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Type | Label | Description |
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Statement | ||
Definition | df-prjsp 39301* | Define the projective space function. In the bijection between 3D lines through the origin and points in the projective plane (see section comment), this is equivalent to making any two 3D points (excluding the origin) equivalent iff one is a multiple of another. This definition does not quite give all the properties needed, since the scalars of a left vector space can be "less dense" than the vectors (for example, equivocating rational multiples of real numbers). (Contributed by BJ and Steven Nguyen, 29-Apr-2023.) |
⊢ ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ⦋((Base‘𝑣) ∖ {(0g‘𝑣)}) / 𝑏⦌(𝑏 / {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))})) | ||
Theorem | prjspval 39302* | Value of the projective space function, which is also known as the projectivization of 𝑉. (Contributed by Steven Nguyen, 29-Apr-2023.) |
⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)}) & ⊢ · = ( ·𝑠 ‘𝑉) & ⊢ 𝑆 = (Scalar‘𝑉) & ⊢ 𝐾 = (Base‘𝑆) ⇒ ⊢ (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = (𝐵 / {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))})) | ||
Theorem | prjsprel 39303* | Utility theorem regarding the relation used in ℙ𝕣𝕠𝕛. (Contributed by Steven Nguyen, 29-Apr-2023.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))} ⇒ ⊢ (𝑋 ∼ 𝑌 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝐾 𝑋 = (𝑚 · 𝑌))) | ||
Theorem | prjspertr 39304* | The relation in ℙ𝕣𝕠𝕛 is transitive. (Contributed by Steven Nguyen, 1-May-2023.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))} & ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)}) & ⊢ 𝑆 = (Scalar‘𝑉) & ⊢ · = ( ·𝑠 ‘𝑉) & ⊢ 𝐾 = (Base‘𝑆) ⇒ ⊢ ((𝑉 ∈ LMod ∧ (𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍)) → 𝑋 ∼ 𝑍) | ||
Theorem | prjsperref 39305* | The relation in ℙ𝕣𝕠𝕛 is reflexive. (Contributed by Steven Nguyen, 30-Apr-2023.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))} & ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)}) & ⊢ 𝑆 = (Scalar‘𝑉) & ⊢ · = ( ·𝑠 ‘𝑉) & ⊢ 𝐾 = (Base‘𝑆) ⇒ ⊢ (𝑉 ∈ LMod → (𝑋 ∈ 𝐵 ↔ 𝑋 ∼ 𝑋)) | ||
Theorem | prjspersym 39306* | The relation in ℙ𝕣𝕠𝕛 is symmetric. (Contributed by Steven Nguyen, 1-May-2023.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))} & ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)}) & ⊢ 𝑆 = (Scalar‘𝑉) & ⊢ · = ( ·𝑠 ‘𝑉) & ⊢ 𝐾 = (Base‘𝑆) ⇒ ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∼ 𝑌) → 𝑌 ∼ 𝑋) | ||
Theorem | prjsper 39307* | The relation in ℙ𝕣𝕠𝕛 is an equivalence relation. (Contributed by Steven Nguyen, 1-May-2023.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))} & ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)}) & ⊢ 𝑆 = (Scalar‘𝑉) & ⊢ · = ( ·𝑠 ‘𝑉) & ⊢ 𝐾 = (Base‘𝑆) ⇒ ⊢ (𝑉 ∈ LVec → ∼ Er 𝐵) | ||
Theorem | prjspreln0 39308* | Two nonzero vectors are equivalent by a nonzero scalar. (Contributed by Steven Nguyen, 31-May-2023.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))} & ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)}) & ⊢ 𝑆 = (Scalar‘𝑉) & ⊢ · = ( ·𝑠 ‘𝑉) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ (𝑉 ∈ LVec → (𝑋 ∼ 𝑌 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ (𝐾 ∖ { 0 })𝑋 = (𝑚 · 𝑌)))) | ||
Theorem | prjspvs 39309* | A nonzero multiple of a vector is equivalent to the vector. (Contributed by Steven Nguyen, 6-Jun-2023.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))} & ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)}) & ⊢ 𝑆 = (Scalar‘𝑉) & ⊢ · = ( ·𝑠 ‘𝑉) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → (𝑁 · 𝑋) ∼ 𝑋) | ||
Theorem | prjsprellsp 39310* | Two vectors are equivalent iff their spans are equal. (Contributed by Steven Nguyen, 31-May-2023.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))} & ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)}) & ⊢ 𝑆 = (Scalar‘𝑉) & ⊢ · = ( ·𝑠 ‘𝑉) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑁 = (LSpan‘𝑉) ⇒ ⊢ ((𝑉 ∈ LVec ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ∼ 𝑌 ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) | ||
Theorem | prjspeclsp 39311* | The vectors equivalent to a vector 𝑋 are the nonzero vectors in the span of 𝑋. (Contributed by Steven Nguyen, 6-Jun-2023.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))} & ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)}) & ⊢ 𝑆 = (Scalar‘𝑉) & ⊢ · = ( ·𝑠 ‘𝑉) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑁 = (LSpan‘𝑉) ⇒ ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵) → [𝑋] ∼ = ((𝑁‘{𝑋}) ∖ {(0g‘𝑉)})) | ||
Theorem | prjspval2 39312* | Alternate definition of projective space. (Contributed by Steven Nguyen, 7-Jun-2023.) |
⊢ 0 = (0g‘𝑉) & ⊢ 𝐵 = ((Base‘𝑉) ∖ { 0 }) & ⊢ 𝑁 = (LSpan‘𝑉) ⇒ ⊢ (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = ∪ 𝑧 ∈ 𝐵 {((𝑁‘{𝑧}) ∖ { 0 })}) | ||
Syntax | cprjspn 39313 | Extend class notation with the n-dimensional projective space function. |
class ℙ𝕣𝕠𝕛n | ||
Definition | df-prjspn 39314* | Define the n-dimensional projective space function. A projective space of dimension 1 is a projective line, and a projective space of dimension 2 is a projective plane. Compare df-ehl 23989. This space is considered n-dimensional because the vector space (𝑘 freeLMod (0...𝑛)) is (n+1)-dimensional and the ℙ𝕣𝕠𝕛 function returns equivalence classes with respect to a linear (1-dimensional) relation. (Contributed by BJ and Steven Nguyen, 29-Apr-2023.) |
⊢ ℙ𝕣𝕠𝕛n = (𝑛 ∈ ℕ0, 𝑘 ∈ DivRing ↦ (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛)))) | ||
Theorem | prjspnval 39315 | Value of the n-dimensional projective space function. (Contributed by Steven Nguyen, 1-May-2023.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁)))) | ||
Theorem | prjspnval2 39316* | Value of the n-dimensional projective space function, expanded. (Contributed by Steven Nguyen, 15-Jul-2023.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))} & ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) & ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) & ⊢ 𝑆 = (Base‘𝐾) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (𝐵 / ∼ )) | ||
Theorem | 0prjspnlem 39317 | Lemma for 0prjspn 39319. The given unit vector is a nonzero vector. (Contributed by Steven Nguyen, 16-Jul-2023.) |
⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) & ⊢ 𝑊 = (𝐾 freeLMod (0...0)) & ⊢ 1 = ((𝐾 unitVec (0...0))‘0) ⇒ ⊢ (𝐾 ∈ DivRing → 1 ∈ 𝐵) | ||
Theorem | 0prjspnrel 39318* | In the zero-dimensional projective space, all vectors are equivalent to the unit vector. (Contributed by Steven Nguyen, 7-Jun-2023.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))} & ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑆 = (Base‘𝐾) & ⊢ 𝑊 = (𝐾 freeLMod (0...0)) & ⊢ 1 = ((𝐾 unitVec (0...0))‘0) ⇒ ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → 𝑋 ∼ 1 ) | ||
Theorem | 0prjspn 39319 | A zero-dimensional projective space has only 1 point. (Contributed by Steven Nguyen, 9-Jun-2023.) |
⊢ 𝑊 = (𝐾 freeLMod (0...0)) & ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) ⇒ ⊢ (𝐾 ∈ DivRing → (0ℙ𝕣𝕠𝕛n𝐾) = {𝐵}) | ||
Theorem | dffltz 39320* | Fermat's Last Theorem (FLT) for nonzero integers is equivalent to the original scope of natural numbers. The backwards direction takes (𝑎↑𝑛) + (𝑏↑𝑛) = (𝑐↑𝑛), and adds the negative of any negative term to both sides, thus creating the corresponding equation with only positive integers. There are six combinations of negativity, so the proof is particularly long. (Contributed by Steven Nguyen, 27-Feb-2023.) |
⊢ (∀𝑛 ∈ (ℤ≥‘3)∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛) ↔ ∀𝑛 ∈ (ℤ≥‘3)∀𝑎 ∈ (ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖ {0})∀𝑐 ∈ (ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛)) | ||
Theorem | fltne 39321 | If a counterexample to FLT exists, its addends are not equal. (Contributed by Steven Nguyen, 1-Jun-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | fltltc 39322 | (𝐶↑𝑁) is the largest term and therefore 𝐵 < 𝐶. (Contributed by Steven Nguyen, 22-Aug-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) ⇒ ⊢ (𝜑 → 𝐵 < 𝐶) | ||
Theorem | fltnltalem 39323 | Lemma for fltnlta 39324. A lower bound for 𝐴 based on pwdif 15223. (Contributed by Steven Nguyen, 22-Aug-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) ⇒ ⊢ (𝜑 → ((𝐶 − 𝐵) · ((𝐶↑(𝑁 − 1)) + ((𝑁 − 1) · (𝐵↑(𝑁 − 1))))) < (𝐴↑𝑁)) | ||
Theorem | fltnlta 39324 | 𝑁 is less than 𝐴. See https://www.youtu.be/EymVXkPWxyc for an outline. (Contributed by Steven Nguyen, 24-Aug-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝑁 < 𝐴) | ||
Theorem | binom2d 39325 | Deduction form of binom2. (Contributed by Igor Ieskov, 14-Dec-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵)↑2) = (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2))) | ||
Theorem | cu3addd 39326 | Cube of sum of three numbers. (Contributed by Igor Ieskov, 14-Dec-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (((𝐴 + 𝐵) + 𝐶)↑3) = (((((𝐴↑3) + (3 · ((𝐴↑2) · 𝐵))) + ((3 · (𝐴 · (𝐵↑2))) + (𝐵↑3))) + (((3 · ((𝐴↑2) · 𝐶)) + (((3 · 2) · (𝐴 · 𝐵)) · 𝐶)) + (3 · ((𝐵↑2) · 𝐶)))) + (((3 · (𝐴 · (𝐶↑2))) + (3 · (𝐵 · (𝐶↑2)))) + (𝐶↑3)))) | ||
Theorem | sqnegd 39327 | The square of the negative of a number. (Contributed by Igor Ieskov, 21-Jan-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (-𝐴↑2) = (𝐴↑2)) | ||
Theorem | negexpidd 39328 | The sum of a real number to the power of N and the negative of the number to the power of N equals zero if N is a nonnegative odd integer. (Contributed by Igor Ieskov, 21-Jan-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → ¬ 2 ∥ 𝑁) ⇒ ⊢ (𝜑 → ((𝐴↑𝑁) + (-𝐴↑𝑁)) = 0) | ||
Theorem | rexlimdv3d 39329* | An extended version of rexlimdvv 3293 to include three set variables. (Contributed by Igor Ieskov, 21-Jan-2024.) |
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓 → 𝜒)) | ||
Theorem | 3cubeslem1 39330 | Lemma for 3cubes 39336. (Contributed by Igor Ieskov, 22-Jan-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℚ) ⇒ ⊢ (𝜑 → 0 < (((𝐴 + 1)↑2) − 𝐴)) | ||
Theorem | 3cubeslem2 39331 | Lemma for 3cubes 39336. Used to show that the denominators in 3cubeslem4 39335 are nonzero. (Contributed by Igor Ieskov, 22-Jan-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℚ) ⇒ ⊢ (𝜑 → ¬ ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3) = 0) | ||
Theorem | 3cubeslem3l 39332 | Lemma for 3cubes 39336. (Contributed by Igor Ieskov, 22-Jan-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℚ) ⇒ ⊢ (𝜑 → (𝐴 · (((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3)↑3)) = (((𝐴↑7) · (3↑9)) + (((𝐴↑6) · (3↑9)) + (((𝐴↑5) · ((3↑8) + (3↑8))) + (((𝐴↑4) · (((3↑7) · 2) + (3↑6))) + (((𝐴↑3) · ((3↑6) + (3↑6))) + (((𝐴↑2) · (3↑5)) + (𝐴 · (3↑3))))))))) | ||
Theorem | 3cubeslem3r 39333 | Lemma for 3cubes 39336. (Contributed by Igor Ieskov, 22-Jan-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℚ) ⇒ ⊢ (𝜑 → ((((((3↑3) · (𝐴↑3)) − 1)↑3) + (((-((3↑3) · (𝐴↑3)) + ((3↑2) · 𝐴)) + 1)↑3)) + ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴))↑3)) = (((𝐴↑7) · (3↑9)) + (((𝐴↑6) · (3↑9)) + (((𝐴↑5) · ((3↑8) + (3↑8))) + (((𝐴↑4) · (((3↑7) · 2) + (3↑6))) + (((𝐴↑3) · ((3↑6) + (3↑6))) + (((𝐴↑2) · (3↑5)) + (𝐴 · (3↑3))))))))) | ||
Theorem | 3cubeslem3 39334 | Lemma for 3cubes 39336. (Contributed by Igor Ieskov, 22-Jan-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℚ) ⇒ ⊢ (𝜑 → (𝐴 · (((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3)↑3)) = ((((((3↑3) · (𝐴↑3)) − 1)↑3) + (((-((3↑3) · (𝐴↑3)) + ((3↑2) · 𝐴)) + 1)↑3)) + ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴))↑3))) | ||
Theorem | 3cubeslem4 39335 | Lemma for 3cubes 39336. This is Ryley's explicit formula for decomposing a rational 𝐴 into a sum of three rational cubes. (Contributed by Igor Ieskov, 22-Jan-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℚ) ⇒ ⊢ (𝜑 → 𝐴 = (((((((3↑3) · (𝐴↑3)) − 1) / ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3))↑3) + ((((-((3↑3) · (𝐴↑3)) + ((3↑2) · 𝐴)) + 1) / ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3))↑3)) + (((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) / ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3))↑3))) | ||
Theorem | 3cubes 39336* | Every rational number is a sum of three rational cubes. (S. Ryley, The Ladies' Diary 122 (1825), 35) (Contributed by Igor Ieskov, 22-Jan-2024.) |
⊢ (𝐴 ∈ ℚ ↔ ∃𝑎 ∈ ℚ ∃𝑏 ∈ ℚ ∃𝑐 ∈ ℚ 𝐴 = (((𝑎↑3) + (𝑏↑3)) + (𝑐↑3))) | ||
Theorem | rntrclfvOAI 39337 | The range of the transitive closure is equal to the range of the relation. (Contributed by OpenAI, 7-Jul-2020.) |
⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) = ran 𝑅) | ||
Theorem | moxfr 39338* | Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.) |
⊢ 𝐴 ∈ V & ⊢ ∃!𝑦 𝑥 = 𝐴 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) | ||
Theorem | imaiinfv 39339* | Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ (𝐹 “ 𝐵)) | ||
Theorem | elrfi 39340* | Elementhood in a set of relative finite intersections. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑣))) | ||
Theorem | elrfirn 39341* | Elementhood in a set of relative finite intersections of an indexed family of sets. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦)))) | ||
Theorem | elrfirn2 39342* | Elementhood in a set of relative finite intersections of an indexed family of sets (implicit). (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran (𝑦 ∈ 𝐼 ↦ 𝐶))) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 𝐶))) | ||
Theorem | cmpfiiin 39343* | In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑆 ∈ (Clsd‘𝐽)) & ⊢ ((𝜑 ∧ (𝑙 ⊆ 𝐼 ∧ 𝑙 ∈ Fin)) → (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆) ≠ ∅) ⇒ ⊢ (𝜑 → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) ≠ ∅) | ||
Theorem | ismrcd1 39344* | Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 16888), isotone (satisfies mrcss 16887), and idempotent (satisfies mrcidm 16890) has a collection of fixed points which is a Moore collection, and itself is the closure operator for that collection. This can be taken as an alternate definition for the closure operators. This is the first half, ismrcd2 39345 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵)) | ||
Theorem | ismrcd2 39345* | Second half of ismrcd1 39344. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → 𝐹 = (mrCls‘dom (𝐹 ∩ I ))) | ||
Theorem | istopclsd 39346* | A closure function which satisfies sscls 21664, clsidm 21675, cls0 21688, and clsun 33676 defines a (unique) topology which it is the closure function on. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)) & ⊢ (𝜑 → (𝐹‘∅) = ∅) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → (𝐹‘(𝑥 ∪ 𝑦)) = ((𝐹‘𝑥) ∪ (𝐹‘𝑦))) & ⊢ 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧)} ⇒ ⊢ (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ (cls‘𝐽) = 𝐹)) | ||
Theorem | ismrc 39347* | A function is a Moore closure operator iff it satisfies mrcssid 16888, mrcss 16887, and mrcidm 16890. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐹 ∈ (mrCls “ (Moore‘𝐵)) ↔ (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))))) | ||
Syntax | cnacs 39348 | Class of Noetherian closure systems. |
class NoeACS | ||
Definition | df-nacs 39349* | Define a closure system of Noetherian type (not standard terminology) as an algebraic system where all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)}) | ||
Theorem | isnacs 39350* | Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔))) | ||
Theorem | nacsfg 39351* | In a Noetherian-type closure system, all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑆 ∈ 𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔)) | ||
Theorem | isnacs2 39352 | Express Noetherian-type closure system with fewer quantifiers. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶)) | ||
Theorem | mrefg2 39353* | Slight variation on finite generation for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹‘𝑔))) | ||
Theorem | mrefg3 39354* | Slight variation on finite generation for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 ⊆ (𝐹‘𝑔))) | ||
Theorem | nacsacs 39355 | A closure system of Noetherian type is algebraic. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋)) | ||
Theorem | isnacs3 39356* | A choice-free order equivalent to the Noetherian condition on a closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠))) | ||
Theorem | incssnn0 39357* | Transitivity induction of subsets, lemma for nacsfix 39358. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ ((∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵)) | ||
Theorem | nacsfix 39358* | An increasing sequence of closed sets in a Noetherian-type closure system eventually fixates. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∃𝑦 ∈ ℕ0 ∀𝑧 ∈ (ℤ≥‘𝑦)(𝐹‘𝑧) = (𝐹‘𝑦)) | ||
Theorem | constmap 39359 |
A constant (represented without dummy variables) is an element of a
function set.
Note: In the following development, we will be quite often quantifying over functions and points in N-dimensional space (which are equivalent to functions from an "index set"). Many of the following theorems exist to transfer standard facts about functions to elements of function sets. (Contributed by Stefan O'Rear, 30-Aug-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}) ∈ (𝐶 ↑m 𝐴)) | ||
Theorem | mapco2g 39360 | Renaming indices in a tuple, with sethood as antecedents. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷) ∈ (𝐵 ↑m 𝐸)) | ||
Theorem | mapco2 39361 | Post-composition (renaming indices) of a mapping viewed as a point. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ 𝐸 ∈ V ⇒ ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷) ∈ (𝐵 ↑m 𝐸)) | ||
Theorem | mapfzcons 39362 | Extending a one-based mapping by adding a tuple at the end results in another mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ (𝐵 ↑m (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → (𝐴 ∪ {〈𝑀, 𝐶〉}) ∈ (𝐵 ↑m (1...𝑀))) | ||
Theorem | mapfzcons1 39363 | Recover prefix mapping from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) ⇒ ⊢ (𝐴 ∈ (𝐵 ↑m (1...𝑁)) → ((𝐴 ∪ {〈𝑀, 𝐶〉}) ↾ (1...𝑁)) = 𝐴) | ||
Theorem | mapfzcons1cl 39364 | A nonempty mapping has a prefix. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) ⇒ ⊢ (𝐴 ∈ (𝐵 ↑m (1...𝑀)) → (𝐴 ↾ (1...𝑁)) ∈ (𝐵 ↑m (1...𝑁))) | ||
Theorem | mapfzcons2 39365 | Recover added element from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) ⇒ ⊢ ((𝐴 ∈ (𝐵 ↑m (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → ((𝐴 ∪ {〈𝑀, 𝐶〉})‘𝑀) = 𝐶) | ||
Theorem | mptfcl 39366* | Interpret range of a maps-to notation as a constraint on the definition. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ ((𝑡 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶 → (𝑡 ∈ 𝐴 → 𝐵 ∈ 𝐶)) | ||
Syntax | cmzpcl 39367 | Extend class notation to include pre-polynomial rings. |
class mzPolyCld | ||
Syntax | cmzp 39368 | Extend class notation to include polynomial rings. |
class mzPoly | ||
Definition | df-mzpcl 39369* | Define the polynomially closed function rings over an arbitrary index set 𝑣. The set (mzPolyCld‘𝑣) contains all sets of functions from (ℤ ↑m 𝑣) to ℤ which include all constants and projections and are closed under addition and multiplication. This is a "temporary" set used to define the polynomial function ring itself (mzPoly‘𝑣); see df-mzp 39370. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ mzPolyCld = (𝑣 ∈ V ↦ {𝑝 ∈ 𝒫 (ℤ ↑m (ℤ ↑m 𝑣)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑣) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑣 (𝑥 ∈ (ℤ ↑m 𝑣) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘f + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑝))}) | ||
Definition | df-mzp 39370 | Polynomials over ℤ with an arbitrary index set, that is, the smallest ring of functions containing all constant functions and all projections. This is almost the most general reasonable definition; to reach full generality, we would need to be able to replace ZZ with an arbitrary (semi)ring (and a coordinate subring), but rings have not been defined yet. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ mzPoly = (𝑣 ∈ V ↦ ∩ (mzPolyCld‘𝑣)) | ||
Theorem | mzpclval 39371* | Substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ (𝑉 ∈ V → (mzPolyCld‘𝑉) = {𝑝 ∈ 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘f + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑝))}) | ||
Theorem | elmzpcl 39372* | Double substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ (𝑉 ∈ V → (𝑃 ∈ (mzPolyCld‘𝑉) ↔ (𝑃 ⊆ (ℤ ↑m (ℤ ↑m 𝑉)) ∧ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑃 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃))))) | ||
Theorem | mzpclall 39373 | The set of all functions with the signature of a polynomial is a polynomially closed set. This is a lemma to show that the intersection in df-mzp 39370 is well-defined. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ (𝑉 ∈ V → (ℤ ↑m (ℤ ↑m 𝑉)) ∈ (mzPolyCld‘𝑉)) | ||
Theorem | mzpcln0 39374 | Corrolary of mzpclall 39373: polynomially closed function sets are not empty. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ (𝑉 ∈ V → (mzPolyCld‘𝑉) ≠ ∅) | ||
Theorem | mzpcl1 39375 | Defining property 1 of a polynomially closed function set 𝑃: it contains all constant functions. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → ((ℤ ↑m 𝑉) × {𝐹}) ∈ 𝑃) | ||
Theorem | mzpcl2 39376* | Defining property 2 of a polynomially closed function set 𝑃: it contains all projections. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃) | ||
Theorem | mzpcl34 39377 | Defining properties 3 and 4 of a polynomially closed function set 𝑃: it is closed under pointwise addition and multiplication. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → ((𝐹 ∘f + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘f · 𝐺) ∈ 𝑃)) | ||
Theorem | mzpval 39378 | Value of the mzPoly function. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ (𝑉 ∈ V → (mzPoly‘𝑉) = ∩ (mzPolyCld‘𝑉)) | ||
Theorem | dmmzp 39379 | mzPoly is defined for all index sets which are sets. This is used with elfvdm 6702 to eliminate sethood antecedents. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ dom mzPoly = V | ||
Theorem | mzpincl 39380 | Polynomial closedness is a universal first-order property and passes to intersections. This is where the closure properties of the polynomial ring itself are proved. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ (𝑉 ∈ V → (mzPoly‘𝑉) ∈ (mzPolyCld‘𝑉)) | ||
Theorem | mzpconst 39381 | Constant functions are polynomial. See also mzpconstmpt 39386. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝑉 ∈ V ∧ 𝐶 ∈ ℤ) → ((ℤ ↑m 𝑉) × {𝐶}) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpf 39382 | A polynomial function is a function from the coordinate space to the integers. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ (𝐹 ∈ (mzPoly‘𝑉) → 𝐹:(ℤ ↑m 𝑉)⟶ℤ) | ||
Theorem | mzpproj 39383* | A projection function is polynomial. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝑉 ∈ V ∧ 𝑋 ∈ 𝑉) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑋)) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpadd 39384 | The pointwise sum of two polynomial functions is a polynomial function. See also mzpaddmpt 39387. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝐴 ∈ (mzPoly‘𝑉) ∧ 𝐵 ∈ (mzPoly‘𝑉)) → (𝐴 ∘f + 𝐵) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpmul 39385 | The pointwise product of two polynomial functions is a polynomial function. See also mzpmulmpt 39388. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝐴 ∈ (mzPoly‘𝑉) ∧ 𝐵 ∈ (mzPoly‘𝑉)) → (𝐴 ∘f · 𝐵) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpconstmpt 39386* | A constant function expressed in maps-to notation is polynomial. This theorem and the several that follow (mzpaddmpt 39387, mzpmulmpt 39388, mzpnegmpt 39390, mzpsubmpt 39389, mzpexpmpt 39391) can be used to build proofs that functions which are "manifestly polynomial", in the sense of being a maps-to containing constants, projections, and simple arithmetic operations, are actually polynomial functions. There is no mzpprojmpt because mzpproj 39383 is already expressed using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ ((𝑉 ∈ V ∧ 𝐶 ∈ ℤ) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐶) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpaddmpt 39387* | Sum of polynomial functions is polynomial. Maps-to version of mzpadd 39384. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ (((𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐴) ∈ (mzPoly‘𝑉) ∧ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐵) ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝐴 + 𝐵)) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpmulmpt 39388* | Product of polynomial functions is polynomial. Maps-to version of mzpmulmpt 39388. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ (((𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐴) ∈ (mzPoly‘𝑉) ∧ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐵) ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝐴 · 𝐵)) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpsubmpt 39389* | The difference of two polynomial functions is polynomial. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (((𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐴) ∈ (mzPoly‘𝑉) ∧ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐵) ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝐴 − 𝐵)) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpnegmpt 39390* | Negation of a polynomial function. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐴) ∈ (mzPoly‘𝑉) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ -𝐴) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpexpmpt 39391* | Raise a polynomial function to a (fixed) exponent. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ (((𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐴) ∈ (mzPoly‘𝑉) ∧ 𝐷 ∈ ℕ0) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝐴↑𝐷)) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpindd 39392* | "Structural" induction to prove properties of all polynomial functions. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → 𝜒) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑉) → 𝜃) & ⊢ ((𝜑 ∧ (𝑓:(ℤ ↑m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑m 𝑉)⟶ℤ ∧ 𝜂)) → 𝜁) & ⊢ ((𝜑 ∧ (𝑓:(ℤ ↑m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑m 𝑉)⟶ℤ ∧ 𝜂)) → 𝜎) & ⊢ (𝑥 = ((ℤ ↑m 𝑉) × {𝑓}) → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂)) & ⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁)) & ⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ (mzPoly‘𝑉)) → 𝜌) | ||
Theorem | mzpmfp 39393 | Relationship between multivariate Z-polynomials and general multivariate polynomial functions. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ (mzPoly‘𝐼) = ran (𝐼 eval ℤring) | ||
Theorem | mzpsubst 39394* | Substituting polynomials for the variables of a polynomial results in a polynomial. 𝐺 is expected to depend on 𝑦 and provide the polynomials which are being substituted. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) | ||
Theorem | mzprename 39395* | Simplified version of mzpsubst 39394 to simply relabel variables in a polynomial. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ 𝑅:𝑉⟶𝑊) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑥 ∘ 𝑅))) ∈ (mzPoly‘𝑊)) | ||
Theorem | mzpresrename 39396* | A polynomial is a polynomial over all larger index sets. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.) |
⊢ ((𝑊 ∈ V ∧ 𝑉 ⊆ 𝑊 ∧ 𝐹 ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑥 ↾ 𝑉))) ∈ (mzPoly‘𝑊)) | ||
Theorem | mzpcompact2lem 39397* | Lemma for mzpcompact2 39398. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∈ (mzPoly‘𝐵) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑐 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑐 ↾ 𝑎))))) | ||
Theorem | mzpcompact2 39398* | Polynomials are finitary objects and can only reference a finite number of variables, even if the index set is infinite. Thus, every polynomial can be expressed as a (uniquely minimal, although we do not prove that) polynomial on a finite number of variables, which is then extended by adding an arbitrary set of ignored variables. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
⊢ (𝐴 ∈ (mzPoly‘𝐵) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑐 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑐 ↾ 𝑎))))) | ||
Theorem | coeq0i 39399 | coeq0 6108 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (𝐴 ∘ 𝐵) = ∅) | ||
Theorem | fzsplit1nn0 39400 | Split a finite 1-based set of integers in the middle, allowing either end to be empty ((1...0)). (Contributed by Stefan O'Rear, 8-Oct-2014.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵))) |
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