P. 441 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44001-44100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sbc4rex 44001*	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	sbcrot3 44002*	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 44003*	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 44004*	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)
21.37.15  Diophantine sets 4: Quantification
Theorem	rexrabdioph 44005*	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 44006*	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 44007*	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 44008*	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 44009*	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 44010*	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 44011*	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))
21.37.16  Diophantine sets 5: Arithmetic sets
Theorem	rabdiophlem1 44012*	Lemma for arithmetic diophantine sets. Convert polynomial-ness of an expression into a constraint suitable for ralimi 3103. (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 44013*	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 44014*	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 44015*	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 44016*	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 44017*	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 44018*	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 44019*	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 44020*	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 44021*	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))
21.37.17  Diophantine sets 6: reusability. renumbering of variables
Theorem	eldioph4b 44022*	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 44023*	Forward-only version of eldioph4b 44022. (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 44024*	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 44025*	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 44026*	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))
21.37.18  Pigeonhole Principle and cardinality helpers
Theorem	fphpd 44027*	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 44028*	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 44029	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 44030*	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)
21.37.19  A non-closed set of reals is infinite
Theorem	rencldnfilem 44031*	Lemma for rencldnfi 44032. (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 44032*	A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem 44031 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)
21.37.20  Lagrange's rational approximation theorem
Theorem	irrapxlem1 44033*	Lemma for irrapx1 44039. Divides the unit interval into 𝐵 half-open sections and using the pigeonhole principle fphpdo 44028 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 44034*	Lemma for irrapx1 44039. 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 44035*	Lemma for irrapx1 44039. 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 44036*	Lemma for irrapx1 44039. 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 44037*	Lemma for irrapx1 44039. 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 44038*	Lemma for irrapx1 44039. 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 44039*	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)
21.37.21  Pell equations 1: A nontrivial solution always exists
Theorem	pellexlem1 44040	Lemma for pellex 44046. 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 44041	Lemma for pellex 44046. 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 44042*	Lemma for pellex 44046. 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 44043*	Lemma for pellex 44046. Invoking irrapx1 44039, 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 44044*	Lemma for pellex 44046. Invoking fiphp3d 44030, 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 44045*	Lemma for pellex 44046. 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 44046*	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)
21.37.22  Pell equations 2: Algebraic number theory of the solution set
Syntax	csquarenn 44047	Extend class notation to include the set of square positive integers.
class ◻NN
Syntax	cpell1qr 44048	Extend class notation to include the class of quadrant-1 Pell solutions.
class Pell1QR
Syntax	cpell1234qr 44049	Extend class notation to include the class of any-quadrant Pell solutions.
class Pell1234QR
Syntax	cpell14qr 44050	Extend class notation to include the class of positive Pell solutions.
class Pell14QR
Syntax	cpellfund 44051	Extend class notation to include the Pell-equation fundamental solution function.
class PellFund
Definition	df-squarenn 44052	Define the set of square positive integers. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ◻NN = {𝑥 ∈ ℕ ∣ (√‘𝑥) ∈ ℚ}
Definition	df-pell1qr 44053*	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 44054*	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 44055*	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 44056*	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 44057*	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 44058*	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 44059*	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 44060*	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 44061*	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 44062*	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 44063	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 44064	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 44065	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 44066	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 44067	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 44068	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 44069	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 44070	A positive Pell solution is a positive number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 < 𝐴)
Theorem	pell14qrrp 44071	A positive Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ+)
Theorem	pell1234qrdich 44072	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)))
Theorem	elpell14qr2 44073	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)))
Theorem	pell14qrmulcl 44074	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))
Theorem	pell14qrreccl 44075	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))
Theorem	pell14qrdivcl 44076	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))
Theorem	pell14qrexpclnn0 44077	Lemma for pell14qrexpcl 44078. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ ℕ0) → (𝐴↑𝐵) ∈ (Pell14QR‘𝐷))
Theorem	pell14qrexpcl 44078	Positive Pell solutions are closed under integer powers. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ ℤ) → (𝐴↑𝐵) ∈ (Pell14QR‘𝐷))
Theorem	pell1qrss14 44079	First-quadrant Pell solutions are a subset of the positive solutions. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷))
Theorem	pell14qrdich 44080	A positive Pell solution is either in the first quadrant, or its reciprocal is. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 ∈ (Pell1QR‘𝐷) ∨ (1 / 𝐴) ∈ (Pell1QR‘𝐷)))
Theorem	pell1qrge1 44081	A Pell solution in the first quadrant is at least 1. (Contributed by Stefan O'Rear, 17-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1QR‘𝐷)) → 1 ≤ 𝐴)
Theorem	pell1qr1 44082	1 is a Pell solution and in the first quadrant as one. (Contributed by Stefan O'Rear, 17-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 ∈ (Pell1QR‘𝐷))
Theorem	elpell1qr2 44083	The first quadrant solutions are precisely the positive Pell solutions which are at least one. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1QR‘𝐷) ↔ (𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 ≤ 𝐴)))
Theorem	pell1qrgaplem 44084	Lemma for pell1qrgap 44085. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ (((𝐷 ∈ ℕ ∧ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0)) ∧ (1 < (𝐴 + ((√‘𝐷) · 𝐵)) ∧ ((𝐴↑2) − (𝐷 · (𝐵↑2))) = 1)) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ (𝐴 + ((√‘𝐷) · 𝐵)))
Theorem	pell1qrgap 44085	First-quadrant Pell solutions are bounded away from 1. (This particular bound allows to prove exact values for the fundamental solution later.) (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1QR‘𝐷) ∧ 1 < 𝐴) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝐴)
Theorem	pell14qrgap 44086	Positive Pell solutions are bounded away from 1. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝐴)
Theorem	pell14qrgapw 44087	Positive Pell solutions are bounded away from 1, with a friendlier bound. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → 2 < 𝐴)
Theorem	pellqrexplicit 44088	Condition for a calculated real to be a Pell solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ((𝐴↑2) − (𝐷 · (𝐵↑2))) = 1) → (𝐴 + ((√‘𝐷) · 𝐵)) ∈ (Pell1QR‘𝐷))
21.37.23  Pell equations 3: characterizing fundamental solution
Theorem	infmrgelbi 44089*	Any lower bound of a nonempty set of real numbers is less than or equal to its infimum, one-direction version. (Contributed by Stefan O'Rear, 1-Sep-2013.) (Revised by AV, 17-Sep-2020.)
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥) → 𝐵 ≤ inf(𝐴, ℝ, < ))
Theorem	pellqrex 44090*	There is a nontrivial solution of a Pell equation in the first quadrant. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ∃𝑥 ∈ (Pell1QR‘𝐷)1 < 𝑥)
Theorem	pellfundval 44091*	Value of the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 18-Sep-2014.) (Revised by AV, 17-Sep-2020.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < ))
Theorem	pellfundre 44092	The fundamental solution of a Pell equation exists as a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ)
Theorem	pellfundge 44093	Lower bound on the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ (PellFund‘𝐷))
Theorem	pellfundgt1 44094	Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < (PellFund‘𝐷))
Theorem	pellfundlb 44095	A nontrivial first quadrant solution is at least as large as the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Proof shortened by AV, 15-Sep-2020.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → (PellFund‘𝐷) ≤ 𝐴)
Theorem	pellfundglb 44096*	If a real is larger than the fundamental solution, there is a nontrivial solution less than it. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ∃𝑥 ∈ (Pell1QR‘𝐷)((PellFund‘𝐷) ≤ 𝑥 ∧ 𝑥 < 𝐴))
Theorem	pellfundex 44097	The fundamental solution as an infimum is itself a solution, showing that the solution set is discrete. Since the fundamental solution is an infimum, there must be an element ge to Fund and lt 2*Fund. If this element is equal to the fundamental solution we're done, otherwise use the infimum again to find another element which must be ge Fund and lt the first element; their ratio is a group element in (1,2), contradicting pell14qrgapw 44087. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell1QR‘𝐷))
Theorem	pellfund14gap 44098	There are no solutions between 1 and the fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) → 𝐴 = 1)
Theorem	pellfundrp 44099	The fundamental Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ+)
Theorem	pellfundne1 44100	The fundamental Pell solution is never 1. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ≠ 1)