P. 80 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dmplp 7901	Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.)
⊢ dom +P = (P × P)
Theorem	dmmp 7902	Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.)
⊢ dom ·P = (P × P)
Theorem	nqprm 7903*	A cut produced from a rational is inhabited. Lemma for nqprlu 7908. (Contributed by Jim Kingdon, 8-Dec-2019.)
⊢ (𝐴 ∈ Q → (∃𝑞 ∈ Q 𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∧ ∃𝑟 ∈ Q 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}))
Theorem	nqprrnd 7904*	A cut produced from a rational is rounded. Lemma for nqprlu 7908. (Contributed by Jim Kingdon, 8-Dec-2019.)
⊢ (𝐴 ∈ Q → (∀𝑞 ∈ Q (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ {𝑥 ∣ 𝑥 <Q 𝐴})) ∧ ∀𝑟 ∈ Q (𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}))))
Theorem	nqprdisj 7905*	A cut produced from a rational is disjoint. Lemma for nqprlu 7908. (Contributed by Jim Kingdon, 8-Dec-2019.)
⊢ (𝐴 ∈ Q → ∀𝑞 ∈ Q ¬ (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}))
Theorem	nqprloc 7906*	A cut produced from a rational is located. Lemma for nqprlu 7908. (Contributed by Jim Kingdon, 8-Dec-2019.)
⊢ (𝐴 ∈ Q → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∨ 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥})))
Theorem	nqprxx 7907*	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 7908*	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 7909*	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 7910	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 7911	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 7912*	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 7913*	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 7914*	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 7915	The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
⊢ 1P ∈ P
Theorem	1prl 7916	The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
⊢ (1st ‘1P) = {𝑥 ∣ 𝑥 <Q 1Q}
Theorem	1pru 7917	The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
⊢ (2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥}
Theorem	addnqprlemrl 7918*	Lemma for addnqpr 7922. 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 7919*	Lemma for addnqpr 7922. 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 7920*	Lemma for addnqpr 7922. 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 7921*	Lemma for addnqpr 7922. 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 7922*	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 7923*	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 7922. (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 7924*	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 7925*	Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7924 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 7926	Calculations for prmuloc 7927. (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 7927*	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 7928*	Positive reals are multiplicatively located. This is a variation of prmuloc 7927 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 7929	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 7930	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 7931	Calculations for mullocpr 7932. (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 7932*	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 7933	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 7934*	Lemma for mulnqpr 7938. 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 7935*	Lemma for mulnqpr 7938. 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 7936*	Lemma for mulnqpr 7938. 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 7937*	Lemma for mulnqpr 7938. 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 7938*	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 7939	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 7940	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 7941	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 7942	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 7943	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 7944	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 7945*	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 7946*	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 7947	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 7948	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 7949	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 7950	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 7951	Lemma for 1idpr 7953. (Contributed by Jim Kingdon, 13-Dec-2019.)
⊢ (𝐴 ∈ P → (1st ‘(𝐴 ·P 1P)) = (1st ‘𝐴))
Theorem	1idpru 7952	Lemma for 1idpr 7953. (Contributed by Jim Kingdon, 13-Dec-2019.)
⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 1P)) = (2nd ‘𝐴))
Theorem	1idpr 7953	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 7954*	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 7955*	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 7956	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 7957. (Contributed by Jim Kingdon, 15-Dec-2019.)
⊢ <P Po P
Theorem	ltsopr 7957	Positive real 'less than' is a weak linear order (in the sense of df-iso 4440). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
⊢ <P Or P
Theorem	ltaddpr 7958	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 7959*	Element in lower cut of the constructed difference. Lemma for ltexpri 7974. (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 7960*	Element in upper cut of the constructed difference. Lemma for ltexpri 7974. (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 7961*	Our constructed difference is inhabited. Lemma for ltexpri 7974. (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 7962*	The lower cut of our constructed difference is open. Lemma for ltexpri 7974. (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 7963*	The lower cut of our constructed difference is lower. Lemma for ltexpri 7974. (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 7964*	The upper cut of our constructed difference is open. Lemma for ltexpri 7974. (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 7965*	The upper cut of our constructed difference is upper. Lemma for ltexpri 7974. (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 7966*	Our constructed difference is rounded. Lemma for ltexpri 7974. (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 7967*	Our constructed difference is disjoint. Lemma for ltexpri 7974. (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 7968*	Our constructed difference is located. Lemma for ltexpri 7974. (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 7969*	Our constructed difference is a positive real. Lemma for ltexpri 7974. (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 7970*	Lemma for ltexpri 7974. 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 7971*	Lemma for ltexpri 7974. 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 7972*	Lemma for ltexpri 7974. 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 7973*	Lemma for ltexpri 7974. 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 7974*	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 7975	Lemma for addcanprg 7977. (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 7976	Lemma for addcanprg 7977. (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 7977	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 7978*	The difference from ltexpri 7974 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
⊢ (𝐴<P 𝐵 → ∃!𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵)
Theorem	ltaprlem 7979	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 7980	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 7981*	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 7982	Strong extensionality of addition (ordering version). This is similar to addext 8932 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 7983*	Membership in the lower cut of 𝐵. Lemma for recexpr 7999. (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 7984*	Membership in the upper cut of 𝐵. Lemma for recexpr 7999. (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 7985*	𝐵 is inhabited. Lemma for recexpr 7999. (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 7986*	The lower cut of 𝐵 is open. Lemma for recexpr 7999. (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 7987*	The lower cut of 𝐵 is lower. Lemma for recexpr 7999. (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 7988*	The upper cut of 𝐵 is open. Lemma for recexpr 7999. (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 7989*	The upper cut of 𝐵 is upper. Lemma for recexpr 7999. (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 7990*	𝐵 is rounded. Lemma for recexpr 7999. (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 7991*	𝐵 is disjoint. Lemma for recexpr 7999. (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 7992*	𝐵 is located. Lemma for recexpr 7999. (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 7993*	𝐵 is a positive real. Lemma for recexpr 7999. (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 7994*	The lower cut of one is a subset of the lower cut of 𝐴 ·P 𝐵. Lemma for recexpr 7999. (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 7995*	The upper cut of one is a subset of the upper cut of 𝐴 ·P 𝐵. Lemma for recexpr 7999. (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 7996*	The lower cut of 𝐴 ·P 𝐵 is a subset of the lower cut of one. Lemma for recexpr 7999. (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 7997*	The upper cut of 𝐴 ·P 𝐵 is a subset of the upper cut of one. Lemma for recexpr 7999. (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 7998*	𝐵 is the reciprocal of 𝐴. Lemma for recexpr 7999. (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 7999*	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 8000	Lemma for aptipr 8002. (Contributed by Jim Kingdon, 28-Jan-2020.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (1st ‘𝐴) ⊆ (1st ‘𝐵))