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Theorem List for Intuitionistic Logic Explorer - 6701-6800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltexprlemopu 6701* The upper cut of our constructed difference is open. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       ((𝐴<P 𝐵𝑟Q𝑟 ∈ (2nd𝐶)) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))

Theoremltexprlemupu 6702* The upper cut of our constructed difference is upper. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       ((𝐴<P 𝐵𝑟Q) → (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) → 𝑟 ∈ (2nd𝐶)))

Theoremltexprlemrnd 6703* Our constructed difference is rounded. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       (𝐴<P 𝐵 → (∀𝑞Q (𝑞 ∈ (1st𝐶) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐶) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))))

Theoremltexprlemdisj 6704* Our constructed difference is disjoint. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       (𝐴<P 𝐵 → ∀𝑞Q ¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)))

Theoremltexprlemloc 6705* Our constructed difference is located. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       (𝐴<P 𝐵 → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶))))

Theoremltexprlempr 6706* Our constructed difference is a positive real. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       (𝐴<P 𝐵𝐶P)

Theoremltexprlemfl 6707* Lemma for ltexpri 6711. One directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       (𝐴<P 𝐵 → (1st ‘(𝐴 +P 𝐶)) ⊆ (1st𝐵))

Theoremltexprlemrl 6708* Lemma for ltexpri 6711. Reverse directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       (𝐴<P 𝐵 → (1st𝐵) ⊆ (1st ‘(𝐴 +P 𝐶)))

Theoremltexprlemfu 6709* Lemma for ltexpri 6711. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       (𝐴<P 𝐵 → (2nd ‘(𝐴 +P 𝐶)) ⊆ (2nd𝐵))

Theoremltexprlemru 6710* Lemma for ltexpri 6711. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       (𝐴<P 𝐵 → (2nd𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶)))

Theoremltexpri 6711* Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
(𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)

(((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐵) ⊆ (1st𝐶))

(((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐵) ⊆ (2nd𝐶))

Theoremaddcanprg 6714 Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by Jim Kingdon, 24-Dec-2019.)
((𝐴P𝐵P𝐶P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶))

Theoremlteupri 6715* The difference from ltexpri 6711 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
(𝐴<P 𝐵 → ∃!𝑥P (𝐴 +P 𝑥) = 𝐵)

Theoremltaprlem 6716 Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.)
(𝐶P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))

Theoremltaprg 6717 Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.)
((𝐴P𝐵P𝐶P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))

Theoremprplnqu 6718* Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.)
(𝜑𝑋P)    &   (𝜑𝑄Q)    &   (𝜑𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))       (𝜑 → ∃𝑦 ∈ (2nd𝑋)(𝑦 +Q 𝑄) = 𝐴)

Theoremaddextpr 6719 Strong extensionality of addition (ordering version). This is similar to addext 7601 but for positive reals and based on less-than rather than apartness. (Contributed by Jim Kingdon, 17-Feb-2020.)
(((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ((𝐴 +P 𝐵)<P (𝐶 +P 𝐷) → (𝐴<P 𝐶𝐵<P 𝐷)))

Theoremrecexprlemell 6720* Membership in the lower cut of 𝐵. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       (𝐶 ∈ (1st𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))

Theoremrecexprlemelu 6721* Membership in the upper cut of 𝐵. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       (𝐶 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝐶 ∧ (*Q𝑦) ∈ (1st𝐴)))

Theoremrecexprlemm 6722* 𝐵 is inhabited. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       (𝐴P → (∃𝑞Q 𝑞 ∈ (1st𝐵) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐵)))

Theoremrecexprlemopl 6723* The lower cut of 𝐵 is open. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 28-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       ((𝐴P𝑞Q𝑞 ∈ (1st𝐵)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐵)))

Theoremrecexprlemlol 6724* The lower cut of 𝐵 is lower. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 28-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       ((𝐴P𝑞Q) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐵)) → 𝑞 ∈ (1st𝐵)))

Theoremrecexprlemopu 6725* The upper cut of 𝐵 is open. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 28-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       ((𝐴P𝑟Q𝑟 ∈ (2nd𝐵)) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐵)))

Theoremrecexprlemupu 6726* The upper cut of 𝐵 is upper. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 28-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       ((𝐴P𝑟Q) → (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐵)) → 𝑟 ∈ (2nd𝐵)))

Theoremrecexprlemrnd 6727* 𝐵 is rounded. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       (𝐴P → (∀𝑞Q (𝑞 ∈ (1st𝐵) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐵))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐵) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐵)))))

Theoremrecexprlemdisj 6728* 𝐵 is disjoint. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       (𝐴P → ∀𝑞Q ¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)))

Theoremrecexprlemloc 6729* 𝐵 is located. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       (𝐴P → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵))))

Theoremrecexprlempr 6730* 𝐵 is a positive real. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       (𝐴P𝐵P)

Theoremrecexprlem1ssl 6731* The lower cut of one is a subset of the lower cut of 𝐴 ·P 𝐵. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       (𝐴P → (1st ‘1P) ⊆ (1st ‘(𝐴 ·P 𝐵)))

Theoremrecexprlem1ssu 6732* The upper cut of one is a subset of the upper cut of 𝐴 ·P 𝐵. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       (𝐴P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))

Theoremrecexprlemss1l 6733* The lower cut of 𝐴 ·P 𝐵 is a subset of the lower cut of one. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       (𝐴P → (1st ‘(𝐴 ·P 𝐵)) ⊆ (1st ‘1P))

Theoremrecexprlemss1u 6734* The upper cut of 𝐴 ·P 𝐵 is a subset of the upper cut of one. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       (𝐴P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))

Theoremrecexprlemex 6735* 𝐵 is the reciprocal of 𝐴. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       (𝐴P → (𝐴 ·P 𝐵) = 1P)

Theoremrecexpr 6736* The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
(𝐴P → ∃𝑥P (𝐴 ·P 𝑥) = 1P)

Theoremaptiprleml 6737 Lemma for aptipr 6739. (Contributed by Jim Kingdon, 28-Jan-2020.)
((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (1st𝐴) ⊆ (1st𝐵))

Theoremaptiprlemu 6738 Lemma for aptipr 6739. (Contributed by Jim Kingdon, 28-Jan-2020.)
((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (2nd𝐵) ⊆ (2nd𝐴))

Theoremaptipr 6739 Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.)
((𝐴P𝐵P ∧ ¬ (𝐴<P 𝐵𝐵<P 𝐴)) → 𝐴 = 𝐵)

Theoremltmprr 6740 Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.)
((𝐴P𝐵P𝐶P) → ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → 𝐴<P 𝐵))

Theoremarchpr 6741* For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer 𝑥 is embedded into the reals as described at nnprlu 6651. (Contributed by Jim Kingdon, 22-Apr-2020.)
(𝐴P → ∃𝑥N 𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩)

Theoremcaucvgprlemcanl 6742* Lemma for cauappcvgprlemladdrl 6755. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.)
(𝜑𝐿P)    &   (𝜑𝑆Q)    &   (𝜑𝑅Q)    &   (𝜑𝑄Q)       (𝜑 → ((𝑅 +Q 𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)) ↔ 𝑅 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))))

Theoremcauappcvgprlemm 6743* Lemma for cauappcvgpr 6760. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿)))

Theoremcauappcvgprlemopl 6744* Lemma for cauappcvgpr 6760. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))

Theoremcauappcvgprlemlol 6745* Lemma for cauappcvgpr 6760. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 4-Aug-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))

Theoremcauappcvgprlemopu 6746* Lemma for cauappcvgpr 6760. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       ((𝜑𝑟 ∈ (2nd𝐿)) → ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))

Theoremcauappcvgprlemupu 6747* Lemma for cauappcvgpr 6760. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 4-Aug-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟 ∈ (2nd𝐿))

Theoremcauappcvgprlemrnd 6748* Lemma for cauappcvgpr 6760. The putative limit is rounded. (Contributed by Jim Kingdon, 18-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       (𝜑 → (∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))))

Theoremcauappcvgprlemdisj 6749* Lemma for cauappcvgpr 6760. The putative limit is disjoint. (Contributed by Jim Kingdon, 18-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))

Theoremcauappcvgprlemloc 6750* Lemma for cauappcvgpr 6760. The putative limit is located. (Contributed by Jim Kingdon, 18-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       (𝜑 → ∀𝑠Q𝑟Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))

Theoremcauappcvgprlemcl 6751* Lemma for cauappcvgpr 6760. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       (𝜑𝐿P)

Theoremcauappcvgprlemladdfu 6752* Lemma for cauappcvgprlemladd 6756. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑𝑆Q)       (𝜑 → (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))

Theoremcauappcvgprlemladdfl 6753* Lemma for cauappcvgprlemladd 6756. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑𝑆Q)       (𝜑 → (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))

Theoremcauappcvgprlemladdru 6754* Lemma for cauappcvgprlemladd 6756. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑𝑆Q)       (𝜑 → (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩) ⊆ (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)))

Theoremcauappcvgprlemladdrl 6755* Lemma for cauappcvgprlemladd 6756. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑𝑆Q)       (𝜑 → (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩) ⊆ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)))

Theoremcauappcvgprlemladd 6756* Lemma for cauappcvgpr 6760. This takes 𝐿 and offsets it by the positive fraction 𝑆. (Contributed by Jim Kingdon, 23-Jun-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑𝑆Q)       (𝜑 → (𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q 𝑆)}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩)

Theoremcauappcvgprlem1 6757* Lemma for cauappcvgpr 6760. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑𝑄Q)    &   (𝜑𝑅Q)       (𝜑 → ⟨{𝑙𝑙 <Q (𝐹𝑄)}, {𝑢 ∣ (𝐹𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩))

Theoremcauappcvgprlem2 6758* Lemma for cauappcvgpr 6760. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑𝑄Q)    &   (𝜑𝑅Q)       (𝜑𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)

Theoremcauappcvgprlemlim 6759* Lemma for cauappcvgpr 6760. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-2020.)
(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩       (𝜑 → ∀𝑞Q𝑟Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))

Theoremcauappcvgpr 6760* A Cauchy approximation has a limit. A Cauchy approximation, here 𝐹, is similar to a Cauchy sequence but is indexed by the desired tolerance (that is, how close together terms needs to be) rather than by natural numbers. This is basically Theorem 11.2.12 of [HoTT], p. (varies) with a few differences such as that we are proving the existence of a limit without anything about how fast it converges (that is, mere existence instead of existence, in HoTT terms), and that the codomain of 𝐹 is Q rather than P. We also specify that every term needs to be larger than a fraction 𝐴, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This proof (including its lemmas) is similar to the proofs of caucvgpr 6780 and caucvgprpr 6810 but is somewhat simpler, so reading this one first may help understanding the other two.

(Contributed by Jim Kingdon, 19-Jun-2020.)

(𝜑𝐹:QQ)    &   (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))       (𝜑 → ∃𝑦P𝑞Q𝑟Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))

Theoremarchrecnq 6761* Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.)
(𝐴Q → ∃𝑗N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝐴)

Theoremarchrecpr 6762* Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.)
(𝐴P → ∃𝑗N ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝐴)

Theoremcaucvgprlemk 6763 Lemma for caucvgpr 6780. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.)
(𝜑𝐽 <N 𝐾)    &   (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑄)       (𝜑 → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑄)

Theoremcaucvgprlemnkj 6764* Lemma for caucvgpr 6780. Part of disjointness. (Contributed by Jim Kingdon, 23-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑𝐾N)    &   (𝜑𝐽N)    &   (𝜑𝑆Q)       (𝜑 → ¬ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))

Theoremcaucvgprlemnbj 6765* Lemma for caucvgpr 6780. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑𝐵N)    &   (𝜑𝐽N)       (𝜑 → ¬ (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽))

Theoremcaucvgprlemm 6766* Lemma for caucvgpr 6780. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿)))

Theoremcaucvgprlemopl 6767* Lemma for caucvgpr 6780. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)))

Theoremcaucvgprlemlol 6768* Lemma for caucvgpr 6780. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))

Theoremcaucvgprlemopu 6769* Lemma for caucvgpr 6780. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       ((𝜑𝑟 ∈ (2nd𝐿)) → ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))

Theoremcaucvgprlemupu 6770* Lemma for caucvgpr 6780. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 20-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟 ∈ (2nd𝐿))

Theoremcaucvgprlemrnd 6771* Lemma for caucvgpr 6780. The putative limit is rounded. (Contributed by Jim Kingdon, 27-Sep-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       (𝜑 → (∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))))

Theoremcaucvgprlemdisj 6772* Lemma for caucvgpr 6780. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))

Theoremcaucvgprlemloc 6773* Lemma for caucvgpr 6780. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       (𝜑 → ∀𝑠Q𝑟Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))

Theoremcaucvgprlemcl 6774* Lemma for caucvgpr 6780. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       (𝜑𝐿P)

Theoremcaucvgprlemladdfu 6775* Lemma for caucvgpr 6780. Adding 𝑆 after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    &   (𝜑𝑆Q)       (𝜑 → (2nd ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ⊆ {𝑢Q ∣ ∃𝑗N (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑢})

Theoremcaucvgprlemladdrl 6776* Lemma for caucvgpr 6780. Adding 𝑆 after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 8-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    &   (𝜑𝑆Q)       (𝜑 → {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹𝑗) +Q 𝑆)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)))

Theoremcaucvgprlem1 6777* Lemma for caucvgpr 6780. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    &   (𝜑𝑄Q)    &   (𝜑𝐽 <N 𝐾)    &   (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑄)       (𝜑 → ⟨{𝑙𝑙 <Q (𝐹𝐾)}, {𝑢 ∣ (𝐹𝐾) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))

Theoremcaucvgprlem2 6778* Lemma for caucvgpr 6780. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    &   (𝜑𝑄Q)    &   (𝜑𝐽 <N 𝐾)    &   (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑄)       (𝜑𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹𝐾) +Q 𝑄) <Q 𝑢}⟩)

Theoremcaucvgprlemlim 6779* Lemma for caucvgpr 6780. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩       (𝜑 → ∀𝑥Q𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)))

Theoremcaucvgpr 6780* A Cauchy sequence of positive fractions with a modulus of convergence converges to a positive real. This is basically Corollary 11.2.13 of [HoTT], p. (varies) (one key difference being that this is for positive reals rather than signed reals). Also, the HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / 𝑛 of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). We also specify that every term needs to be larger than a fraction 𝐴, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This proof (including its lemmas) is similar to the proofs of cauappcvgpr 6760 and caucvgprpr 6810. Reading cauappcvgpr 6760 first (the simplest of the three) might help understanding the other two.

(Contributed by Jim Kingdon, 18-Jun-2020.)

(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))       (𝜑 → ∃𝑦P𝑥Q𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)))

Theoremcaucvgprprlemk 6781* Lemma for caucvgprpr 6810. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.)
(𝜑𝐽 <N 𝐾)    &   (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)       (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)

Theoremcaucvgprprlemloccalc 6782* Lemma for caucvgprpr 6810. Rearranging some expressions for caucvgprprlemloc 6801. (Contributed by Jim Kingdon, 8-Feb-2021.)
(𝜑𝑆 <Q 𝑇)    &   (𝜑𝑌Q)    &   (𝜑 → (𝑆 +Q 𝑌) = 𝑇)    &   (𝜑𝑋Q)    &   (𝜑 → (𝑋 +Q 𝑋) <Q 𝑌)    &   (𝜑𝑀N)    &   (𝜑 → (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑋)       (𝜑 → (⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝑇}, {𝑢𝑇 <Q 𝑢}⟩)

Theoremcaucvgprprlemell 6783* Lemma for caucvgprpr 6810. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.)
𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝑋 ∈ (1st𝐿) ↔ (𝑋Q ∧ ∃𝑏N ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))

Theoremcaucvgprprlemelu 6784* Lemma for caucvgprpr 6810. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-Jan-2021.)
𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝑋 ∈ (2nd𝐿) ↔ (𝑋Q ∧ ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩))

Theoremcaucvgprprlemcbv 6785* Lemma for caucvgprpr 6810. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))       (𝜑 → ∀𝑎N𝑏N (𝑎 <N 𝑏 → ((𝐹𝑎)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))

Theoremcaucvgprprlemval 6786* Lemma for caucvgprpr 6810. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))       ((𝜑𝐴 <N 𝐵) → ((𝐹𝐴)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹𝐵)<P ((𝐹𝐴) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))

Theoremcaucvgprprlemnkltj 6787* Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑𝐾N)    &   (𝜑𝐽N)    &   (𝜑𝑆Q)       ((𝜑𝐾 <N 𝐽) → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))

Theoremcaucvgprprlemnkeqj 6788* Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑𝐾N)    &   (𝜑𝐽N)    &   (𝜑𝑆Q)       ((𝜑𝐾 = 𝐽) → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))

Theoremcaucvgprprlemnjltk 6789* Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑𝐾N)    &   (𝜑𝐽N)    &   (𝜑𝑆Q)       ((𝜑𝐽 <N 𝐾) → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))

Theoremcaucvgprprlemnkj 6790* Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 20-Jan-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑𝐾N)    &   (𝜑𝐽N)    &   (𝜑𝑆Q)       (𝜑 → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))

Theoremcaucvgprprlemnbj 6791* Lemma for caucvgprpr 6810. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-2021.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑𝐵N)    &   (𝜑𝐽N)       (𝜑 → ¬ (((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P (𝐹𝐽))

Theoremcaucvgprprlemml 6792* Lemma for caucvgprpr 6810. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝜑 → ∃𝑠Q 𝑠 ∈ (1st𝐿))

Theoremcaucvgprprlemmu 6793* Lemma for caucvgprpr 6810. The upper cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝜑 → ∃𝑡Q 𝑡 ∈ (2nd𝐿))

Theoremcaucvgprprlemm 6794* Lemma for caucvgprpr 6810. The putative limit is inhabited. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑡Q 𝑡 ∈ (2nd𝐿)))

Theoremcaucvgprprlemopl 6795* Lemma for caucvgprpr 6810. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)))

Theoremcaucvgprprlemlol 6796* Lemma for caucvgprpr 6810. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       ((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))

Theoremcaucvgprprlemopu 6797* Lemma for caucvgprpr 6810. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       ((𝜑𝑡 ∈ (2nd𝐿)) → ∃𝑠Q (𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)))

Theoremcaucvgprprlemupu 6798* Lemma for caucvgprpr 6810. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       ((𝜑𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)) → 𝑡 ∈ (2nd𝐿))

Theoremcaucvgprprlemrnd 6799* Lemma for caucvgprpr 6810. The putative limit is rounded. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝜑 → (∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿))) ∧ ∀𝑡Q (𝑡 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)))))

Theoremcaucvgprprlemdisj 6800* Lemma for caucvgprpr 6810. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩       (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))

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