1 37842* | Uniqueness of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) | ||
| Theorem | idrval 37843* | The value of the identity element. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝐴 → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))) | ||
| Theorem | iorlid 37844 | A magma right and left identity belongs to the underlying set of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) ⇒ ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 ∈ 𝑋) | ||
| Theorem | cmpidelt 37845 | A magma right and left identity element keeps the other elements unchanged. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) ⇒ ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴)) | ||
| Syntax | csem 37846 | Extend class notation with the class of all semigroups. |
| class SemiGrp | ||
| Definition | df-sgrOLD 37847 | Obsolete version of df-sgrp 18744 as of 3-Feb-2020. A semigroup is an associative magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) |
| ⊢ SemiGrp = (Magma ∩ Ass) | ||
| Theorem | smgrpismgmOLD 37848 | Obsolete version of sgrpmgm 18749 as of 3-Feb-2020. A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma) | ||
| Theorem | issmgrpOLD 37849* | Obsolete version of issgrp 18745 as of 3-Feb-2020. The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑋 = dom dom 𝐺 ⇒ ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ SemiGrp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))) | ||
| Theorem | smgrpmgm 37850 | A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) |
| ⊢ 𝑋 = dom dom 𝐺 ⇒ ⊢ (𝐺 ∈ SemiGrp → 𝐺:(𝑋 × 𝑋)⟶𝑋) | ||
| Theorem | smgrpassOLD 37851* | Obsolete version of sgrpass 18750 as of 3-Feb-2020. A semigroup is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑋 = dom dom 𝐺 ⇒ ⊢ (𝐺 ∈ SemiGrp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))) | ||
| Syntax | cmndo 37852 | Extend class notation with the class of all monoids. |
| class MndOp | ||
| Definition | df-mndo 37853 | A monoid is a semigroup with an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) |
| ⊢ MndOp = (SemiGrp ∩ ExId ) | ||
| Theorem | mndoissmgrpOLD 37854 | Obsolete version of mndsgrp 18765 as of 3-Feb-2020. A monoid is a semigroup. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp) | ||
| Theorem | mndoisexid 37855 | A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) |
| ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId ) | ||
| Theorem | mndoismgmOLD 37856 | Obsolete version of mndmgm 18766 as of 3-Feb-2020. A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ Magma) | ||
| Theorem | mndomgmid 37857 | A monoid is a magma with an identity element. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.) |
| ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId )) | ||
| Theorem | ismndo 37858* | The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = dom dom 𝐺 ⇒ ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp ↔ (𝐺 ∈ SemiGrp ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))) | ||
| Theorem | ismndo1 37859* | The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = dom dom 𝐺 ⇒ ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))) | ||
| Theorem | ismndo2 37860* | The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))) | ||
| Theorem | grpomndo 37861 | A group is a monoid. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
| ⊢ (𝐺 ∈ GrpOp → 𝐺 ∈ MndOp) | ||
| Theorem | exidcl 37862 | Closure of the binary operation of a magma with identity. (Contributed by Jeff Madsen, 16-Jun-2011.) |
| ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) | ||
| Theorem | exidreslem 37863* | Lemma for exidres 37864 and exidresid 37865. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.) |
| ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) & ⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌)) ⇒ ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))) | ||
| Theorem | exidres 37864 | The restriction of a binary operation with identity to a subset containing the identity has an identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.) |
| ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) & ⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌)) ⇒ ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → 𝐻 ∈ ExId ) | ||
| Theorem | exidresid 37865 | The restriction of a binary operation with identity to a subset containing the identity has the same identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.) |
| ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) & ⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌)) ⇒ ⊢ (((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) ∧ 𝐻 ∈ Magma) → (GId‘𝐻) = 𝑈) | ||
| Theorem | ablo4pnp 37866 | A commutative/associative law for Abelian groups. (Contributed by Jeff Madsen, 11-Jun-2010.) |
| ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ AbelOp ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐹 ∈ 𝑋))) → ((𝐴𝐺𝐵)𝐷(𝐶𝐺𝐹)) = ((𝐴𝐷𝐶)𝐺(𝐵𝐷𝐹))) | ||
| Theorem | grpoeqdivid 37867 | Two group elements are equal iff their quotient is the identity. (Contributed by Jeff Madsen, 6-Jan-2011.) |
| ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐺) & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 = 𝐵 ↔ (𝐴𝐷𝐵) = 𝑈)) | ||
| Theorem | grposnOLD 37868 | The group operation for the singleton group. Obsolete, use grp1 19077. instead. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ {⟨⟨𝐴, 𝐴⟩, 𝐴⟩} ∈ GrpOp | ||
| Syntax | cghomOLD 37869 | Obsolete version of cghm 19242 as of 15-Mar-2020. Extend class notation to include the class of group homomorphisms. (New usage is discouraged.) |
| class GrpOpHom | ||
| Definition | df-ghomOLD 37870* | Obsolete version of df-ghm 19243 as of 15-Mar-2020. Define the set of group homomorphisms from 𝑔 to ℎ. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) |
| ⊢ GrpOpHom = (𝑔 ∈ GrpOp, ℎ ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ℎ ∧ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝑔𝑦)))}) | ||
| Theorem | elghomlem1OLD 37871* | Obsolete as of 15-Mar-2020. Lemma for elghomOLD 37873. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)))} ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆) | ||
| Theorem | elghomlem2OLD 37872* | Obsolete as of 15-Mar-2020. Lemma for elghomOLD 37873. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)))} ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦))))) | ||
| Theorem | elghomOLD 37873* | Obsolete version of isghm 19245 as of 15-Mar-2020. Membership in the set of group homomorphisms from 𝐺 to 𝐻. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑊 = ran 𝐻 ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:𝑋⟶𝑊 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦))))) | ||
| Theorem | ghomlinOLD 37874 | Obsolete version of ghmlin 19251 as of 15-Mar-2020. Linearity of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐹‘𝐴)𝐻(𝐹‘𝐵)) = (𝐹‘(𝐴𝐺𝐵))) | ||
| Theorem | ghomidOLD 37875 | Obsolete version of ghmid 19252 as of 15-Mar-2020. A group homomorphism maps identity element to identity element. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑈 = (GId‘𝐺) & ⊢ 𝑇 = (GId‘𝐻) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑈) = 𝑇) | ||
| Theorem | ghomf 37876 | Mapping property of a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.) |
| ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑊 = ran 𝐻 ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋⟶𝑊) | ||
| Theorem | ghomco 37877 | The composition of two group homomorphisms is a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 27-Dec-2014.) |
| ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐾 ∈ GrpOp) ∧ (𝑆 ∈ (𝐺 GrpOpHom 𝐻) ∧ 𝑇 ∈ (𝐻 GrpOpHom 𝐾))) → (𝑇 ∘ 𝑆) ∈ (𝐺 GrpOpHom 𝐾)) | ||
| Theorem | ghomdiv 37878 | Group homomorphisms preserve division. (Contributed by Jeff Madsen, 16-Jun-2011.) |
| ⊢ 𝑋 = ran 𝐺 & ⊢ 𝐷 = ( /𝑔 ‘𝐺) & ⊢ 𝐶 = ( /𝑔 ‘𝐻) ⇒ ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐷𝐵)) = ((𝐹‘𝐴)𝐶(𝐹‘𝐵))) | ||
| Theorem | grpokerinj 37879 | A group homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.) |
| ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑊 = (GId‘𝐺) & ⊢ 𝑌 = ran 𝐻 & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋–1-1→𝑌 ↔ (◡𝐹 “ {𝑈}) = {𝑊})) | ||
| Syntax | crngo 37880 | Extend class notation with the class of all unital rings. |
| class RingOps | ||
| Definition | df-rngo 37881* | Define the class of all unital rings. (Contributed by Jeff Hankins, 21-Nov-2006.) (New usage is discouraged.) |
| ⊢ RingOps = {⟨𝑔, ℎ⟩ ∣ ((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦)))} | ||
| Theorem | relrngo 37882 | The class of all unital rings is a relation. (Contributed by FL, 31-Aug-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| ⊢ Rel RingOps | ||
| Theorem | isrngo 37883* | The predicate "is a (unital) ring." Definition of "ring with unit" in [Schechter] p. 187. (Contributed by Jeff Hankins, 21-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))) | ||
| Theorem | isrngod 37884* | Conditions that determine a ring. (Changed label from isringd 20304 to isrngod 37884-NM 2-Aug-2013.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| ⊢ (𝜑 → 𝐺 ∈ AbelOp) & ⊢ (𝜑 → 𝑋 = ran 𝐺) & ⊢ (𝜑 → 𝐻:(𝑋 × 𝑋)⟶𝑋) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) & ⊢ (𝜑 → 𝑈 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑈𝐻𝑦) = 𝑦) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑦𝐻𝑈) = 𝑦) ⇒ ⊢ (𝜑 → ⟨𝐺, 𝐻⟩ ∈ RingOps) | ||
| Theorem | rngoi 37885* | The properties of a unital ring. (Contributed by Steve Rodriguez, 8-Sep-2007.) (Proof shortened by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))) | ||
| Theorem | rngosm 37886 | Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋) | ||
| Theorem | rngocl 37887 | Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋) | ||
| Theorem | rngoid 37888* | The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴)) | ||
| Theorem | rngoideu 37889* | The unity element of a ring is unique. (Contributed by NM, 4-Apr-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) | ||
| Theorem | rngodi 37890 | Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶))) | ||
| Theorem | rngodir 37891 | Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(𝐵𝐻𝐶))) | ||
| Theorem | rngoass 37892 | Associative law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶))) | ||
| Theorem | rngo2 37893* | A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴)) | ||
| Theorem | rngoablo 37894 | A ring's addition operation is an Abelian group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝐺 = (1st ‘𝑅) ⇒ ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp) | ||
| Theorem | rngoablo2 37895 | In a unital ring the addition is an abelian group. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.) |
| ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ AbelOp) | ||
| Theorem | rngogrpo 37896 | A ring's addition operation is a group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.) |
| ⊢ 𝐺 = (1st ‘𝑅) ⇒ ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) | ||
| Theorem | rngone0 37897 | The base set of a ring is not empty. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → 𝑋 ≠ ∅) | ||
| Theorem | rngogcl 37898 | Closure law for the addition (group) operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) | ||
| Theorem | rngocom 37899 | The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)) | ||
| Theorem | rngoaass 37900 | The addition operation of a ring is associative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶))) | ||
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