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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | bj-isrvec2 37301 | The predicate "is a real vector space". (Contributed by BJ, 6-Jan-2024.) |
| ⊢ (𝜑 → (Scalar‘𝑉) = 𝐾) ⇒ ⊢ (𝜑 → (𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LVec ∧ 𝐾 = ℝfld))) | ||
| Theorem | bj-rvecssvec 37302 | Real vector spaces are vector spaces. (Contributed by BJ, 6-Jan-2024.) |
| ⊢ ℝ-Vec ⊆ LVec | ||
| Theorem | bj-rveccmod 37303 | Real vector spaces are subcomplex modules (elemental version). (Contributed by BJ, 6-Jan-2024.) |
| ⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂMod) | ||
| Theorem | bj-rvecsscmod 37304 | Real vector spaces are subcomplex modules. (Contributed by BJ, 6-Jan-2024.) |
| ⊢ ℝ-Vec ⊆ ℂMod | ||
| Theorem | bj-rvecsscvec 37305 | Real vector spaces are subcomplex vector spaces. (Contributed by BJ, 6-Jan-2024.) |
| ⊢ ℝ-Vec ⊆ ℂVec | ||
| Theorem | bj-rveccvec 37306 | Real vector spaces are subcomplex vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.) |
| ⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂVec) | ||
| Theorem | bj-rvecssabl 37307 | (The additive groups of) real vector spaces are commutative groups. (Contributed by BJ, 9-Jun-2019.) |
| ⊢ ℝ-Vec ⊆ Abel | ||
| Theorem | bj-rvecabl 37308 | (The additive groups of) real vector spaces are commutative groups (elemental version). (Contributed by BJ, 9-Jun-2019.) |
| ⊢ (𝐴 ∈ ℝ-Vec → 𝐴 ∈ Abel) | ||
Some lemmas to ease algebraic manipulations. | ||
| Theorem | bj-subcom 37309 | A consequence of commutativity of multiplication. (Contributed by BJ, 6-Jun-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) − (𝐵 · 𝐴)) = 0) | ||
| Theorem | bj-lineqi 37310 | Solution of a (scalar) linear equation. (Contributed by BJ, 6-Jun-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑌 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → ((𝐴 · 𝑋) + 𝐵) = 𝑌) ⇒ ⊢ (𝜑 → 𝑋 = ((𝑌 − 𝐵) / 𝐴)) | ||
Lemmas about barycentric coordinates. For the moment, this is limited to the one-dimensional case (complex line), where existence and uniqueness of barycentric coordinates are proved by bj-bary1 37313 (which computes them). It would be nice to prove the two-dimensional case (is it easier to use ad hoc computations, or Cramer formulas?), in order to do some planar geometry. | ||
| Theorem | bj-bary1lem 37311 | Lemma for bj-bary1 37313: expression for a barycenter of two points in one dimension (complex line). (Contributed by BJ, 6-Jun-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → 𝑋 = ((((𝐵 − 𝑋) / (𝐵 − 𝐴)) · 𝐴) + (((𝑋 − 𝐴) / (𝐵 − 𝐴)) · 𝐵))) | ||
| Theorem | bj-bary1lem1 37312 | Lemma for bj-bary1 37313: computation of one of the two barycentric coordinates of a barycenter of two points in one dimension (complex line). (Contributed by BJ, 6-Jun-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ ℂ) & ⊢ (𝜑 → 𝑇 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ (𝑆 + 𝑇) = 1) → 𝑇 = ((𝑋 − 𝐴) / (𝐵 − 𝐴)))) | ||
| Theorem | bj-bary1 37313 | Barycentric coordinates in one dimension (complex line). In the statement, 𝑋 is the barycenter of the two points 𝐴, 𝐵 with respective normalized coefficients 𝑆, 𝑇. (Contributed by BJ, 6-Jun-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ ℂ) & ⊢ (𝜑 → 𝑇 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ (𝑆 + 𝑇) = 1) ↔ (𝑆 = ((𝐵 − 𝑋) / (𝐵 − 𝐴)) ∧ 𝑇 = ((𝑋 − 𝐴) / (𝐵 − 𝐴))))) | ||
| Syntax | cend 37314 | Token for the monoid of endomorphisms. |
| class End | ||
| Definition | df-bj-end 37315* | The monoid of endomorphisms on an object of a category. (Contributed by BJ, 4-Apr-2024.) |
| ⊢ End = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ {〈(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)〉, 〈(+g‘ndx), (〈𝑥, 𝑥〉(comp‘𝑐)𝑥)〉})) | ||
| Theorem | bj-endval 37316 | Value of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.) (Proof modification is discouraged.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) ⇒ ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) = {〈(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)〉, 〈(+g‘ndx), (〈𝑋, 𝑋〉(comp‘𝐶)𝑋)〉}) | ||
| Theorem | bj-endbase 37317 | Base set of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.) (Proof modification is discouraged.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) ⇒ ⊢ (𝜑 → (Base‘((End ‘𝐶)‘𝑋)) = (𝑋(Hom ‘𝐶)𝑋)) | ||
| Theorem | bj-endcomp 37318 | Composition law of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.) (Proof modification is discouraged.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) ⇒ ⊢ (𝜑 → (+g‘((End ‘𝐶)‘𝑋)) = (〈𝑋, 𝑋〉(comp‘𝐶)𝑋)) | ||
| Theorem | bj-endmnd 37319 | The monoid of endomorphisms on an object of a category is a monoid. (Contributed by BJ, 5-Apr-2024.) (Proof modification is discouraged.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) ⇒ ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) ∈ Mnd) | ||
| Theorem | taupilem3 37320 | Lemma for tau-related theorems. (Contributed by Jim Kingdon, 16-Feb-2019.) |
| ⊢ (𝐴 ∈ (ℝ+ ∩ (◡cos “ {1})) ↔ (𝐴 ∈ ℝ+ ∧ (cos‘𝐴) = 1)) | ||
| Theorem | taupilemrplb 37321* | A set of positive reals has (in the reals) a lower bound. (Contributed by Jim Kingdon, 19-Feb-2019.) |
| ⊢ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)𝑥 ≤ 𝑦 | ||
| Theorem | taupilem1 37322 | Lemma for taupi 37324. A positive real whose cosine is one is at least 2 · π. (Contributed by Jim Kingdon, 19-Feb-2019.) |
| ⊢ ((𝐴 ∈ ℝ+ ∧ (cos‘𝐴) = 1) → (2 · π) ≤ 𝐴) | ||
| Theorem | taupilem2 37323 | Lemma for taupi 37324. The smallest positive real whose cosine is one is at most 2 · π. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.) |
| ⊢ τ ≤ (2 · π) | ||
| Theorem | taupi 37324 | Relationship between τ and π. This can be seen as connecting the ratio of a circle's circumference to its radius and the ratio of a circle's circumference to its diameter. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.) |
| ⊢ τ = (2 · π) | ||
| Theorem | dfgcd3 37325* | Alternate definition of the gcd operator. (Contributed by Jim Kingdon, 31-Dec-2021.) |
| ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (℩𝑑 ∈ ℕ0 ∀𝑧 ∈ ℤ (𝑧 ∥ 𝑑 ↔ (𝑧 ∥ 𝑀 ∧ 𝑧 ∥ 𝑁)))) | ||
| Theorem | irrdifflemf 37326 | Lemma for irrdiff 37327. The forward direction. (Contributed by Jim Kingdon, 20-May-2024.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → ¬ 𝐴 ∈ ℚ) & ⊢ (𝜑 → 𝑄 ∈ ℚ) & ⊢ (𝜑 → 𝑅 ∈ ℚ) & ⊢ (𝜑 → 𝑄 ≠ 𝑅) ⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) ≠ (abs‘(𝐴 − 𝑅))) | ||
| Theorem | irrdiff 37327* | The irrationals are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 19-May-2024.) |
| ⊢ (𝐴 ∈ ℝ → (¬ 𝐴 ∈ ℚ ↔ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟))))) | ||
| Theorem | iccioo01 37328 | The closed unit interval is equinumerous to the open unit interval. Based on a Mastodon post by Michael Kinyon. (Contributed by Jim Kingdon, 4-Jun-2024.) |
| ⊢ (0[,]1) ≈ (0(,)1) | ||
| Theorem | csbrecsg 37329 | Move class substitution in and out of recs. (Contributed by ML, 25-Oct-2020.) |
| ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌recs(𝐹) = recs(⦋𝐴 / 𝑥⦌𝐹)) | ||
| Theorem | csbrdgg 37330 | Move class substitution in and out of the recursive function generator. (Contributed by ML, 25-Oct-2020.) |
| ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌rec(𝐹, 𝐼) = rec(⦋𝐴 / 𝑥⦌𝐹, ⦋𝐴 / 𝑥⦌𝐼)) | ||
| Theorem | csboprabg 37331* | Move class substitution in and out of class abstractions of nested ordered pairs. (Contributed by ML, 25-Oct-2020.) |
| ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{〈〈𝑦, 𝑧〉, 𝑑〉 ∣ 𝜑} = {〈〈𝑦, 𝑧〉, 𝑑〉 ∣ [𝐴 / 𝑥]𝜑}) | ||
| Theorem | csbmpo123 37332* | Move class substitution in and out of maps-to notation for operations. (Contributed by ML, 25-Oct-2020.) |
| ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐷) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌, 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍 ↦ ⦋𝐴 / 𝑥⦌𝐷)) | ||
| Theorem | con1bii2 37333 | A contraposition inference. (Contributed by ML, 18-Oct-2020.) |
| ⊢ (¬ 𝜑 ↔ 𝜓) ⇒ ⊢ (𝜑 ↔ ¬ 𝜓) | ||
| Theorem | con2bii2 37334 | A contraposition inference. (Contributed by ML, 18-Oct-2020.) |
| ⊢ (𝜑 ↔ ¬ 𝜓) ⇒ ⊢ (¬ 𝜑 ↔ 𝜓) | ||
| Theorem | vtoclefex 37335* | Implicit substitution of a class for a setvar variable. (Contributed by ML, 17-Oct-2020.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝑥 = 𝐴 → 𝜑) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜑) | ||
| Theorem | rnmptsn 37336* | The range of a function mapping to singletons. (Contributed by ML, 15-Jul-2020.) |
| ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} | ||
| Theorem | f1omptsnlem 37337* | This is the core of the proof of f1omptsn 37338, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 15-Jul-2020.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) & ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} ⇒ ⊢ 𝐹:𝐴–1-1-onto→𝑅 | ||
| Theorem | f1omptsn 37338* | A function mapping to singletons is bijective onto a set of singletons. (Contributed by ML, 16-Jul-2020.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) & ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} ⇒ ⊢ 𝐹:𝐴–1-1-onto→𝑅 | ||
| Theorem | mptsnunlem 37339* | This is the core of the proof of mptsnun 37340, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) & ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} ⇒ ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵)) | ||
| Theorem | mptsnun 37340* | A class 𝐵 is equal to the union of the class of all singletons of elements of 𝐵. (Contributed by ML, 16-Jul-2020.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) & ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} ⇒ ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵)) | ||
| Theorem | dissneqlem 37341* | This is the core of the proof of dissneq 37342, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.) |
| ⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} ⇒ ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴) | ||
| Theorem | dissneq 37342* | Any topology that contains every single-point set is the discrete topology. (Contributed by ML, 16-Jul-2020.) |
| ⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} ⇒ ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴) | ||
| Theorem | exlimim 37343* | Closed form of exlimimd 37344. (Contributed by ML, 17-Jul-2020.) |
| ⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → 𝜓) | ||
| Theorem | exlimimd 37344* | Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.) |
| ⊢ (𝜑 → ∃𝑥𝜓) & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → 𝜒) | ||
| Theorem | exellim 37345* | Closed form of exellimddv 37346. See also exlimim 37343 for a more general theorem. (Contributed by ML, 17-Jul-2020.) |
| ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑) | ||
| Theorem | exellimddv 37346* | Eliminate an antecedent when the antecedent is elementhood, deduction version. See exellim 37345 for the closed form, which requires the use of a universal quantifier. (Contributed by ML, 17-Jul-2020.) |
| ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ⇒ ⊢ (𝜑 → 𝜓) | ||
| Theorem | topdifinfindis 37347* | Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is the trivial topology when 𝐴 is finite. (Contributed by ML, 14-Jul-2020.) |
| ⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} ⇒ ⊢ (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴}) | ||
| Theorem | topdifinffinlem 37348* | This is the core of the proof of topdifinffin 37349, but to avoid the distinct variables on the definition, we need to split this proof into two. (Contributed by ML, 17-Jul-2020.) |
| ⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} ⇒ ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin) | ||
| Theorem | topdifinffin 37349* | Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is a topology only if 𝐴 is finite. (Contributed by ML, 17-Jul-2020.) |
| ⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} ⇒ ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin) | ||
| Theorem | topdifinf 37350* | Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is a topology if and only if 𝐴 is finite, in which case it is the trivial topology. (Contributed by ML, 17-Jul-2020.) |
| ⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} ⇒ ⊢ ((𝑇 ∈ (TopOn‘𝐴) ↔ 𝐴 ∈ Fin) ∧ (𝑇 ∈ (TopOn‘𝐴) → 𝑇 = {∅, 𝐴})) | ||
| Theorem | topdifinfeq 37351* | Two different ways of defining the collection from Exercise 3 of [Munkres] p. 83. (Contributed by ML, 18-Jul-2020.) |
| ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ ((𝐴 ∖ 𝑥) = ∅ ∨ (𝐴 ∖ 𝑥) = 𝐴))} = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} | ||
| Theorem | icorempo 37352* | Closed-below, open-above intervals of reals. (Contributed by ML, 26-Jul-2020.) |
| ⊢ 𝐹 = ([,) ↾ (ℝ × ℝ)) ⇒ ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | ||
| Theorem | icoreresf 37353 | Closed-below, open-above intervals of reals map to subsets of reals. (Contributed by ML, 25-Jul-2020.) |
| ⊢ ([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ | ||
| Theorem | icoreval 37354* | Value of the closed-below, open-above interval function on reals. (Contributed by ML, 26-Jul-2020.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) = {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)}) | ||
| Theorem | icoreelrnab 37355* | Elementhood in the set of closed-below, open-above intervals of reals. (Contributed by ML, 27-Jul-2020.) |
| ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) | ||
| Theorem | isbasisrelowllem1 37356* | Lemma for isbasisrelowl 37359. (Contributed by ML, 27-Jul-2020.) |
| ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑)) → (𝑥 ∩ 𝑦) ∈ 𝐼) | ||
| Theorem | isbasisrelowllem2 37357* | Lemma for isbasisrelowl 37359. (Contributed by ML, 27-Jul-2020.) |
| ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) → (𝑥 ∩ 𝑦) ∈ 𝐼) | ||
| Theorem | icoreclin 37358* | The set of closed-below, open-above intervals of reals is closed under finite intersection. (Contributed by ML, 27-Jul-2020.) |
| ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∩ 𝑦) ∈ 𝐼) | ||
| Theorem | isbasisrelowl 37359 | The set of all closed-below, open-above intervals of reals form a basis. (Contributed by ML, 27-Jul-2020.) |
| ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ 𝐼 ∈ TopBases | ||
| Theorem | icoreunrn 37360 | The union of all closed-below, open-above intervals of reals is the set of reals. (Contributed by ML, 27-Jul-2020.) |
| ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ ℝ = ∪ 𝐼 | ||
| Theorem | istoprelowl 37361 | The set of all closed-below, open-above intervals of reals generate a topology on the reals. (Contributed by ML, 27-Jul-2020.) |
| ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ (topGen‘𝐼) ∈ (TopOn‘ℝ) | ||
| Theorem | icoreelrn 37362* | A class abstraction which is an element of the set of closed-below, open-above intervals of reals. (Contributed by ML, 1-Aug-2020.) |
| ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)} ∈ 𝐼) | ||
| Theorem | iooelexlt 37363* | An element of an open interval is not its smallest element. (Contributed by ML, 2-Aug-2020.) |
| ⊢ (𝑋 ∈ (𝐴(,)𝐵) → ∃𝑦 ∈ (𝐴(,)𝐵)𝑦 < 𝑋) | ||
| Theorem | relowlssretop 37364 | The lower limit topology on the reals is finer than the standard topology. (Contributed by ML, 1-Aug-2020.) |
| ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ (topGen‘ran (,)) ⊆ (topGen‘𝐼) | ||
| Theorem | relowlpssretop 37365 | The lower limit topology on the reals is strictly finer than the standard topology. (Contributed by ML, 2-Aug-2020.) |
| ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ (topGen‘ran (,)) ⊊ (topGen‘𝐼) | ||
| Theorem | sucneqond 37366 | Inequality of an ordinal set with its successor. Does not use the axiom of regularity. (Contributed by ML, 18-Oct-2020.) |
| ⊢ (𝜑 → 𝑋 = suc 𝑌) & ⊢ (𝜑 → 𝑌 ∈ On) ⇒ ⊢ (𝜑 → 𝑋 ≠ 𝑌) | ||
| Theorem | sucneqoni 37367 | Inequality of an ordinal set with its successor. Does not use the axiom of regularity. (Contributed by ML, 18-Oct-2020.) |
| ⊢ 𝑋 = suc 𝑌 & ⊢ 𝑌 ∈ On ⇒ ⊢ 𝑋 ≠ 𝑌 | ||
| Theorem | onsucuni3 37368 | If an ordinal number has a predecessor, then it is successor of that predecessor. (Contributed by ML, 17-Oct-2020.) |
| ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 = suc ∪ 𝐵) | ||
| Theorem | 1oequni2o 37369 | The ordinal number 1o is the predecessor of the ordinal number 2o. (Contributed by ML, 19-Oct-2020.) |
| ⊢ 1o = ∪ 2o | ||
| Theorem | rdgsucuni 37370 | If an ordinal number has a predecessor, the value of the recursive definition generator at that number in terms of its predecessor. (Contributed by ML, 17-Oct-2020.) |
| ⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (rec(𝐹, 𝐼)‘𝐵) = (𝐹‘(rec(𝐹, 𝐼)‘∪ 𝐵))) | ||
| Theorem | rdgeqoa 37371 | If a recursive function with an initial value 𝐴 at step 𝑁 is equal to itself with an initial value 𝐵 at step 𝑀, then every finite number of successor steps will also be equal. (Contributed by ML, 21-Oct-2020.) |
| ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋)))) | ||
| Theorem | elxp8 37372 | Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 8049. (Contributed by ML, 19-Oct-2020.) |
| ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) | ||
| Theorem | cbveud 37373* | Deduction used to change bound variables in an existential uniqueness quantifier, using implicit substitution. (Contributed by ML, 27-Mar-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑦𝜒)) | ||
| Theorem | cbvreud 37374* | Deduction used to change bound variables in a restricted existential uniqueness quantifier. (Contributed by ML, 27-Mar-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑦 ∈ 𝐴 𝜒)) | ||
| Theorem | difunieq 37375 | The difference of unions is a subset of the union of the difference. (Contributed by ML, 29-Mar-2021.) |
| ⊢ (∪ 𝐴 ∖ ∪ 𝐵) ⊆ ∪ (𝐴 ∖ 𝐵) | ||
| Theorem | inunissunidif 37376 | Theorem about subsets of the difference of unions. (Contributed by ML, 29-Mar-2021.) |
| ⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶))) | ||
| Theorem | rdgellim 37377 | Elementhood in a recursive definition at a limit ordinal. (Contributed by ML, 30-Mar-2022.) |
| ⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵))) | ||
| Theorem | rdglimss 37378 | A recursive definition at a limit ordinal is a superset of itself at any smaller ordinal. (Contributed by ML, 30-Mar-2022.) |
| ⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (rec(𝐹, 𝐴)‘𝐶) ⊆ (rec(𝐹, 𝐴)‘𝐵)) | ||
| Theorem | rdgssun 37379* | In a recursive definition where each step expands on the previous one using a union, every previous step is a subset of every later step. (Contributed by ML, 1-Apr-2022.) |
| ⊢ 𝐹 = (𝑤 ∈ V ↦ (𝑤 ∪ 𝐵)) & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ 𝑋) → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)) | ||
| Theorem | exrecfnlem 37380* | Lemma for exrecfn 37381. (Contributed by ML, 30-Mar-2022.) |
| ⊢ 𝐹 = (𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵))) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦 𝐵 ∈ 𝑊) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥)) | ||
| Theorem | exrecfn 37381* | Theorem about the existence of infinite recursive sets. 𝑦 should usually be free in 𝐵. (Contributed by ML, 30-Mar-2022.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦 𝐵 ∈ 𝑊) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥)) | ||
| Theorem | exrecfnpw 37382* | For any base set, a set which contains the powerset of all of its own elements exists. (Contributed by ML, 30-Mar-2022.) |
| ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ∈ 𝑥)) | ||
| Theorem | finorwe 37383 | If the Axiom of Infinity is denied, every total order is a well-order. The notion of a well-order cannot be usefully expressed without the Axiom of Infinity due to the inability to quantify over proper classes. (Contributed by ML, 5-Oct-2023.) |
| ⊢ (¬ ω ∈ V → ( < Or 𝐴 → < We 𝐴)) | ||
| Syntax | cfinxp 37384 | Extend the definition of a class to include Cartesian exponentiation. |
| class (𝑈↑↑𝑁) | ||
| Definition | df-finxp 37385* |
Define Cartesian exponentiation on a class.
Note that this definition is limited to finite exponents, since it is defined using nested ordered pairs. If tuples of infinite length are needed, or if they might be needed in the future, use df-ixp 8938 or df-map 8868 instead. The main advantage of this definition is that it integrates better with functions and relations. For example if 𝑅 is a subset of (𝐴↑↑2o), then df-br 5144 can be used on it, and df-fv 6569 can also be used, and so on. It's also worth keeping in mind that ((𝑈↑↑𝑀) × (𝑈↑↑𝑁)) is generally not equal to (𝑈↑↑(𝑀 +o 𝑁)). This definition is technical. Use finxp1o 37393 and finxpsuc 37399 for a more standard recursive experience. (Contributed by ML, 16-Oct-2020.) |
| ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈𝑁, 𝑦〉)‘𝑁))} | ||
| Theorem | dffinxpf 37386* | This theorem is the same as Definition df-finxp 37385, except that the large function is replaced by a class variable for brevity. (Contributed by ML, 24-Oct-2020.) |
| ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) ⇒ ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, 〈𝑁, 𝑦〉)‘𝑁))} | ||
| Theorem | finxpeq1 37387 | Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.) |
| ⊢ (𝑈 = 𝑉 → (𝑈↑↑𝑁) = (𝑉↑↑𝑁)) | ||
| Theorem | finxpeq2 37388 | Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.) |
| ⊢ (𝑀 = 𝑁 → (𝑈↑↑𝑀) = (𝑈↑↑𝑁)) | ||
| Theorem | csbfinxpg 37389* | Distribute proper substitution through Cartesian exponentiation. (Contributed by ML, 25-Oct-2020.) |
| ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑈↑↑𝑁) = (⦋𝐴 / 𝑥⦌𝑈↑↑⦋𝐴 / 𝑥⦌𝑁)) | ||
| Theorem | finxpreclem1 37390* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.) |
| ⊢ (𝑋 ∈ 𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉)))‘〈1o, 𝑋〉)) | ||
| Theorem | finxpreclem2 37391* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.) |
| ⊢ ((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉)))‘〈1o, 𝑋〉)) | ||
| Theorem | finxp0 37392 | The value of Cartesian exponentiation at zero. (Contributed by ML, 24-Oct-2020.) |
| ⊢ (𝑈↑↑∅) = ∅ | ||
| Theorem | finxp1o 37393 | The value of Cartesian exponentiation at one. (Contributed by ML, 17-Oct-2020.) |
| ⊢ (𝑈↑↑1o) = 𝑈 | ||
| Theorem | finxpreclem3 37394* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 20-Oct-2020.) |
| ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) ⇒ ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 〈∪ 𝑁, (1st ‘𝑋)〉 = (𝐹‘〈𝑁, 𝑋〉)) | ||
| Theorem | finxpreclem4 37395* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 23-Oct-2020.) |
| ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) ⇒ ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, 〈𝑁, 𝑦〉)‘𝑁) = (rec(𝐹, 〈∪ 𝑁, (1st ‘𝑦)〉)‘∪ 𝑁)) | ||
| Theorem | finxpreclem5 37396* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.) |
| ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) ⇒ ⊢ ((𝑛 ∈ ω ∧ 1o ∈ 𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘〈𝑛, 𝑥〉) = 〈𝑛, 𝑥〉)) | ||
| Theorem | finxpreclem6 37397* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.) |
| ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) ⇒ ⊢ ((𝑁 ∈ ω ∧ 1o ∈ 𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)) | ||
| Theorem | finxpsuclem 37398* | Lemma for finxpsuc 37399. (Contributed by ML, 24-Oct-2020.) |
| ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) ⇒ ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈)) | ||
| Theorem | finxpsuc 37399 | The value of Cartesian exponentiation at a successor. (Contributed by ML, 24-Oct-2020.) |
| ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈)) | ||
| Theorem | finxp2o 37400 | The value of Cartesian exponentiation at two. (Contributed by ML, 19-Oct-2020.) |
| ⊢ (𝑈↑↑2o) = (𝑈 × 𝑈) | ||
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