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Theorem List for Metamath Proof Explorer - 39501-39600   *Has distinct variable group(s)
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20.29.11  Diophantine sets 2 miscellanea

Theoremellz1 39501 Membership in a lower set of integers. (Contributed by Stefan O'Rear, 9-Oct-2014.)
(𝐵 ∈ ℤ → (𝐴 ∈ (ℤ ∖ (ℤ‘(𝐵 + 1))) ↔ (𝐴 ∈ ℤ ∧ 𝐴𝐵)))

Theoremlzunuz 39502 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))) ∪ (ℤ𝐵)) = ℤ)

Theoremfz1eqin 39503 Express a one-based finite range as the intersection of lower integers with . (Contributed by Stefan O'Rear, 9-Oct-2014.)
(𝑁 ∈ ℕ0 → (1...𝑁) = ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ))

Theoremlzenom 39504 Lower integers are countably infinite. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝑁 ∈ ℤ → (ℤ ∖ (ℤ‘(𝑁 + 1))) ≈ ω)

Theoremelmapresaunres2 39505 fresaunres2 6526 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.)
((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)

20.29.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra

Theoremdiophin 39506 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)) → (𝐴𝐵) ∈ (Dioph‘𝑁))

Theoremdiophun 39507 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)) → (𝐴𝐵) ∈ (Dioph‘𝑁))

Theoremeldiophss 39508 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 ⊆ (ℕ0m (1...𝐵)))

20.29.13  Diophantine sets 3: construction

Theoremdiophrex 39509* Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)
((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑡 ∣ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))

Theoremeq0rabdioph 39510* 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 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 = 0} ∈ (Dioph‘𝑁))

Theoremeqrabdioph 39511* 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 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 = 𝐵} ∈ (Dioph‘𝑁))

Theorem0dioph 39512 The null set is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝐴 ∈ ℕ0 → ∅ ∈ (Dioph‘𝐴))

Theoremvdioph 39513 The "universal" set (as large as possible given eldiophss 39508) is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝐴 ∈ ℕ0 → (ℕ0m (1...𝐴)) ∈ (Dioph‘𝐴))

Theoremanrabdioph 39514* Diophantine set builder for conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(({𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ (𝜑𝜓)} ∈ (Dioph‘𝑁))

Theoremorrabdioph 39515* Diophantine set builder for disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(({𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ (𝜑𝜓)} ∈ (Dioph‘𝑁))

Theorem3anrabdioph 39516* Diophantine set builder for ternary conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(({𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝜒} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ (𝜑𝜓𝜒)} ∈ (Dioph‘𝑁))

Theorem3orrabdioph 39517* Diophantine set builder for ternary disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(({𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝜒} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ (𝜑𝜓𝜒)} ∈ (Dioph‘𝑁))

20.29.14  Diophantine sets 4 miscellanea

Theorem2sbcrex 39518* 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]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)

TheoremsbcrexgOLD 39519* 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 3836 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))

Theorem2sbcrexOLD 39520* Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7175 instead. (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       ([𝐴 / 𝑎][𝐵 / 𝑏]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)

Theoremsbc2rex 39521* Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)
([𝐴 / 𝑎]𝑏𝐵𝑐𝐶 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝜑)

Theoremsbc2rexgOLD 39522* Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7175 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝜑))

Theoremsbc4rex 39523* Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)
([𝐴 / 𝑎]𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 [𝐴 / 𝑎]𝜑)

Theoremsbc4rexgOLD 39524* Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7175 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 [𝐴 / 𝑎]𝜑))

Theoremsbcrot3 39525* 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)

Theoremsbcrot5 39526* 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)

Theoremsbccomieg 39527* 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 = 𝐴𝐵 = 𝐶)       ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑)

20.29.15  Diophantine sets 4: Quantification

Theoremrexrabdioph 39528* Diophantine set builder for existential quantification. (Contributed by Stefan O'Rear, 10-Oct-2014.)
𝑀 = (𝑁 + 1)    &   (𝑣 = (𝑡𝑀) → (𝜓𝜒))    &   (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝜒𝜑))       ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} ∈ (Dioph‘𝑁))

Theoremrexfrabdioph 39529* 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 ∈ (ℕ0m (1...𝑀)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))

Theorem2rexfrabdioph 39530* 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 ∈ (ℕ0m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → {𝑢 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))

Theorem3rexfrabdioph 39531* 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 ∈ (ℕ0m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → {𝑢 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0𝑥 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))

Theorem4rexfrabdioph 39532* 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 ∈ (ℕ0m (1...𝐽)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑡𝐽) / 𝑦]𝜑} ∈ (Dioph‘𝐽)) → {𝑢 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0𝑥 ∈ ℕ0𝑦 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))

Theorem6rexfrabdioph 39533* 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 ∈ (ℕ0m (1...𝐻)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑡𝐽) / 𝑦][(𝑡𝐼) / 𝑧][(𝑡𝐻) / 𝑝]𝜑} ∈ (Dioph‘𝐻)) → {𝑢 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0𝑥 ∈ ℕ0𝑦 ∈ ℕ0𝑧 ∈ ℕ0𝑝 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))

Theorem7rexfrabdioph 39534* 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 ∈ (ℕ0m (1...𝐺)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑡𝐽) / 𝑦][(𝑡𝐼) / 𝑧][(𝑡𝐻) / 𝑝][(𝑡𝐺) / 𝑞]𝜑} ∈ (Dioph‘𝐺)) → {𝑢 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0𝑥 ∈ ℕ0𝑦 ∈ ℕ0𝑧 ∈ ℕ0𝑝 ∈ ℕ0𝑞 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))

20.29.16  Diophantine sets 5: Arithmetic sets

Theoremrabdiophlem1 39535* Lemma for arithmetic diophantine sets. Convert polynomial-ness of an expression into a constraint suitable for ralimi 3147. (Contributed by Stefan O'Rear, 10-Oct-2014.)
((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0m (1...𝑁))𝐴 ∈ ℤ)

Theoremrabdiophlem2 39536* 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)) / 𝑢𝐴) ∈ (mzPoly‘(1...𝑀)))

Theoremelnn0rabdioph 39537* 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 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} ∈ (Dioph‘𝑁))

Theoremrexzrexnn0 39538* Rewrite a quantification over integers into a quantification over naturals. (Contributed by Stefan O'Rear, 11-Oct-2014.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = -𝑦 → (𝜑𝜒))       (∃𝑥 ∈ ℤ 𝜑 ↔ ∃𝑦 ∈ ℕ0 (𝜓𝜒))

Theoremlerabdioph 39539* 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 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴𝐵} ∈ (Dioph‘𝑁))

Theoremeluzrabdioph 39540* Diophantine set builder for membership in a fixed upper set of integers. (Contributed by Stefan O'Rear, 11-Oct-2014.)
((𝑁 ∈ ℕ0𝑀 ∈ ℤ ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 ∈ (ℤ𝑀)} ∈ (Dioph‘𝑁))

Theoremelnnrabdioph 39541* 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 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 ∈ ℕ} ∈ (Dioph‘𝑁))

Theoremltrabdioph 39542* 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 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴 < 𝐵} ∈ (Dioph‘𝑁))

Theoremnerabdioph 39543* 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 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴𝐵} ∈ (Dioph‘𝑁))

Theoremdvdsrabdioph 39544* 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 ∈ (ℕ0m (1...𝑁)) ∣ 𝐴𝐵} ∈ (Dioph‘𝑁))

20.29.17  Diophantine sets 6: reusability. renumbering of variables

Theoremeldioph4b 39545* 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 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0m 𝑊)(𝑝‘(𝑡𝑤)) = 0}))

Theoremeldioph4i 39546* Forward-only version of eldioph4b 39545. (Contributed by Stefan O'Rear, 16-Oct-2014.)
𝑊 ∈ V    &    ¬ 𝑊 ∈ Fin    &   (𝑊 ∩ ℕ) = ∅       ((𝑁 ∈ ℕ0𝑃 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))) → {𝑡 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0m 𝑊)(𝑃‘(𝑡𝑤)) = 0} ∈ (Dioph‘𝑁))

Theoremdiophren 39547* 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 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))

Theoremrabrenfdioph 39548* Change variable numbers in a Diophantine class abstraction using explicit substitution. (Contributed by Stefan O'Rear, 17-Oct-2014.)
((𝐵 ∈ ℕ0𝐹:(1...𝐴)⟶(1...𝐵) ∧ {𝑎 ∈ (ℕ0m (1...𝐴)) ∣ 𝜑} ∈ (Dioph‘𝐴)) → {𝑏 ∈ (ℕ0m (1...𝐵)) ∣ [(𝑏𝐹) / 𝑎]𝜑} ∈ (Dioph‘𝐵))

Theoremrabren3dioph 39549* Change variable numbers in a 3-variable Diophantine class abstraction. (Contributed by Stefan O'Rear, 17-Oct-2014.)
(((𝑎‘1) = (𝑏𝑋) ∧ (𝑎‘2) = (𝑏𝑌) ∧ (𝑎‘3) = (𝑏𝑍)) → (𝜑𝜓))    &   𝑋 ∈ (1...𝑁)    &   𝑌 ∈ (1...𝑁)    &   𝑍 ∈ (1...𝑁)       ((𝑁 ∈ ℕ0 ∧ {𝑎 ∈ (ℕ0m (1...3)) ∣ 𝜑} ∈ (Dioph‘3)) → {𝑏 ∈ (ℕ0m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁))

20.29.18  Pigeonhole Principle and cardinality helpers

Theoremfphpd 39550* 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.)
(𝜑𝐵𝐴)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   (𝑥 = 𝑦𝐶 = 𝐷)       (𝜑 → ∃𝑥𝐴𝑦𝐴 (𝑥𝑦𝐶 = 𝐷))

Theoremfphpdo 39551* Pigeonhole principle for sets of real numbers with implicit output reordering. (Contributed by Stefan O'Rear, 12-Sep-2014.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐵𝐴)    &   ((𝜑𝑧𝐴) → 𝐶𝐵)    &   (𝑧 = 𝑥𝐶 = 𝐷)    &   (𝑧 = 𝑦𝐶 = 𝐸)       (𝜑 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸))

Theoremctbnfien 39552 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) ∧ (𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin)) → 𝐴𝑌)

Theoremfiphp3d 39553* Infinite pigeonhole principle for partitioning an infinite set between finitely many buckets. (Contributed by Stefan O'Rear, 18-Oct-2014.)
(𝜑𝐴 ≈ ℕ)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐷𝐵)       (𝜑 → ∃𝑦𝐵 {𝑥𝐴𝐷 = 𝑦} ≈ ℕ)

20.29.19  A non-closed set of reals is infinite

Theoremrencldnfilem 39554* Lemma for rencldnfi 39555. (Contributed by Stefan O'Rear, 18-Oct-2014.)
(((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 ≠ ∅ ∧ ¬ 𝐵𝐴)) ∧ ∀𝑥 ∈ ℝ+𝑦𝐴 (abs‘(𝑦𝐵)) < 𝑥) → ¬ 𝐴 ∈ Fin)

Theoremrencldnfi 39555* A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem 39554 using infima; this theorem removes the requirement that A be nonempty. (Contributed by Stefan O'Rear, 19-Oct-2014.)
(((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ 𝐵𝐴) ∧ ∀𝑥 ∈ ℝ+𝑦𝐴 (abs‘(𝑦𝐵)) < 𝑥) → ¬ 𝐴 ∈ Fin)

20.29.20  Lagrange's rational approximation theorem

Theoremirrapxlem1 39556* Lemma for irrapx1 39562. Divides the unit interval into 𝐵 half-open sections and using the pigeonhole principle fphpdo 39551 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)))))

Theoremirrapxlem2 39557* Lemma for irrapx1 39562. 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)))

Theoremirrapxlem3 39558* Lemma for irrapx1 39562. 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))

Theoremirrapxlem4 39559* Lemma for irrapx1 39562. 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(𝑥𝐵, 𝐵, 𝑥)))

Theoremirrapxlem5 39560* Lemma for irrapx1 39562. Switching to real intervals and fraction syntax. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℚ (0 < 𝑥 ∧ (abs‘(𝑥𝐴)) < 𝐵 ∧ (abs‘(𝑥𝐴)) < ((denom‘𝑥)↑-2)))

Theoremirrapxlem6 39561* Lemma for irrapx1 39562. 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‘(𝑦𝐴)) < ((denom‘𝑦)↑-2))} (abs‘(𝑥𝐴)) < 𝐵)

Theoremirrapx1 39562* 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‘(𝑦𝐴)) < ((denom‘𝑦)↑-2))} ≈ ℕ)

20.29.21  Pell equations 1: A nontrivial solution always exists

Theorempellexlem1 39563 Lemma for pellex 39569. 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)

Theorempellexlem2 39564 Lemma for pellex 39569. 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))))

Theorempellexlem3 39565* Lemma for pellex 39569. 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)))))})

Theorempellexlem4 39566* Lemma for pellex 39569. Invoking irrapx1 39562, 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)

Theorempellexlem5 39567* Lemma for pellex 39569. Invoking fiphp3d 39553, 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))

Theorempellexlem6 39568* Lemma for pellex 39569. 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.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℕ)    &   (𝜑 → ¬ (√‘𝐷) ∈ ℚ)    &   (𝜑𝐸 ∈ ℕ)    &   (𝜑𝐹 ∈ ℕ)    &   (𝜑 → ¬ (𝐴 = 𝐸𝐵 = 𝐹))    &   (𝜑𝐶 ≠ 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)

Theorempellex 39569* 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)

20.29.22  Pell equations 2: Algebraic number theory of the solution set

Syntaxcsquarenn 39570 Extend class notation to include the set of square positive integers.
class NN

Syntaxcpell1qr 39571 Extend class notation to include the class of quadrant-1 Pell solutions.
class Pell1QR

Syntaxcpell1234qr 39572 Extend class notation to include the class of any-quadrant Pell solutions.
class Pell1234QR

Syntaxcpell14qr 39573 Extend class notation to include the class of positive Pell solutions.
class Pell14QR

Syntaxcpellfund 39574 Extend class notation to include the Pell-equation fundamental solution function.
class PellFund

Definitiondf-squarenn 39575 Define the set of square positive integers. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN = {𝑥 ∈ ℕ ∣ (√‘𝑥) ∈ ℚ}

Definitiondf-pell1qr 39576* 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 ∈ ℕ0𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})

Definitiondf-pell14qr 39577* Define the positive solutions of a Pell equation. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Pell14QR = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})

Definitiondf-pell1234qr 39578* 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)})

Definitiondf-pellfund 39579* 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, < ))

Theorempell1qrval 39580* 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 ∈ ℕ0𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})

Theoremelpell1qr 39581* 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 ∈ ℕ0𝑤 ∈ ℕ0 (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))))

Theorempell14qrval 39582* Value of the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})

Theoremelpell14qr 39583* 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 ∈ ℕ0𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))))

Theorempell1234qrval 39584* 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)})

Theoremelpell1234qr 39585* 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))))

Theorempell1234qrre 39586 General Pell solutions are (coded as) real numbers. (Contributed by Stefan O'Rear, 17-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → 𝐴 ∈ ℝ)

Theorempell1234qrne0 39587 No solution to a Pell equation is zero. (Contributed by Stefan O'Rear, 17-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → 𝐴 ≠ 0)

Theorempell1234qrreccl 39588 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))

Theorempell1234qrmulcl 39589 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))

Theorempell14qrss1234 39590 A positive Pell solution is a general Pell solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) ⊆ (Pell1234QR‘𝐷))

Theorempell14qrre 39591 A positive Pell solution is a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ)

Theorempell14qrne0 39592 A positive Pell solution is a nonzero number. (Contributed by Stefan O'Rear, 17-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ≠ 0)

Theorempell14qrgt0 39593 A positive Pell solution is a positive number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 < 𝐴)

Theorempell14qrrp 39594 A positive Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ+)

Theorempell1234qrdich 39595 A general Pell solution is either a positive solution, or its negation is. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → (𝐴 ∈ (Pell14QR‘𝐷) ∨ -𝐴 ∈ (Pell14QR‘𝐷)))

Theoremelpell14qr2 39596 A number is a positive Pell solution iff it is positive and a Pell solution, justifying our name choice. (Contributed by Stefan O'Rear, 19-Oct-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ (𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴)))

Theorempell14qrmulcl 39597 Positive Pell solutions are closed under multiplication. (Contributed by Stefan O'Rear, 17-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → (𝐴 · 𝐵) ∈ (Pell14QR‘𝐷))

Theorempell14qrreccl 39598 Positive Pell solutions are closed under reciprocal. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (1 / 𝐴) ∈ (Pell14QR‘𝐷))

Theorempell14qrdivcl 39599 Positive Pell solutions are closed under division. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → (𝐴 / 𝐵) ∈ (Pell14QR‘𝐷))

Theorempell14qrexpclnn0 39600 Lemma for pell14qrexpcl 39601. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ ℕ0) → (𝐴𝐵) ∈ (Pell14QR‘𝐷))

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