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Theorem List for Intuitionistic Logic Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmpvlu 7001* Value of multiplication on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
((𝐴P𝐵P) → (𝐴 ·P 𝐵) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 ·Q 𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 ·Q 𝑧)}⟩)
 
Theoremdmplp 7002 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.)
dom +P = (P × P)
 
Theoremdmmp 7003 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.)
dom ·P = (P × P)
 
Theoremnqprm 7004* A cut produced from a rational is inhabited. Lemma for nqprlu 7009. (Contributed by Jim Kingdon, 8-Dec-2019.)
(𝐴Q → (∃𝑞Q 𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∧ ∃𝑟Q 𝑟 ∈ {𝑥𝐴 <Q 𝑥}))
 
Theoremnqprrnd 7005* A cut produced from a rational is rounded. Lemma for nqprlu 7009. (Contributed by Jim Kingdon, 8-Dec-2019.)
(𝐴Q → (∀𝑞Q (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ {𝑥𝑥 <Q 𝐴})) ∧ ∀𝑟Q (𝑟 ∈ {𝑥𝐴 <Q 𝑥} ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ {𝑥𝐴 <Q 𝑥}))))
 
Theoremnqprdisj 7006* A cut produced from a rational is disjoint. Lemma for nqprlu 7009. (Contributed by Jim Kingdon, 8-Dec-2019.)
(𝐴Q → ∀𝑞Q ¬ (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∧ 𝑞 ∈ {𝑥𝐴 <Q 𝑥}))
 
Theoremnqprloc 7007* A cut produced from a rational is located. Lemma for nqprlu 7009. (Contributed by Jim Kingdon, 8-Dec-2019.)
(𝐴Q → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∨ 𝑟 ∈ {𝑥𝐴 <Q 𝑥})))
 
Theoremnqprxx 7008* The canonical embedding of the rationals into the reals, expressed with the same variable for the lower and upper cuts. (Contributed by Jim Kingdon, 8-Dec-2019.)
(𝐴Q → ⟨{𝑥𝑥 <Q 𝐴}, {𝑥𝐴 <Q 𝑥}⟩ ∈ P)
 
Theoremnqprlu 7009* The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.)
(𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
 
Theoremrecnnpr 7010* The reciprocal of a positive integer, as a positive real. (Contributed by Jim Kingdon, 27-Feb-2021.)
(𝐴N → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
 
Theoremltnqex 7011 The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
{𝑥𝑥 <Q 𝐴} ∈ V
 
Theoremgtnqex 7012 The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
{𝑥𝐴 <Q 𝑥} ∈ V
 
Theoremnqprl 7013* 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 7014* 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 7015* 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 7016 The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
1PP
 
Theorem1prl 7017 The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
(1st ‘1P) = {𝑥𝑥 <Q 1Q}
 
Theorem1pru 7018 The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
(2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥}
 
Theoremaddnqprlemrl 7019* Lemma for addnqpr 7023. 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 7020* Lemma for addnqpr 7023. 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 7021* Lemma for addnqpr 7023. 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 7022* Lemma for addnqpr 7023. 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 7023* 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 7024* 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 7023. (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 7025* 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 7026* Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7025 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 7027 Calculations for prmuloc 7028. (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 7028* 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 7029* Positive reals are multiplicatively located. This is a variation of prmuloc 7028 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 7030 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 7031 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 7032 Calculations for mullocpr 7033. (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 7033* 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 7034 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 7035* Lemma for mulnqpr 7039. 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 7036* Lemma for mulnqpr 7039. 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 7037* Lemma for mulnqpr 7039. 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 7038* Lemma for mulnqpr 7039. 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 7039* 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 7040 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 7041 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 7042 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 7043 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 7044 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 7045 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 7046* 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 7047* 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 7048 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 7049 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 7050 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 7051 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 7052 Lemma for 1idpr 7054. (Contributed by Jim Kingdon, 13-Dec-2019.)
(𝐴P → (1st ‘(𝐴 ·P 1P)) = (1st𝐴))
 
Theorem1idpru 7053 Lemma for 1idpr 7054. (Contributed by Jim Kingdon, 13-Dec-2019.)
(𝐴P → (2nd ‘(𝐴 ·P 1P)) = (2nd𝐴))
 
Theorem1idpr 7054 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 7055* 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 7056* 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 7057 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 7058. (Contributed by Jim Kingdon, 15-Dec-2019.)
<P Po P
 
Theoremltsopr 7058 Positive real 'less than' is a weak linear order (in the sense of df-iso 4088). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
<P Or P
 
Theoremltaddpr 7059 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 7060* Element in lower cut of the constructed difference. Lemma for ltexpri 7075. (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 7061* Element in upper cut of the constructed difference. Lemma for ltexpri 7075. (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 7062* Our constructed difference is inhabited. Lemma for ltexpri 7075. (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 7063* The lower cut of our constructed difference is open. Lemma for ltexpri 7075. (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 7064* The lower cut of our constructed difference is lower. Lemma for ltexpri 7075. (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 7065* The upper cut of our constructed difference is open. Lemma for ltexpri 7075. (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 7066* The upper cut of our constructed difference is upper. Lemma for ltexpri 7075. (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 7067* Our constructed difference is rounded. Lemma for ltexpri 7075. (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 7068* Our constructed difference is disjoint. Lemma for ltexpri 7075. (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 7069* Our constructed difference is located. Lemma for ltexpri 7075. (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 7070* Our constructed difference is a positive real. Lemma for ltexpri 7075. (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 7071* Lemma for ltexpri 7075. 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 7072* Lemma for ltexpri 7075. 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 7073* Lemma for ltexpri 7075. 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 7074* Lemma for ltexpri 7075. 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 7075* 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)
 
Theoremaddcanprleml 7076 Lemma for addcanprg 7078. (Contributed by Jim Kingdon, 25-Dec-2019.)
(((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐵) ⊆ (1st𝐶))
 
Theoremaddcanprlemu 7077 Lemma for addcanprg 7078. (Contributed by Jim Kingdon, 25-Dec-2019.)
(((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐵) ⊆ (2nd𝐶))
 
Theoremaddcanprg 7078 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 7079* The difference from ltexpri 7075 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
(𝐴<P 𝐵 → ∃!𝑥P (𝐴 +P 𝑥) = 𝐵)
 
Theoremltaprlem 7080 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 7081 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 7082* 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 7083 Strong extensionality of addition (ordering version). This is similar to addext 7987 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 7084* Membership in the lower cut of 𝐵. Lemma for recexpr 7100. (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 7085* Membership in the upper cut of 𝐵. Lemma for recexpr 7100. (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 7086* 𝐵 is inhabited. Lemma for recexpr 7100. (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 7087* The lower cut of 𝐵 is open. Lemma for recexpr 7100. (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 7088* The lower cut of 𝐵 is lower. Lemma for recexpr 7100. (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 7089* The upper cut of 𝐵 is open. Lemma for recexpr 7100. (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 7090* The upper cut of 𝐵 is upper. Lemma for recexpr 7100. (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 7091* 𝐵 is rounded. Lemma for recexpr 7100. (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 7092* 𝐵 is disjoint. Lemma for recexpr 7100. (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 7093* 𝐵 is located. Lemma for recexpr 7100. (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 7094* 𝐵 is a positive real. Lemma for recexpr 7100. (Contributed by Jim Kingdon, 27-Dec-2019.)
𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩       (𝐴P𝐵P)
 
Theoremrecexprlem1ssl 7095* The lower cut of one is a subset of the lower cut of 𝐴 ·P 𝐵. Lemma for recexpr 7100. (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 7096* The upper cut of one is a subset of the upper cut of 𝐴 ·P 𝐵. Lemma for recexpr 7100. (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 7097* The lower cut of 𝐴 ·P 𝐵 is a subset of the lower cut of one. Lemma for recexpr 7100. (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 7098* The upper cut of 𝐴 ·P 𝐵 is a subset of the upper cut of one. Lemma for recexpr 7100. (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 7099* 𝐵 is the reciprocal of 𝐴. Lemma for recexpr 7100. (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 7100* 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)
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