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Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltnqex 6801 The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
{𝑥𝑥 <Q 𝐴} ∈ V

Theoremgtnqex 6802 The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
{𝑥𝐴 <Q 𝑥} ∈ V

Theoremnqprl 6803* Comparing a fraction to a real can be done by whether it is an element of the lower cut, or by <P. (Contributed by Jim Kingdon, 8-Jul-2020.)
((𝐴Q𝐵P) → (𝐴 ∈ (1st𝐵) ↔ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))

Theoremnqpru 6804* Comparing a fraction to a real can be done by whether it is an element of the upper cut, or by <P. (Contributed by Jim Kingdon, 29-Nov-2020.)
((𝐴Q𝐵P) → (𝐴 ∈ (2nd𝐵) ↔ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))

Theoremnnprlu 6805* The canonical embedding of positive integers into the positive reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
(𝐴N → ⟨{𝑙𝑙 <Q [⟨𝐴, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐴, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)

Theorem1pr 6806 The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
1PP

Theorem1prl 6807 The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
(1st ‘1P) = {𝑥𝑥 <Q 1Q}

Theorem1pru 6808 The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
(2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥}

Theoremaddnqprlemrl 6809* Lemma for addnqpr 6813. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴Q𝐵Q) → (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))

Theoremaddnqprlemru 6810* Lemma for addnqpr 6813. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴Q𝐵Q) → (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))

Theoremaddnqprlemfl 6811* Lemma for addnqpr 6813. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴Q𝐵Q) → (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))

Theoremaddnqprlemfu 6812* Lemma for addnqpr 6813. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴Q𝐵Q) → (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))

Theoremaddnqpr 6813* Addition of fractions embedded into positive reals. One can either add the fractions as fractions, or embed them into positive reals and add them as positive reals, and get the same result. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴Q𝐵Q) → ⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))

Theoremaddnqpr1 6814* Addition of one to a fraction embedded into a positive real. One can either add the fraction one to the fraction, or the positive real one to the positive real, and get the same result. Special case of addnqpr 6813. (Contributed by Jim Kingdon, 26-Apr-2020.)
(𝐴Q → ⟨{𝑙𝑙 <Q (𝐴 +Q 1Q)}, {𝑢 ∣ (𝐴 +Q 1Q) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P 1P))

Theoremappdivnq 6815* Approximate division for positive rationals. Proposition 12.7 of [BauerTaylor], p. 55 (a special case where 𝐴 and 𝐵 are positive, as well as 𝐶). Our proof is simpler than the one in BauerTaylor because we have reciprocals. (Contributed by Jim Kingdon, 8-Dec-2019.)
((𝐴 <Q 𝐵𝐶Q) → ∃𝑚Q (𝐴 <Q (𝑚 ·Q 𝐶) ∧ (𝑚 ·Q 𝐶) <Q 𝐵))

Theoremappdiv0nq 6816* Approximate division for positive rationals. This can be thought of as a variation of appdivnq 6815 in which 𝐴 is zero, although it can be stated and proved in terms of positive rationals alone, without zero as such. (Contributed by Jim Kingdon, 9-Dec-2019.)
((𝐵Q𝐶Q) → ∃𝑚Q (𝑚 ·Q 𝐶) <Q 𝐵)

Theoremprmuloclemcalc 6817 Calculations for prmuloc 6818. (Contributed by Jim Kingdon, 9-Dec-2019.)
(𝜑𝑅 <Q 𝑈)    &   (𝜑𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑 → (𝐴 +Q 𝑋) = 𝐵)    &   (𝜑 → (𝑃 ·Q 𝐵) <Q (𝑅 ·Q 𝑋))    &   (𝜑𝐴Q)    &   (𝜑𝐵Q)    &   (𝜑𝐷Q)    &   (𝜑𝑃Q)    &   (𝜑𝑋Q)       (𝜑 → (𝑈 ·Q 𝐴) <Q (𝐷 ·Q 𝐵))

Theoremprmuloc 6818* Positive reals are multiplicatively located. Lemma 12.8 of [BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.)
((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) → ∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))

Theoremprmuloc2 6819* Positive reals are multiplicatively located. This is a variation of prmuloc 6818 which only constructs one (named) point and is therefore often easier to work with. It states that given a ratio 𝐵, there are elements of the lower and upper cut which have exactly that ratio between them. (Contributed by Jim Kingdon, 28-Dec-2019.)
((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)

Theoremmulnqprl 6820 Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → 𝑋 ∈ (1st ‘(𝐴 ·P 𝐵))))

Theoremmulnqpru 6821 Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) <Q 𝑋𝑋 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Theoremmullocprlem 6822 Calculations for mullocpr 6823. (Contributed by Jim Kingdon, 10-Dec-2019.)
(𝜑 → (𝐴P𝐵P))    &   (𝜑 → (𝑈 ·Q 𝑄) <Q (𝐸 ·Q (𝐷 ·Q 𝑈)))    &   (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈)))    &   (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅))    &   (𝜑 → (𝑄Q𝑅Q))    &   (𝜑 → (𝐷Q𝑈Q))    &   (𝜑 → (𝐷 ∈ (1st𝐴) ∧ 𝑈 ∈ (2nd𝐴)))    &   (𝜑 → (𝐸Q𝑇Q))       (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Theoremmullocpr 6823* Locatedness of multiplication on positive reals. Lemma 12.9 in [BauerTaylor], p. 56 (but where both 𝐴 and 𝐵 are positive, not just 𝐴). (Contributed by Jim Kingdon, 8-Dec-2019.)
((𝐴P𝐵P) → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))

Theoremmulclpr 6824 Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.)
((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)

Theoremmulnqprlemrl 6825* Lemma for mulnqpr 6829. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴Q𝐵Q) → (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))

Theoremmulnqprlemru 6826* Lemma for mulnqpr 6829. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴Q𝐵Q) → (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))

Theoremmulnqprlemfl 6827* Lemma for mulnqpr 6829. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴Q𝐵Q) → (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))

Theoremmulnqprlemfu 6828* Lemma for mulnqpr 6829. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴Q𝐵Q) → (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))

Theoremmulnqpr 6829* Multiplication of fractions embedded into positive reals. One can either multiply the fractions as fractions, or embed them into positive reals and multiply them as positive reals, and get the same result. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴Q𝐵Q) → ⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))

Theoremaddcomprg 6830 Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)
((𝐴P𝐵P) → (𝐴 +P 𝐵) = (𝐵 +P 𝐴))

Theoremaddassprg 6831 Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)
((𝐴P𝐵P𝐶P) → ((𝐴 +P 𝐵) +P 𝐶) = (𝐴 +P (𝐵 +P 𝐶)))

Theoremmulcomprg 6832 Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
((𝐴P𝐵P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴))

Theoremmulassprg 6833 Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
((𝐴P𝐵P𝐶P) → ((𝐴 ·P 𝐵) ·P 𝐶) = (𝐴 ·P (𝐵 ·P 𝐶)))

Theoremdistrlem1prl 6834 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
((𝐴P𝐵P𝐶P) → (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))

Theoremdistrlem1pru 6835 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
((𝐴P𝐵P𝐶P) → (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))

Theoremdistrlem4prl 6836* Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
(((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))

Theoremdistrlem4pru 6837* Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
(((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))

Theoremdistrlem5prl 6838 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
((𝐴P𝐵P𝐶P) → (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))

Theoremdistrlem5pru 6839 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
((𝐴P𝐵P𝐶P) → (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))

Theoremdistrprg 6840 Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.)
((𝐴P𝐵P𝐶P) → (𝐴 ·P (𝐵 +P 𝐶)) = ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))

Theoremltprordil 6841 If a positive real is less than a second positive real, its lower cut is a subset of the second's lower cut. (Contributed by Jim Kingdon, 23-Dec-2019.)
(𝐴<P 𝐵 → (1st𝐴) ⊆ (1st𝐵))

Theorem1idprl 6842 Lemma for 1idpr 6844. (Contributed by Jim Kingdon, 13-Dec-2019.)
(𝐴P → (1st ‘(𝐴 ·P 1P)) = (1st𝐴))

Theorem1idpru 6843 Lemma for 1idpr 6844. (Contributed by Jim Kingdon, 13-Dec-2019.)
(𝐴P → (2nd ‘(𝐴 ·P 1P)) = (2nd𝐴))

Theorem1idpr 6844 1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.)
(𝐴P → (𝐴 ·P 1P) = 𝐴)

Theoremltnqpr 6845* We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 19-Jun-2021.)
((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))

Theoremltnqpri 6846* We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 8-Jan-2021.)
(𝐴 <Q 𝐵 → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)

Theoremltpopr 6847 Positive real 'less than' is a partial ordering. Remark ("< is transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p. (varies). Lemma for ltsopr 6848. (Contributed by Jim Kingdon, 15-Dec-2019.)
<P Po P

Theoremltsopr 6848 Positive real 'less than' is a weak linear order (in the sense of df-iso 4060). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
<P Or P

Theoremltaddpr 6849 The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
((𝐴P𝐵P) → 𝐴<P (𝐴 +P 𝐵))

Theoremltexprlemell 6850* Element in lower cut of the constructed difference. Lemma for ltexpri 6865. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))

Theoremltexprlemelu 6851* Element in upper cut of the constructed difference. Lemma for ltexpri 6865. (Contributed by Jim Kingdon, 21-Dec-2019.)
𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩       (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))

Theoremltexprlemm 6852* Our constructed difference is inhabited. Lemma for ltexpri 6865. (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 𝑟 ∈ (2nd𝐶)))

Theoremltexprlemopl 6853* The lower cut of our constructed difference is open. Lemma for ltexpri 6865. (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 𝐵𝑞Q𝑞 ∈ (1st𝐶)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)))

Theoremltexprlemlol 6854* The lower cut of our constructed difference is lower. Lemma for ltexpri 6865. (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 𝐵𝑞Q) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) → 𝑞 ∈ (1st𝐶)))

Theoremltexprlemopu 6855* The upper cut of our constructed difference is open. Lemma for ltexpri 6865. (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 6856* The upper cut of our constructed difference is upper. Lemma for ltexpri 6865. (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 6857* Our constructed difference is rounded. Lemma for ltexpri 6865. (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 6858* Our constructed difference is disjoint. Lemma for ltexpri 6865. (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 6859* Our constructed difference is located. Lemma for ltexpri 6865. (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 6860* Our constructed difference is a positive real. Lemma for ltexpri 6865. (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 6861* Lemma for ltexpri 6865. 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 6862* Lemma for ltexpri 6865. 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 6863* Lemma for ltexpri 6865. 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 6864* Lemma for ltexpri 6865. 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 6865* 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 6868 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 6869* The difference from ltexpri 6865 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
(𝐴<P 𝐵 → ∃!𝑥P (𝐴 +P 𝑥) = 𝐵)

Theoremltaprlem 6870 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 6871 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 6872* 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 6873 Strong extensionality of addition (ordering version). This is similar to addext 7777 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 6874* Membership in the lower cut of 𝐵. Lemma for recexpr 6890. (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 6875* Membership in the upper cut of 𝐵. Lemma for recexpr 6890. (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 6876* 𝐵 is inhabited. Lemma for recexpr 6890. (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 6877* The lower cut of 𝐵 is open. Lemma for recexpr 6890. (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 6878* The lower cut of 𝐵 is lower. Lemma for recexpr 6890. (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 6879* The upper cut of 𝐵 is open. Lemma for recexpr 6890. (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 6880* The upper cut of 𝐵 is upper. Lemma for recexpr 6890. (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 6881* 𝐵 is rounded. Lemma for recexpr 6890. (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 6882* 𝐵 is disjoint. Lemma for recexpr 6890. (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 6883* 𝐵 is located. Lemma for recexpr 6890. (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 6884* 𝐵 is a positive real. Lemma for recexpr 6890. (Contributed by Jim Kingdon, 27-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       (𝐴P𝐵P)

Theoremrecexprlem1ssl 6885* The lower cut of one is a subset of the lower cut of 𝐴 ·P 𝐵. Lemma for recexpr 6890. (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 6886* The upper cut of one is a subset of the upper cut of 𝐴 ·P 𝐵. Lemma for recexpr 6890. (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 6887* The lower cut of 𝐴 ·P 𝐵 is a subset of the lower cut of one. Lemma for recexpr 6890. (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 6888* The upper cut of 𝐴 ·P 𝐵 is a subset of the upper cut of one. Lemma for recexpr 6890. (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 6889* 𝐵 is the reciprocal of 𝐴. Lemma for recexpr 6890. (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 6890* 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 6891 Lemma for aptipr 6893. (Contributed by Jim Kingdon, 28-Jan-2020.)
((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (1st𝐴) ⊆ (1st𝐵))

Theoremaptiprlemu 6892 Lemma for aptipr 6893. (Contributed by Jim Kingdon, 28-Jan-2020.)
((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (2nd𝐵) ⊆ (2nd𝐴))

Theoremaptipr 6893 Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.)
((𝐴P𝐵P ∧ ¬ (𝐴<P 𝐵𝐵<P 𝐴)) → 𝐴 = 𝐵)

Theoremltmprr 6894 Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.)
((𝐴P𝐵P𝐶P) → ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → 𝐴<P 𝐵))

Theoremarchpr 6895* 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 6805. (Contributed by Jim Kingdon, 22-Apr-2020.)
(𝐴P → ∃𝑥N 𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩)

Theoremcaucvgprlemcanl 6896* Lemma for cauappcvgprlemladdrl 6909. 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 6897* Lemma for cauappcvgpr 6914. 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 6898* Lemma for cauappcvgpr 6914. 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 6899* Lemma for cauappcvgpr 6914. 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 6900* Lemma for cauappcvgpr 6914. 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)))

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