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Type | Label | Description |
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Statement | ||
Theorem | eldioph2lem1 39701* | Lemma for eldioph2 39703. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ∃𝑑 ∈ (ℤ≥‘𝑁)∃𝑒 ∈ V (𝑒:(1...𝑑)–1-1-onto→𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) | ||
Theorem | eldioph2lem2 39702* | Lemma for eldioph2 39703. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
⊢ (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ≥‘𝑁))) → ∃𝑐(𝑐:(1...𝐴)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) | ||
Theorem | eldioph2 39703* | Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 39693. (Contributed by Stefan O'Rear, 8-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁)) | ||
Theorem | eldioph2b 39704* | While Diophantine sets were defined to have a finite number of witness variables consequtively following the observable variables, this is not necessary; they can equivalently be taken to use any witness set (𝑆 ∖ (1...𝑁)). For instance, in diophin 39713 we use this to take the two input sets to have disjoint witness sets. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) | ||
Theorem | eldiophelnn0 39705 | Remove antecedent on 𝐵 from Diophantine set constructors. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (𝐴 ∈ (Dioph‘𝐵) → 𝐵 ∈ ℕ0) | ||
Theorem | eldioph3b 39706* | Define Diophantine sets in terms of polynomials with variables indexed by ℕ. This avoids a quantifier over the number of witness variables and will be easier to use than eldiophb 39698 in most cases. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (𝐴 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘ℕ)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) | ||
Theorem | eldioph3 39707* | Inference version of eldioph3b 39706 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘ℕ)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁)) | ||
Theorem | ellz1 39708 | Membership in a lower set of integers. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
⊢ (𝐵 ∈ ℤ → (𝐴 ∈ (ℤ ∖ (ℤ≥‘(𝐵 + 1))) ↔ (𝐴 ∈ ℤ ∧ 𝐴 ≤ 𝐵))) | ||
Theorem | lzunuz 39709 | The union of a lower set of integers and an upper set of integers which abut or overlap is all of the integers. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≤ (𝐴 + 1)) → ((ℤ ∖ (ℤ≥‘(𝐴 + 1))) ∪ (ℤ≥‘𝐵)) = ℤ) | ||
Theorem | fz1eqin 39710 | Express a one-based finite range as the intersection of lower integers with ℕ. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
⊢ (𝑁 ∈ ℕ0 → (1...𝑁) = ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ)) | ||
Theorem | lzenom 39711 | Lower integers are countably infinite. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (𝑁 ∈ ℤ → (ℤ ∖ (ℤ≥‘(𝑁 + 1))) ≈ ω) | ||
Theorem | elmapresaunres2 39712 | fresaunres2 6524 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺) | ||
Theorem | diophin 39713 | If two sets are Diophantine, so is their intersection. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∩ 𝐵) ∈ (Dioph‘𝑁)) | ||
Theorem | diophun 39714 | If two sets are Diophantine, so is their union. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁)) | ||
Theorem | eldiophss 39715 | Diophantine sets are sets of tuples of nonnegative integers. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ (𝐴 ∈ (Dioph‘𝐵) → 𝐴 ⊆ (ℕ0 ↑m (1...𝐵))) | ||
Theorem | diophrex 39716* | Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑡 ∣ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))} ∈ (Dioph‘𝑁)) | ||
Theorem | eq0rabdioph 39717* | This is the first of a number of theorems which allow sets to be proven Diophantine by syntactic induction, and models the correspondence between Diophantine sets and monotone existential first-order logic. This first theorem shows that the zero set of an implicit polynomial is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 = 0} ∈ (Dioph‘𝑁)) | ||
Theorem | eqrabdioph 39718* | Diophantine set builder for equality of polynomial expressions. Note that the two expressions need not be nonnegative; only variables are so constrained. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 = 𝐵} ∈ (Dioph‘𝑁)) | ||
Theorem | 0dioph 39719 | The null set is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (𝐴 ∈ ℕ0 → ∅ ∈ (Dioph‘𝐴)) | ||
Theorem | vdioph 39720 | The "universal" set (as large as possible given eldiophss 39715) is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (𝐴 ∈ ℕ0 → (ℕ0 ↑m (1...𝐴)) ∈ (Dioph‘𝐴)) | ||
Theorem | anrabdioph 39721* | Diophantine set builder for conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (({𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝜑 ∧ 𝜓)} ∈ (Dioph‘𝑁)) | ||
Theorem | orrabdioph 39722* | Diophantine set builder for disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (({𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝜑 ∨ 𝜓)} ∈ (Dioph‘𝑁)) | ||
Theorem | 3anrabdioph 39723* | Diophantine set builder for ternary conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (({𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜒} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝜑 ∧ 𝜓 ∧ 𝜒)} ∈ (Dioph‘𝑁)) | ||
Theorem | 3orrabdioph 39724* | Diophantine set builder for ternary disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (({𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜒} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝜑 ∨ 𝜓 ∨ 𝜒)} ∈ (Dioph‘𝑁)) | ||
Theorem | 2sbcrex 39725* | Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.) |
⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑) | ||
Theorem | sbcrexgOLD 39726* | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcrex 3804 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) | ||
Theorem | 2sbcrexOLD 39727* | Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7177 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑) | ||
Theorem | sbc2rex 39728* | Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.) |
⊢ ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎]𝜑) | ||
Theorem | sbc2rexgOLD 39729* | Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7177 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎]𝜑)) | ||
Theorem | sbc4rex 39730* | Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.) |
⊢ ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 [𝐴 / 𝑎]𝜑) | ||
Theorem | sbc4rexgOLD 39731* | Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7177 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 [𝐴 / 𝑎]𝜑)) | ||
Theorem | sbcrot3 39732* | Rotate a sequence of three explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.) |
⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑) | ||
Theorem | sbcrot5 39733* | Rotate a sequence of five explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.) |
⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑) | ||
Theorem | sbccomieg 39734* | Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.) |
⊢ (𝑎 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑 ↔ [𝐶 / 𝑏][𝐴 / 𝑎]𝜑) | ||
Theorem | rexrabdioph 39735* | Diophantine set builder for existential quantification. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ 𝑀 = (𝑁 + 1) & ⊢ (𝑣 = (𝑡‘𝑀) → (𝜓 ↔ 𝜒)) & ⊢ (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝜒 ↔ 𝜑)) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} ∈ (Dioph‘𝑁)) | ||
Theorem | rexfrabdioph 39736* | Diophantine set builder for existential quantifier, explicit substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑀)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁)) | ||
Theorem | 2rexfrabdioph 39737* | Diophantine set builder for existential quantifier, explicit substitution, two variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) & ⊢ 𝐿 = (𝑀 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 ∃𝑤 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁)) | ||
Theorem | 3rexfrabdioph 39738* | Diophantine set builder for existential quantifier, explicit substitution, two variables. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) & ⊢ 𝐿 = (𝑀 + 1) & ⊢ 𝐾 = (𝐿 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 ∃𝑤 ∈ ℕ0 ∃𝑥 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁)) | ||
Theorem | 4rexfrabdioph 39739* | Diophantine set builder for existential quantifier, explicit substitution, four variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) & ⊢ 𝐿 = (𝑀 + 1) & ⊢ 𝐾 = (𝐿 + 1) & ⊢ 𝐽 = (𝐾 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐽)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑡‘𝐽) / 𝑦]𝜑} ∈ (Dioph‘𝐽)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 ∃𝑤 ∈ ℕ0 ∃𝑥 ∈ ℕ0 ∃𝑦 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁)) | ||
Theorem | 6rexfrabdioph 39740* | Diophantine set builder for existential quantifier, explicit substitution, six variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) & ⊢ 𝐿 = (𝑀 + 1) & ⊢ 𝐾 = (𝐿 + 1) & ⊢ 𝐽 = (𝐾 + 1) & ⊢ 𝐼 = (𝐽 + 1) & ⊢ 𝐻 = (𝐼 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐻)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑡‘𝐽) / 𝑦][(𝑡‘𝐼) / 𝑧][(𝑡‘𝐻) / 𝑝]𝜑} ∈ (Dioph‘𝐻)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 ∃𝑤 ∈ ℕ0 ∃𝑥 ∈ ℕ0 ∃𝑦 ∈ ℕ0 ∃𝑧 ∈ ℕ0 ∃𝑝 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁)) | ||
Theorem | 7rexfrabdioph 39741* | Diophantine set builder for existential quantifier, explicit substitution, seven variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) & ⊢ 𝐿 = (𝑀 + 1) & ⊢ 𝐾 = (𝐿 + 1) & ⊢ 𝐽 = (𝐾 + 1) & ⊢ 𝐼 = (𝐽 + 1) & ⊢ 𝐻 = (𝐼 + 1) & ⊢ 𝐺 = (𝐻 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐺)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑡‘𝐽) / 𝑦][(𝑡‘𝐼) / 𝑧][(𝑡‘𝐻) / 𝑝][(𝑡‘𝐺) / 𝑞]𝜑} ∈ (Dioph‘𝐺)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 ∃𝑤 ∈ ℕ0 ∃𝑥 ∈ ℕ0 ∃𝑦 ∈ ℕ0 ∃𝑧 ∈ ℕ0 ∃𝑝 ∈ ℕ0 ∃𝑞 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁)) | ||
Theorem | rabdiophlem1 39742* | Lemma for arithmetic diophantine sets. Convert polynomial-ness of an expression into a constraint suitable for ralimi 3128. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐴 ∈ ℤ) | ||
Theorem | rabdiophlem2 39743* | Lemma for arithmetic diophantine sets. Reuse a polynomial expression under a new quantifier. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ 𝑀 = (𝑁 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑m (1...𝑀)) ↦ ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴) ∈ (mzPoly‘(1...𝑀))) | ||
Theorem | elnn0rabdioph 39744* | Diophantine set builder for nonnegativity constraints. The first builder which uses a witness variable internally; an expression is nonnegative if there is a nonnegative integer equal to it. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} ∈ (Dioph‘𝑁)) | ||
Theorem | rexzrexnn0 39745* | Rewrite an existential quantification restricted to integers into an existential quantification restricted to naturals. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (∃𝑥 ∈ ℤ 𝜑 ↔ ∃𝑦 ∈ ℕ0 (𝜓 ∨ 𝜒)) | ||
Theorem | lerabdioph 39746* | Diophantine set builder for the "less than or equal to" relation. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ≤ 𝐵} ∈ (Dioph‘𝑁)) | ||
Theorem | eluzrabdioph 39747* | Diophantine set builder for membership in a fixed upper set of integers. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ (ℤ≥‘𝑀)} ∈ (Dioph‘𝑁)) | ||
Theorem | elnnrabdioph 39748* | Diophantine set builder for positivity. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ ℕ} ∈ (Dioph‘𝑁)) | ||
Theorem | ltrabdioph 39749* | Diophantine set builder for the strict less than relation. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 < 𝐵} ∈ (Dioph‘𝑁)) | ||
Theorem | nerabdioph 39750* | Diophantine set builder for inequality. This not quite trivial theorem touches on something important; Diophantine sets are not closed under negation, but they contain an important subclass that is, namely the recursive sets. With this theorem and De Morgan's laws, all quantifier-free formulas can be negated. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ≠ 𝐵} ∈ (Dioph‘𝑁)) | ||
Theorem | dvdsrabdioph 39751* | Divisibility is a Diophantine relation. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∥ 𝐵} ∈ (Dioph‘𝑁)) | ||
Theorem | eldioph4b 39752* | Membership in Dioph expressed using a quantified union to add witness variables instead of a restriction to remove them. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
⊢ 𝑊 ∈ V & ⊢ ¬ 𝑊 ∈ Fin & ⊢ (𝑊 ∩ ℕ) = ∅ ⇒ ⊢ (𝑆 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0})) | ||
Theorem | eldioph4i 39753* | Forward-only version of eldioph4b 39752. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
⊢ 𝑊 ∈ V & ⊢ ¬ 𝑊 ∈ Fin & ⊢ (𝑊 ∩ ℕ) = ∅ ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑃‘(𝑡 ∪ 𝑤)) = 0} ∈ (Dioph‘𝑁)) | ||
Theorem | diophren 39754* | Change variables in a Diophantine set, using class notation. This allows already proved Diophantine sets to be reused in contexts with more variables. (Contributed by Stefan O'Rear, 16-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.) |
⊢ ((𝑆 ∈ (Dioph‘𝑁) ∧ 𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)) | ||
Theorem | rabrenfdioph 39755* | Change variable numbers in a Diophantine class abstraction using explicit substitution. (Contributed by Stefan O'Rear, 17-Oct-2014.) |
⊢ ((𝐵 ∈ ℕ0 ∧ 𝐹:(1...𝐴)⟶(1...𝐵) ∧ {𝑎 ∈ (ℕ0 ↑m (1...𝐴)) ∣ 𝜑} ∈ (Dioph‘𝐴)) → {𝑏 ∈ (ℕ0 ↑m (1...𝐵)) ∣ [(𝑏 ∘ 𝐹) / 𝑎]𝜑} ∈ (Dioph‘𝐵)) | ||
Theorem | rabren3dioph 39756* | Change variable numbers in a 3-variable Diophantine class abstraction. (Contributed by Stefan O'Rear, 17-Oct-2014.) |
⊢ (((𝑎‘1) = (𝑏‘𝑋) ∧ (𝑎‘2) = (𝑏‘𝑌) ∧ (𝑎‘3) = (𝑏‘𝑍)) → (𝜑 ↔ 𝜓)) & ⊢ 𝑋 ∈ (1...𝑁) & ⊢ 𝑌 ∈ (1...𝑁) & ⊢ 𝑍 ∈ (1...𝑁) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑎 ∈ (ℕ0 ↑m (1...3)) ∣ 𝜑} ∈ (Dioph‘3)) → {𝑏 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁)) | ||
Theorem | fphpd 39757* | Pigeonhole principle expressed with implicit substitution. If the range is smaller than the domain, two inputs must be mapped to the same output. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ (𝜑 → 𝐵 ≺ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ 𝐶 = 𝐷)) | ||
Theorem | fphpdo 39758* | Pigeonhole principle for sets of real numbers with implicit output reordering. (Contributed by Stefan O'Rear, 12-Sep-2014.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐵 ≺ 𝐴) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ (𝑧 = 𝑥 → 𝐶 = 𝐷) & ⊢ (𝑧 = 𝑦 → 𝐶 = 𝐸) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 < 𝑦 ∧ 𝐷 = 𝐸)) | ||
Theorem | ctbnfien 39759 | An infinite subset of a countable set is countable, without using choice. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ (((𝑋 ≈ ω ∧ 𝑌 ≈ ω) ∧ (𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin)) → 𝐴 ≈ 𝑌) | ||
Theorem | fiphp3d 39760* | Infinite pigeonhole principle for partitioning an infinite set between finitely many buckets. (Contributed by Stefan O'Rear, 18-Oct-2014.) |
⊢ (𝜑 → 𝐴 ≈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 {𝑥 ∈ 𝐴 ∣ 𝐷 = 𝑦} ≈ ℕ) | ||
Theorem | rencldnfilem 39761* | Lemma for rencldnfi 39762. (Contributed by Stefan O'Rear, 18-Oct-2014.) |
⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 ≠ ∅ ∧ ¬ 𝐵 ∈ 𝐴)) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐴 (abs‘(𝑦 − 𝐵)) < 𝑥) → ¬ 𝐴 ∈ Fin) | ||
Theorem | rencldnfi 39762* | A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem 39761 using infima; this theorem removes the requirement that A be nonempty. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ 𝐴) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐴 (abs‘(𝑦 − 𝐵)) < 𝑥) → ¬ 𝐴 ∈ Fin) | ||
Theorem | irrapxlem1 39763* | Lemma for irrapx1 39769. Divides the unit interval into 𝐵 half-open sections and using the pigeonhole principle fphpdo 39758 finds two multiples of 𝐴 in the same section mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ (0...𝐵)∃𝑦 ∈ (0...𝐵)(𝑥 < 𝑦 ∧ (⌊‘(𝐵 · ((𝐴 · 𝑥) mod 1))) = (⌊‘(𝐵 · ((𝐴 · 𝑦) mod 1))))) | ||
Theorem | irrapxlem2 39764* | Lemma for irrapx1 39769. Two multiples in the same bucket means they are very close mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ (0...𝐵)∃𝑦 ∈ (0...𝐵)(𝑥 < 𝑦 ∧ (abs‘(((𝐴 · 𝑥) mod 1) − ((𝐴 · 𝑦) mod 1))) < (1 / 𝐵))) | ||
Theorem | irrapxlem3 39765* | Lemma for irrapx1 39769. By subtraction, there is a multiple very close to an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ (1...𝐵)∃𝑦 ∈ ℕ0 (abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / 𝐵)) | ||
Theorem | irrapxlem4 39766* | Lemma for irrapx1 39769. Eliminate ranges, use positivity of the input to force positivity of the output by increasing 𝐵 as needed. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ (abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / if(𝑥 ≤ 𝐵, 𝐵, 𝑥))) | ||
Theorem | irrapxlem5 39767* | Lemma for irrapx1 39769. Switching to real intervals and fraction syntax. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℚ (0 < 𝑥 ∧ (abs‘(𝑥 − 𝐴)) < 𝐵 ∧ (abs‘(𝑥 − 𝐴)) < ((denom‘𝑥)↑-2))) | ||
Theorem | irrapxlem6 39768* | Lemma for irrapx1 39769. Explicit description of a non-closed set. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} (abs‘(𝑥 − 𝐴)) < 𝐵) | ||
Theorem | irrapx1 39769* | Dirichlet's approximation theorem. Every positive irrational number has infinitely many rational approximations which are closer than the inverse squares of their reduced denominators. Lemma 61 in [vandenDries] p. 42. (Contributed by Stefan O'Rear, 14-Sep-2014.) |
⊢ (𝐴 ∈ (ℝ+ ∖ ℚ) → {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦 − 𝐴)) < ((denom‘𝑦)↑-2))} ≈ ℕ) | ||
Theorem | pellexlem1 39770 | Lemma for pellex 39776. Arithmetical core of pellexlem3, norm lower bound. This begins Dirichlet's proof of the Pell equation solution existence; the proof here follows theorem 62 of [vandenDries] p. 43. (Contributed by Stefan O'Rear, 14-Sep-2014.) |
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ¬ (√‘𝐷) ∈ ℚ) → ((𝐴↑2) − (𝐷 · (𝐵↑2))) ≠ 0) | ||
Theorem | pellexlem2 39771 | Lemma for pellex 39776. Arithmetical core of pellexlem3, norm upper bound. (Contributed by Stefan O'Rear, 14-Sep-2014.) |
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) < (1 + (2 · (√‘𝐷)))) | ||
Theorem | pellexlem3 39772* | Lemma for pellex 39776. To each good rational approximation of (√‘𝐷), there exists a near-solution. (Contributed by Stefan O'Rear, 14-Sep-2014.) |
⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} ≼ {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))}) | ||
Theorem | pellexlem4 39773* | Lemma for pellex 39776. Invoking irrapx1 39769, we have infinitely many near-solutions. (Contributed by Stefan O'Rear, 14-Sep-2014.) |
⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ≈ ℕ) | ||
Theorem | pellexlem5 39774* | Lemma for pellex 39776. Invoking fiphp3d 39760, we have infinitely many near-solutions for some specific norm. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑥 ∈ ℤ (𝑥 ≠ 0 ∧ {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ)) | ||
Theorem | pellexlem6 39775* | Lemma for pellex 39776. Doing a field division between near solutions get us to norm 1, and the modularity constraint ensures we still have an integer. Returning NN guarantees that we are not returning the trivial solution (1,0). We are not explicitly defining the Pell-field, Pell-ring, and Pell-norm explicitly because after this construction is done we will never use them. This is mostly basic algebraic number theory and could be simplified if a generic framework for that were in place. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℕ) & ⊢ (𝜑 → ¬ (√‘𝐷) ∈ ℚ) & ⊢ (𝜑 → 𝐸 ∈ ℕ) & ⊢ (𝜑 → 𝐹 ∈ ℕ) & ⊢ (𝜑 → ¬ (𝐴 = 𝐸 ∧ 𝐵 = 𝐹)) & ⊢ (𝜑 → 𝐶 ≠ 0) & ⊢ (𝜑 → ((𝐴↑2) − (𝐷 · (𝐵↑2))) = 𝐶) & ⊢ (𝜑 → ((𝐸↑2) − (𝐷 · (𝐹↑2))) = 𝐶) & ⊢ (𝜑 → (𝐴 mod (abs‘𝐶)) = (𝐸 mod (abs‘𝐶))) & ⊢ (𝜑 → (𝐵 mod (abs‘𝐶)) = (𝐹 mod (abs‘𝐶))) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1) | ||
Theorem | pellex 39776* | Every Pell equation has a nontrivial solution. Theorem 62 in [vandenDries] p. 43. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1) | ||
Syntax | csquarenn 39777 | Extend class notation to include the set of square positive integers. |
class ◻NN | ||
Syntax | cpell1qr 39778 | Extend class notation to include the class of quadrant-1 Pell solutions. |
class Pell1QR | ||
Syntax | cpell1234qr 39779 | Extend class notation to include the class of any-quadrant Pell solutions. |
class Pell1234QR | ||
Syntax | cpell14qr 39780 | Extend class notation to include the class of positive Pell solutions. |
class Pell14QR | ||
Syntax | cpellfund 39781 | Extend class notation to include the Pell-equation fundamental solution function. |
class PellFund | ||
Definition | df-squarenn 39782 | Define the set of square positive integers. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
⊢ ◻NN = {𝑥 ∈ ℕ ∣ (√‘𝑥) ∈ ℚ} | ||
Definition | df-pell1qr 39783* | Define the solutions of a Pell equation in the first quadrant. To avoid pair pain, we represent this via the canonical embedding into the reals. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
⊢ Pell1QR = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)}) | ||
Definition | df-pell14qr 39784* | Define the positive solutions of a Pell equation. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
⊢ Pell14QR = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)}) | ||
Definition | df-pell1234qr 39785* | Define the general solutions of a Pell equation. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
⊢ Pell1234QR = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)}) | ||
Definition | df-pellfund 39786* | A function mapping Pell discriminants to the corresponding fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.) (Revised by AV, 17-Sep-2020.) |
⊢ PellFund = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ inf({𝑧 ∈ (Pell14QR‘𝑥) ∣ 1 < 𝑧}, ℝ, < )) | ||
Theorem | pell1qrval 39787* | Value of the set of first-quadrant Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)}) | ||
Theorem | elpell1qr 39788* | Membership in a first-quadrant Pell solution set. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))) | ||
Theorem | pell14qrval 39789* | Value of the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)}) | ||
Theorem | elpell14qr 39790* | Membership in the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))) | ||
Theorem | pell1234qrval 39791* | Value of the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1234QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)}) | ||
Theorem | elpell1234qr 39792* | Membership in the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1234QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))) | ||
Theorem | pell1234qrre 39793 | General Pell solutions are (coded as) real numbers. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → 𝐴 ∈ ℝ) | ||
Theorem | pell1234qrne0 39794 | No solution to a Pell equation is zero. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → 𝐴 ≠ 0) | ||
Theorem | pell1234qrreccl 39795 | General solutions of the Pell equation are closed under reciprocals. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → (1 / 𝐴) ∈ (Pell1234QR‘𝐷)) | ||
Theorem | pell1234qrmulcl 39796 | General solutions of the Pell equation are closed under multiplication. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷) ∧ 𝐵 ∈ (Pell1234QR‘𝐷)) → (𝐴 · 𝐵) ∈ (Pell1234QR‘𝐷)) | ||
Theorem | pell14qrss1234 39797 | A positive Pell solution is a general Pell solution. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) ⊆ (Pell1234QR‘𝐷)) | ||
Theorem | pell14qrre 39798 | A positive Pell solution is a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ) | ||
Theorem | pell14qrne0 39799 | A positive Pell solution is a nonzero number. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ≠ 0) | ||
Theorem | pell14qrgt0 39800 | A positive Pell solution is a positive number. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 < 𝐴) |
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