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	elprnql 7701	An element of a positive real's lower cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐵 ∈ 𝐿) → 𝐵 ∈ Q)
Theorem	elprnqu 7702	An element of a positive real's upper cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐵 ∈ 𝑈) → 𝐵 ∈ Q)
Theorem	0npr 7703	The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
⊢ ¬ ∅ ∈ P
Theorem	prcdnql 7704	A lower cut is closed downwards under the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐵 ∈ 𝐿) → (𝐶 <Q 𝐵 → 𝐶 ∈ 𝐿))
Theorem	prcunqu 7705	An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐶 ∈ 𝑈) → (𝐶 <Q 𝐵 → 𝐵 ∈ 𝑈))
Theorem	prubl 7706	A positive fraction not in a lower cut is an upper bound. (Contributed by Jim Kingdon, 29-Sep-2019.)
⊢ (((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐵 ∈ 𝐿) ∧ 𝐶 ∈ Q) → (¬ 𝐶 ∈ 𝐿 → 𝐵 <Q 𝐶))
Theorem	prltlu 7707	An element of a lower cut is less than an element of the corresponding upper cut. (Contributed by Jim Kingdon, 15-Oct-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐵 ∈ 𝐿 ∧ 𝐶 ∈ 𝑈) → 𝐵 <Q 𝐶)
Theorem	prnmaxl 7708*	A lower cut has no largest member. (Contributed by Jim Kingdon, 29-Sep-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐵 ∈ 𝐿) → ∃𝑥 ∈ 𝐿 𝐵 <Q 𝑥)
Theorem	prnminu 7709*	An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐵 ∈ 𝑈) → ∃𝑥 ∈ 𝑈 𝑥 <Q 𝐵)
Theorem	prnmaddl 7710*	A lower cut has no largest member. Addition version. (Contributed by Jim Kingdon, 29-Sep-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐵 ∈ 𝐿) → ∃𝑥 ∈ Q (𝐵 +Q 𝑥) ∈ 𝐿)
Theorem	prloc 7711	A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 <Q 𝐵) → (𝐴 ∈ 𝐿 ∨ 𝐵 ∈ 𝑈))
Theorem	prdisj 7712	A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ Q) → ¬ (𝐴 ∈ 𝐿 ∧ 𝐴 ∈ 𝑈))
Theorem	prarloclemlt 7713	Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7723. (Contributed by Jim Kingdon, 10-Nov-2019.)
⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → (𝐴 +Q ([⟨(𝑦 +o 1o), 1o⟩] ~Q ·Q 𝑃)) <Q (𝐴 +Q ([⟨((𝑦 +o 2o) +o 𝑋), 1o⟩] ~Q ·Q 𝑃)))
Theorem	prarloclemlo 7714*	Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 7723. (Contributed by Jim Kingdon, 10-Nov-2019.)
⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → ((𝐴 +Q ([⟨(𝑦 +o 1o), 1o⟩] ~Q ·Q 𝑃)) ∈ 𝐿 → (((𝐴 +Q0 ([⟨𝑦, 1o⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +o 2o) +o suc 𝑋), 1o⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1o⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +o 2o) +o 𝑋), 1o⟩] ~Q ·Q 𝑃)) ∈ 𝑈))))
Theorem	prarloclemup 7715	Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 7723. (Contributed by Jim Kingdon, 10-Nov-2019.)
⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → ((𝐴 +Q ([⟨((𝑦 +o 2o) +o 𝑋), 1o⟩] ~Q ·Q 𝑃)) ∈ 𝑈 → (((𝐴 +Q0 ([⟨𝑦, 1o⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +o 2o) +o suc 𝑋), 1o⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1o⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +o 2o) +o 𝑋), 1o⟩] ~Q ·Q 𝑃)) ∈ 𝑈))))
Theorem	prarloclem3step 7716*	Induction step for prarloclem3 7717. (Contributed by Jim Kingdon, 9-Nov-2019.)
⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1o⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +o 2o) +o suc 𝑋), 1o⟩] ~Q ·Q 𝑃)) ∈ 𝑈)) → ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1o⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +o 2o) +o 𝑋), 1o⟩] ~Q ·Q 𝑃)) ∈ 𝑈))
Theorem	prarloclem3 7717*	Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7723. (Contributed by Jim Kingdon, 27-Oct-2019.)
⊢ (((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿) ∧ (𝑋 ∈ ω ∧ 𝑃 ∈ Q) ∧ ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1o⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +o 2o) +o 𝑋), 1o⟩] ~Q ·Q 𝑃)) ∈ 𝑈)) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([⟨𝑗, 1o⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨(𝑗 +o 2o), 1o⟩] ~Q ·Q 𝑃)) ∈ 𝑈))
Theorem	prarloclem4 7718*	A slight rearrangement of prarloclem3 7717. Lemma for prarloc 7723. (Contributed by Jim Kingdon, 4-Nov-2019.)
⊢ (((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿) ∧ 𝑃 ∈ Q) → (∃𝑥 ∈ ω ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1o⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +o 2o) +o 𝑥), 1o⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([⟨𝑗, 1o⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨(𝑗 +o 2o), 1o⟩] ~Q ·Q 𝑃)) ∈ 𝑈)))
Theorem	prarloclemn 7719*	Subtracting two from a positive integer. Lemma for prarloc 7723. (Contributed by Jim Kingdon, 5-Nov-2019.)
⊢ ((𝑁 ∈ N ∧ 1o <N 𝑁) → ∃𝑥 ∈ ω (2o +o 𝑥) = 𝑁)
Theorem	prarloclem5 7720*	A substitution of zero for 𝑦 and 𝑁 minus two for 𝑥. Lemma for prarloc 7723. (Contributed by Jim Kingdon, 4-Nov-2019.)
⊢ (((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿) ∧ (𝑁 ∈ N ∧ 𝑃 ∈ Q ∧ 1o <N 𝑁) ∧ (𝐴 +Q ([⟨𝑁, 1o⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑥 ∈ ω ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1o⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +o 2o) +o 𝑥), 1o⟩] ~Q ·Q 𝑃)) ∈ 𝑈))
Theorem	prarloclem 7721*	A special case of Lemma 6.16 from [BauerTaylor], p. 32. Given evenly spaced rational numbers from 𝐴 to 𝐴 +Q (𝑁 ·Q 𝑃) (which are in the lower and upper cuts, respectively, of a real number), there are a pair of numbers, two positions apart in the even spacing, which straddle the cut. (Contributed by Jim Kingdon, 22-Oct-2019.)
⊢ (((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿) ∧ (𝑁 ∈ N ∧ 𝑃 ∈ Q ∧ 1o <N 𝑁) ∧ (𝐴 +Q ([⟨𝑁, 1o⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([⟨𝑗, 1o⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨(𝑗 +o 2o), 1o⟩] ~Q ·Q 𝑃)) ∈ 𝑈))
Theorem	prarloclemcalc 7722	Some calculations for prarloc 7723. (Contributed by Jim Kingdon, 26-Oct-2019.)
⊢ (((𝐴 = (𝑋 +Q0 ([⟨𝑀, 1o⟩] ~Q0 ·Q0 𝑄)) ∧ 𝐵 = (𝑋 +Q ([⟨(𝑀 +o 2o), 1o⟩] ~Q ·Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +Q 𝑄) <Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝐵 <Q (𝐴 +Q 𝑃))
Theorem	prarloc 7723*	A Dedekind cut is arithmetically located. Part of Proposition 11.15 of [BauerTaylor], p. 52, slightly modified. It states that given a tolerance 𝑃, there are elements of the lower and upper cut which are within that tolerance of each other. Usually, proofs will be shorter if they use prarloc2 7724 instead. (Contributed by Jim Kingdon, 22-Oct-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃))
Theorem	prarloc2 7724*	A Dedekind cut is arithmetically located. This is a variation of prarloc 7723 which only constructs one (named) point and is therefore often easier to work with. It states that given a tolerance 𝑃, there are elements of the lower and upper cut which are exactly that tolerance from each other. (Contributed by Jim Kingdon, 26-Dec-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎 ∈ 𝐿 (𝑎 +Q 𝑃) ∈ 𝑈)
Theorem	ltrelpr 7725	Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
⊢ <P ⊆ (P × P)
Theorem	ltdfpr 7726*	More convenient form of df-iltp 7690. (Contributed by Jim Kingdon, 15-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵))))
Theorem	genpdflem 7727*	Simplification of upper or lower cut expression. Lemma for genpdf 7728. (Contributed by Jim Kingdon, 30-Sep-2019.)
⊢ ((𝜑 ∧ 𝑟 ∈ 𝐴) → 𝑟 ∈ Q)    &   ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ∈ Q)    ⇒   ⊢ (𝜑 → {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐵 ∧ 𝑞 = (𝑟𝐺𝑠))} = {𝑞 ∈ Q ∣ ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐵 𝑞 = (𝑟𝐺𝑠)})
Theorem	genpdf 7728*	Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑤) ∧ 𝑠 ∈ (1st ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑤) ∧ 𝑠 ∈ (2nd ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}⟩)    ⇒   ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1st ‘𝑤)∃𝑠 ∈ (1st ‘𝑣)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2nd ‘𝑤)∃𝑠 ∈ (2nd ‘𝑣)𝑞 = (𝑟𝐺𝑠)}⟩)
Theorem	genipv 7729*	Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1st ‘𝐴)∃𝑠 ∈ (1st ‘𝐵)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2nd ‘𝐴)∃𝑠 ∈ (2nd ‘𝐵)𝑞 = (𝑟𝐺𝑠)}⟩)
Theorem	genplt2i 7730*	Operating on both sides of two inequalities, when the operation is consistent with <Q. (Contributed by Jim Kingdon, 6-Oct-2019.)
⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    ⇒   ⊢ ((𝐴 <Q 𝐵 ∧ 𝐶 <Q 𝐷) → (𝐴𝐺𝐶) <Q (𝐵𝐺𝐷))
Theorem	genpelxp 7731*	Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q))
Theorem	genpelvl 7732*	Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1st ‘𝐴)∃ℎ ∈ (1st ‘𝐵)𝐶 = (𝑔𝐺ℎ)))
Theorem	genpelvu 7733*	Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝐶 = (𝑔𝐺ℎ)))
Theorem	genpprecll 7734*	Pre-closure law for general operation on lower cuts. (Contributed by Jim Kingdon, 2-Oct-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐶 ∈ (1st ‘𝐴) ∧ 𝐷 ∈ (1st ‘𝐵)) → (𝐶𝐺𝐷) ∈ (1st ‘(𝐴𝐹𝐵))))
Theorem	genppreclu 7735*	Pre-closure law for general operation on upper cuts. (Contributed by Jim Kingdon, 7-Nov-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐶 ∈ (2nd ‘𝐴) ∧ 𝐷 ∈ (2nd ‘𝐵)) → (𝐶𝐺𝐷) ∈ (2nd ‘(𝐴𝐹𝐵))))
Theorem	genipdm 7736*	Domain of general operation on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ dom 𝐹 = (P × P)
Theorem	genpml 7737*	The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
Theorem	genpmu 7738*	The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
Theorem	genpcdl 7739*	Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-Oct-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (1st ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1st ‘𝐵))) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔𝐺ℎ) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (1st ‘(𝐴𝐹𝐵)) → (𝑥 <Q 𝑓 → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵)))))
Theorem	genpcuu 7740*	Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2nd ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑔𝐺ℎ) <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
Theorem	genprndl 7741*	The lower cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    &   ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (1st ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1st ‘𝐵))) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔𝐺ℎ) → 𝑥 ∈ (1st ‘(𝐴𝐹𝐵))))    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘(𝐴𝐹𝐵)))))
Theorem	genprndu 7742*	The upper cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    &   ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2nd ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑔𝐺ℎ) <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))))
Theorem	genpdisj 7743*	The lower and upper cuts produced by addition or multiplication on positive reals are disjoint. (Contributed by Jim Kingdon, 15-Oct-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))
Theorem	genpassl 7744*	Associativity of lower cuts. Lemma for genpassg 7746. (Contributed by Jim Kingdon, 11-Dec-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   ⊢ dom 𝐹 = (P × P)    &   ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) ∈ P)    &   ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ)))    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶))))
Theorem	genpassu 7745*	Associativity of upper cuts. Lemma for genpassg 7746. (Contributed by Jim Kingdon, 11-Dec-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   ⊢ dom 𝐹 = (P × P)    &   ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) ∈ P)    &   ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ)))    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) = (2nd ‘(𝐴𝐹(𝐵𝐹𝐶))))
Theorem	genpassg 7746*	Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   ⊢ dom 𝐹 = (P × P)    &   ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) ∈ P)    &   ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ)))    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
Theorem	addnqprllem 7747	Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.)
⊢ (((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐺 ∈ 𝐿) ∧ 𝑋 ∈ Q) → (𝑋 <Q 𝑆 → ((𝑋 ·Q (*Q‘𝑆)) ·Q 𝐺) ∈ 𝐿))
Theorem	addnqprulem 7748	Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.)
⊢ (((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐺 ∈ 𝑈) ∧ 𝑋 ∈ Q) → (𝑆 <Q 𝑋 → ((𝑋 ·Q (*Q‘𝑆)) ·Q 𝐺) ∈ 𝑈))
Theorem	addnqprl 7749	Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-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	addnqpru 7750	Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-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	addlocprlemlt 7751	Lemma for addlocpr 7756. The 𝑄 <Q (𝐷 +Q 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ P)    &   ⊢ (𝜑 → 𝐵 ∈ P)    &   ⊢ (𝜑 → 𝑄 <Q 𝑅)    &   ⊢ (𝜑 → 𝑃 ∈ Q)    &   ⊢ (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))    &   ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))    &   ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))    &   ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))    &   ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))    ⇒   ⊢ (𝜑 → (𝑄 <Q (𝐷 +Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 +P 𝐵))))
Theorem	addlocprlemeqgt 7752	Lemma for addlocpr 7756. This is a step used in both the 𝑄 = (𝐷 +Q 𝐸) and (𝐷 +Q 𝐸) <Q 𝑄 cases. (Contributed by Jim Kingdon, 7-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ P)    &   ⊢ (𝜑 → 𝐵 ∈ P)    &   ⊢ (𝜑 → 𝑄 <Q 𝑅)    &   ⊢ (𝜑 → 𝑃 ∈ Q)    &   ⊢ (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))    &   ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))    &   ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))    &   ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))    &   ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))    ⇒   ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
Theorem	addlocprlemeq 7753	Lemma for addlocpr 7756. The 𝑄 = (𝐷 +Q 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ P)    &   ⊢ (𝜑 → 𝐵 ∈ P)    &   ⊢ (𝜑 → 𝑄 <Q 𝑅)    &   ⊢ (𝜑 → 𝑃 ∈ Q)    &   ⊢ (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))    &   ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))    &   ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))    &   ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))    &   ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))    ⇒   ⊢ (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
Theorem	addlocprlemgt 7754	Lemma for addlocpr 7756. The (𝐷 +Q 𝐸) <Q 𝑄 case. (Contributed by Jim Kingdon, 6-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ P)    &   ⊢ (𝜑 → 𝐵 ∈ P)    &   ⊢ (𝜑 → 𝑄 <Q 𝑅)    &   ⊢ (𝜑 → 𝑃 ∈ Q)    &   ⊢ (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))    &   ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))    &   ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))    &   ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))    &   ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))    ⇒   ⊢ (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
Theorem	addlocprlem 7755	Lemma for addlocpr 7756. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ P)    &   ⊢ (𝜑 → 𝐵 ∈ P)    &   ⊢ (𝜑 → 𝑄 <Q 𝑅)    &   ⊢ (𝜑 → 𝑃 ∈ Q)    &   ⊢ (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))    &   ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))    &   ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))    &   ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))    &   ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))    ⇒   ⊢ (𝜑 → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
Theorem	addlocpr 7756*	Locatedness of addition on positive reals. Lemma 11.16 in [BauerTaylor], p. 53. The proof in BauerTaylor relies on signed rationals, so we replace it with another proof which applies prarloc 7723 to both 𝐴 and 𝐵, and uses nqtri3or 7616 rather than prloc 7711 to decide whether 𝑞 is too big to be in the lower cut of 𝐴 +P 𝐵 (and deduce that if it is, then 𝑟 must be in the upper cut). What the two proofs have in common is that they take the difference between 𝑞 and 𝑟 to determine how tight a range they need around the real numbers. (Contributed by Jim Kingdon, 5-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
Theorem	addclpr 7757	Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123. Combination of Lemma 11.13 and Lemma 11.16 in [BauerTaylor], p. 53. (Contributed by NM, 13-Mar-1996.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P)
Theorem	plpvlu 7758*	Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦 +Q 𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦 +Q 𝑧)}⟩)
Theorem	mpvlu 7759*	Value of multiplication on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦 ·Q 𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦 ·Q 𝑧)}⟩)
Theorem	dmplp 7760	Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.)
⊢ dom +P = (P × P)
Theorem	dmmp 7761	Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.)
⊢ dom ·P = (P × P)
Theorem	nqprm 7762*	A cut produced from a rational is inhabited. Lemma for nqprlu 7767. (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 7763*	A cut produced from a rational is rounded. Lemma for nqprlu 7767. (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 7764*	A cut produced from a rational is disjoint. Lemma for nqprlu 7767. (Contributed by Jim Kingdon, 8-Dec-2019.)
⊢ (𝐴 ∈ Q → ∀𝑞 ∈ Q ¬ (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}))
Theorem	nqprloc 7765*	A cut produced from a rational is located. Lemma for nqprlu 7767. (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 7766*	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 7767*	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 7768*	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 7769	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 7770	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 7771*	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 7772*	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 7773*	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 7774	The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
⊢ 1P ∈ P
Theorem	1prl 7775	The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
⊢ (1st ‘1P) = {𝑥 ∣ 𝑥 <Q 1Q}
Theorem	1pru 7776	The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
⊢ (2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥}
Theorem	addnqprlemrl 7777*	Lemma for addnqpr 7781. 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 7778*	Lemma for addnqpr 7781. 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 7779*	Lemma for addnqpr 7781. 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 7780*	Lemma for addnqpr 7781. 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 7781*	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 7782*	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 7781. (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 7783*	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 7784*	Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7783 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 7785	Calculations for prmuloc 7786. (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 7786*	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 7787*	Positive reals are multiplicatively located. This is a variation of prmuloc 7786 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 7788	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 7789	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 7790	Calculations for mullocpr 7791. (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 7791*	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 7792	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 7793*	Lemma for mulnqpr 7797. 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 7794*	Lemma for mulnqpr 7797. 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 7795*	Lemma for mulnqpr 7797. 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 7796*	Lemma for mulnqpr 7797. 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 7797*	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 7798	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 7799	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 7800	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))