P. 78 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nqprxx 7701*	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.)
⊢ (𝐴 ∈ Q → ⟨{𝑥 ∣ 𝑥 <Q 𝐴}, {𝑥 ∣ 𝐴 <Q 𝑥}⟩ ∈ P)
Theorem	nqprlu 7702*	The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.)
⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
Theorem	recnnpr 7703*	The reciprocal of a positive integer, as a positive real. (Contributed by Jim Kingdon, 27-Feb-2021.)
⊢ (𝐴 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
Theorem	ltnqex 7704	The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
⊢ {𝑥 ∣ 𝑥 <Q 𝐴} ∈ V
Theorem	gtnqex 7705	The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
⊢ {𝑥 ∣ 𝐴 <Q 𝑥} ∈ V
Theorem	nqprl 7706*	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.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐴 ∈ (1st ‘𝐵) ↔ ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩<P 𝐵))
Theorem	nqpru 7707*	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.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐴 ∈ (2nd ‘𝐵) ↔ 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))
Theorem	nnprlu 7708*	The canonical embedding of positive integers into the positive reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
⊢ (𝐴 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝐴, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐴, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
Theorem	1pr 7709	The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
⊢ 1P ∈ P
Theorem	1prl 7710	The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
⊢ (1st ‘1P) = {𝑥 ∣ 𝑥 <Q 1Q}
Theorem	1pru 7711	The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
⊢ (2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥}
Theorem	addnqprlemrl 7712*	Lemma for addnqpr 7716. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
Theorem	addnqprlemru 7713*	Lemma for addnqpr 7716. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
Theorem	addnqprlemfl 7714*	Lemma for addnqpr 7716. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
Theorem	addnqprlemfu 7715*	Lemma for addnqpr 7716. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
Theorem	addnqpr 7716*	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.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))
Theorem	addnqpr1 7717*	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 7716. (Contributed by Jim Kingdon, 26-Apr-2020.)
⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 1Q)}, {𝑢 ∣ (𝐴 +Q 1Q) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P 1P))
Theorem	appdivnq 7718*	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 𝐵 ∧ 𝐶 ∈ Q) → ∃𝑚 ∈ Q (𝐴 <Q (𝑚 ·Q 𝐶) ∧ (𝑚 ·Q 𝐶) <Q 𝐵))
Theorem	appdiv0nq 7719*	Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7718 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.)
⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ∃𝑚 ∈ Q (𝑚 ·Q 𝐶) <Q 𝐵)
Theorem	prmuloclemcalc 7720	Calculations for prmuloc 7721. (Contributed by Jim Kingdon, 9-Dec-2019.)
⊢ (𝜑 → 𝑅 <Q 𝑈)    &   ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   ⊢ (𝜑 → (𝐴 +Q 𝑋) = 𝐵)    &   ⊢ (𝜑 → (𝑃 ·Q 𝐵) <Q (𝑅 ·Q 𝑋))    &   ⊢ (𝜑 → 𝐴 ∈ Q)    &   ⊢ (𝜑 → 𝐵 ∈ Q)    &   ⊢ (𝜑 → 𝐷 ∈ Q)    &   ⊢ (𝜑 → 𝑃 ∈ Q)    &   ⊢ (𝜑 → 𝑋 ∈ Q)    ⇒   ⊢ (𝜑 → (𝑈 ·Q 𝐴) <Q (𝐷 ·Q 𝐵))
Theorem	prmuloc 7721*	Positive reals are multiplicatively located. Lemma 12.8 of [BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 <Q 𝐵) → ∃𝑑 ∈ Q ∃𝑢 ∈ Q (𝑑 ∈ 𝐿 ∧ 𝑢 ∈ 𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
Theorem	prmuloc2 7722*	Positive reals are multiplicatively located. This is a variation of prmuloc 7721 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) → ∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)
Theorem	mulnqprl 7723	Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → 𝑋 ∈ (1st ‘(𝐴 ·P 𝐵))))
Theorem	mulnqpru 7724	Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → 𝑋 ∈ (2nd ‘(𝐴 ·P 𝐵))))
Theorem	mullocprlem 7725	Calculations for mullocpr 7726. (Contributed by Jim Kingdon, 10-Dec-2019.)
⊢ (𝜑 → (𝐴 ∈ P ∧ 𝐵 ∈ P))    &   ⊢ (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 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q))    &   ⊢ (𝜑 → (𝐷 ∈ Q ∧ 𝑈 ∈ Q))    &   ⊢ (𝜑 → (𝐷 ∈ (1st ‘𝐴) ∧ 𝑈 ∈ (2nd ‘𝐴)))    &   ⊢ (𝜑 → (𝐸 ∈ Q ∧ 𝑇 ∈ Q))    ⇒   ⊢ (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
Theorem	mullocpr 7726*	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.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
Theorem	mulclpr 7727	Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
Theorem	mulnqprlemrl 7728*	Lemma for mulnqpr 7732. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
Theorem	mulnqprlemru 7729*	Lemma for mulnqpr 7732. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
Theorem	mulnqprlemfl 7730*	Lemma for mulnqpr 7732. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
Theorem	mulnqprlemfu 7731*	Lemma for mulnqpr 7732. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
Theorem	mulnqpr 7732*	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.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))
Theorem	addcomprg 7733	Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) = (𝐵 +P 𝐴))
Theorem	addassprg 7734	Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 +P 𝐵) +P 𝐶) = (𝐴 +P (𝐵 +P 𝐶)))
Theorem	mulcomprg 7735	Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴))
Theorem	mulassprg 7736	Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 ·P 𝐵) ·P 𝐶) = (𝐴 ·P (𝐵 ·P 𝐶)))
Theorem	distrlem1prl 7737	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
Theorem	distrlem1pru 7738	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
Theorem	distrlem4prl 7739*	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵)) ∧ (𝑓 ∈ (1st ‘𝐴) ∧ 𝑧 ∈ (1st ‘𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
Theorem	distrlem4pru 7740*	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
Theorem	distrlem5prl 7741	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
Theorem	distrlem5pru 7742	Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
Theorem	distrprg 7743	Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) = ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
Theorem	ltprordil 7744	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 → (1st ‘𝐴) ⊆ (1st ‘𝐵))
Theorem	1idprl 7745	Lemma for 1idpr 7747. (Contributed by Jim Kingdon, 13-Dec-2019.)
⊢ (𝐴 ∈ P → (1st ‘(𝐴 ·P 1P)) = (1st ‘𝐴))
Theorem	1idpru 7746	Lemma for 1idpr 7747. (Contributed by Jim Kingdon, 13-Dec-2019.)
⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 1P)) = (2nd ‘𝐴))
Theorem	1idpr 7747	1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.)
⊢ (𝐴 ∈ P → (𝐴 ·P 1P) = 𝐴)
Theorem	ltnqpr 7748*	We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 19-Jun-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))
Theorem	ltnqpri 7749*	We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 8-Jan-2021.)
⊢ (𝐴 <Q 𝐵 → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)
Theorem	ltpopr 7750	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 7751. (Contributed by Jim Kingdon, 15-Dec-2019.)
⊢ <P Po P
Theorem	ltsopr 7751	Positive real 'less than' is a weak linear order (in the sense of df-iso 4365). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
⊢ <P Or P
Theorem	ltaddpr 7752	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.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 𝐴<P (𝐴 +P 𝐵))
Theorem	ltexprlemell 7753*	Element in lower cut of the constructed difference. Lemma for ltexpri 7768. (Contributed by Jim Kingdon, 21-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝑞 ∈ (1st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
Theorem	ltexprlemelu 7754*	Element in upper cut of the constructed difference. Lemma for ltexpri 7768. (Contributed by Jim Kingdon, 21-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝑟 ∈ (2nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))))
Theorem	ltexprlemm 7755*	Our constructed difference is inhabited. Lemma for ltexpri 7768. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶)))
Theorem	ltexprlemopl 7756*	The lower cut of our constructed difference is open. Lemma for ltexpri 7768. (Contributed by Jim Kingdon, 21-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ ((𝐴<P 𝐵 ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1st ‘𝐶)) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶)))
Theorem	ltexprlemlol 7757*	The lower cut of our constructed difference is lower. Lemma for ltexpri 7768. (Contributed by Jim Kingdon, 21-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ ((𝐴<P 𝐵 ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶)) → 𝑞 ∈ (1st ‘𝐶)))
Theorem	ltexprlemopu 7758*	The upper cut of our constructed difference is open. Lemma for ltexpri 7768. (Contributed by Jim Kingdon, 21-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)))
Theorem	ltexprlemupu 7759*	The upper cut of our constructed difference is upper. Lemma for ltexpri 7768. (Contributed by Jim Kingdon, 21-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)) → 𝑟 ∈ (2nd ‘𝐶)))
Theorem	ltexprlemrnd 7760*	Our constructed difference is rounded. Lemma for ltexpri 7768. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → (∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)))))
Theorem	ltexprlemdisj 7761*	Our constructed difference is disjoint. Lemma for ltexpri 7768. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐶) ∧ 𝑞 ∈ (2nd ‘𝐶)))
Theorem	ltexprlemloc 7762*	Our constructed difference is located. Lemma for ltexpri 7768. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶))))
Theorem	ltexprlempr 7763*	Our constructed difference is a positive real. Lemma for ltexpri 7768. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → 𝐶 ∈ P)
Theorem	ltexprlemfl 7764*	Lemma for ltexpri 7768. One direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → (1st ‘(𝐴 +P 𝐶)) ⊆ (1st ‘𝐵))
Theorem	ltexprlemrl 7765*	Lemma for ltexpri 7768. Reverse direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → (1st ‘𝐵) ⊆ (1st ‘(𝐴 +P 𝐶)))
Theorem	ltexprlemfu 7766*	Lemma for ltexpri 7768. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → (2nd ‘(𝐴 +P 𝐶)) ⊆ (2nd ‘𝐵))
Theorem	ltexprlemru 7767*	Lemma for ltexpri 7768. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ⊢ (𝐴<P 𝐵 → (2nd ‘𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶)))
Theorem	ltexpri 7768*	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 → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵)
Theorem	addcanprleml 7769	Lemma for addcanprg 7771. (Contributed by Jim Kingdon, 25-Dec-2019.)
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st ‘𝐵) ⊆ (1st ‘𝐶))
Theorem	addcanprlemu 7770	Lemma for addcanprg 7771. (Contributed by Jim Kingdon, 25-Dec-2019.)
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd ‘𝐵) ⊆ (2nd ‘𝐶))
Theorem	addcanprg 7771	Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by Jim Kingdon, 24-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶))
Theorem	lteupri 7772*	The difference from ltexpri 7768 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
⊢ (𝐴<P 𝐵 → ∃!𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵)
Theorem	ltaprlem 7773	Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.)
⊢ (𝐶 ∈ P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
Theorem	ltaprg 7774	Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
Theorem	prplnqu 7775*	Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.)
⊢ (𝜑 → 𝑋 ∈ P)    &   ⊢ (𝜑 → 𝑄 ∈ Q)    &   ⊢ (𝜑 → 𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)))    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ (2nd ‘𝑋)(𝑦 +Q 𝑄) = 𝐴)
Theorem	addextpr 7776	Strong extensionality of addition (ordering version). This is similar to addext 8725 but for positive reals and based on less-than rather than apartness. (Contributed by Jim Kingdon, 17-Feb-2020.)
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ((𝐴 +P 𝐵)<P (𝐶 +P 𝐷) → (𝐴<P 𝐶 ∨ 𝐵<P 𝐷)))
Theorem	recexprlemell 7777*	Membership in the lower cut of 𝐵. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 27-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ (𝐶 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))
Theorem	recexprlemelu 7778*	Membership in the upper cut of 𝐵. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 27-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ (𝐶 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝐶 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)))
Theorem	recexprlemm 7779*	𝐵 is inhabited. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 27-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ (𝐴 ∈ P → (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐵) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐵)))
Theorem	recexprlemopl 7780*	The lower cut of 𝐵 is open. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 28-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1st ‘𝐵)) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵)))
Theorem	recexprlemlol 7781*	The lower cut of 𝐵 is lower. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 28-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵)) → 𝑞 ∈ (1st ‘𝐵)))
Theorem	recexprlemopu 7782*	The upper cut of 𝐵 is open. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 28-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐵)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)))
Theorem	recexprlemupu 7783*	The upper cut of 𝐵 is upper. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 28-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)) → 𝑟 ∈ (2nd ‘𝐵)))
Theorem	recexprlemrnd 7784*	𝐵 is rounded. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 27-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ (𝐴 ∈ P → (∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐵) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)))))
Theorem	recexprlemdisj 7785*	𝐵 is disjoint. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 27-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ (𝐴 ∈ P → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐵) ∧ 𝑞 ∈ (2nd ‘𝐵)))
Theorem	recexprlemloc 7786*	𝐵 is located. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 27-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ (𝐴 ∈ P → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵))))
Theorem	recexprlempr 7787*	𝐵 is a positive real. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 27-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
Theorem	recexprlem1ssl 7788*	The lower cut of one is a subset of the lower cut of 𝐴 ·P 𝐵. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 27-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ (𝐴 ∈ P → (1st ‘1P) ⊆ (1st ‘(𝐴 ·P 𝐵)))
Theorem	recexprlem1ssu 7789*	The upper cut of one is a subset of the upper cut of 𝐴 ·P 𝐵. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 27-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ (𝐴 ∈ P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))
Theorem	recexprlemss1l 7790*	The lower cut of 𝐴 ·P 𝐵 is a subset of the lower cut of one. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 27-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ (𝐴 ∈ P → (1st ‘(𝐴 ·P 𝐵)) ⊆ (1st ‘1P))
Theorem	recexprlemss1u 7791*	The upper cut of 𝐴 ·P 𝐵 is a subset of the upper cut of one. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 27-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))
Theorem	recexprlemex 7792*	𝐵 is the reciprocal of 𝐴. Lemma for recexpr 7793. (Contributed by Jim Kingdon, 27-Dec-2019.)
⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ⊢ (𝐴 ∈ P → (𝐴 ·P 𝐵) = 1P)
Theorem	recexpr 7793*	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.)
⊢ (𝐴 ∈ P → ∃𝑥 ∈ P (𝐴 ·P 𝑥) = 1P)
Theorem	aptiprleml 7794	Lemma for aptipr 7796. (Contributed by Jim Kingdon, 28-Jan-2020.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (1st ‘𝐴) ⊆ (1st ‘𝐵))
Theorem	aptiprlemu 7795	Lemma for aptipr 7796. (Contributed by Jim Kingdon, 28-Jan-2020.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (2nd ‘𝐵) ⊆ (2nd ‘𝐴))
Theorem	aptipr 7796	Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐴 = 𝐵)
Theorem	ltmprr 7797	Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → 𝐴<P 𝐵))
Theorem	archpr 7798*	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 7708. (Contributed by Jim Kingdon, 22-Apr-2020.)
⊢ (𝐴 ∈ P → ∃𝑥 ∈ N 𝐴<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)
Theorem	caucvgprlemcanl 7799*	Lemma for cauappcvgprlemladdrl 7812. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.)
⊢ (𝜑 → 𝐿 ∈ P)    &   ⊢ (𝜑 → 𝑆 ∈ Q)    &   ⊢ (𝜑 → 𝑅 ∈ Q)    &   ⊢ (𝜑 → 𝑄 ∈ Q)    ⇒   ⊢ (𝜑 → ((𝑅 +Q 𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)) ↔ 𝑅 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))))
Theorem	cauappcvgprlemm 7800*	Lemma for cauappcvgpr 7817. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q)    &   ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    ⇒   ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))