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	recexnq 7701*	Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.)
⊢ (𝐴 ∈ Q → ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q))
Theorem	recmulnqg 7702	Relationship between reciprocal and multiplication on positive fractions. (Contributed by Jim Kingdon, 19-Sep-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))
Theorem	recclnq 7703	Closure law for positive fraction reciprocal. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.)
⊢ (𝐴 ∈ Q → (*Q‘𝐴) ∈ Q)
Theorem	recidnq 7704	A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.)
⊢ (𝐴 ∈ Q → (𝐴 ·Q (*Q‘𝐴)) = 1Q)
Theorem	recrecnq 7705	Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.)
⊢ (𝐴 ∈ Q → (*Q‘(*Q‘𝐴)) = 𝐴)
Theorem	rec1nq 7706	Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
⊢ (*Q‘1Q) = 1Q
Theorem	nqtri3or 7707	Trichotomy for positive fractions. (Contributed by Jim Kingdon, 21-Sep-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <Q 𝐴))
Theorem	ltdcnq 7708	Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → DECID 𝐴 <Q 𝐵)
Theorem	ltsonq 7709	'Less than' is a strict ordering on positive fractions. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.)
⊢ <Q Or Q
Theorem	nqtric 7710	Trichotomy for positive fractions. (Contributed by Jim Kingdon, 21-Sep-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <Q 𝐴)))
Theorem	ltanqg 7711	Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120. (Contributed by Jim Kingdon, 22-Sep-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))
Theorem	ltmnqg 7712	Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by Jim Kingdon, 22-Sep-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
Theorem	ltanqi 7713	Ordering property of addition for positive fractions. One direction of ltanqg 7711. (Contributed by Jim Kingdon, 9-Dec-2019.)
⊢ ((𝐴 <Q 𝐵 ∧ 𝐶 ∈ Q) → (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵))
Theorem	ltmnqi 7714	Ordering property of multiplication for positive fractions. One direction of ltmnqg 7712. (Contributed by Jim Kingdon, 9-Dec-2019.)
⊢ ((𝐴 <Q 𝐵 ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵))
Theorem	lt2addnq 7715	Ordering property of addition for positive fractions. (Contributed by Jim Kingdon, 7-Dec-2019.)
⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → ((𝐴 <Q 𝐵 ∧ 𝐶 <Q 𝐷) → (𝐴 +Q 𝐶) <Q (𝐵 +Q 𝐷)))
Theorem	lt2mulnq 7716	Ordering property of multiplication for positive fractions. (Contributed by Jim Kingdon, 18-Jul-2021.)
⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → ((𝐴 <Q 𝐵 ∧ 𝐶 <Q 𝐷) → (𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐷)))
Theorem	1lt2nq 7717	One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
⊢ 1Q <Q (1Q +Q 1Q)
Theorem	ltaddnq 7718	The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝐵))
Theorem	ltexnqq 7719*	Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by Jim Kingdon, 23-Sep-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵))
Theorem	ltexnqi 7720*	Ordering on positive fractions in terms of existence of sum. (Contributed by Jim Kingdon, 30-Apr-2020.)
⊢ (𝐴 <Q 𝐵 → ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵)
Theorem	halfnqq 7721*	One-half of any positive fraction is a fraction. (Contributed by Jim Kingdon, 23-Sep-2019.)
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝐴)
Theorem	halfnq 7722*	One-half of any positive fraction exists. Lemma for Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 16-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
⊢ (𝐴 ∈ Q → ∃𝑥(𝑥 +Q 𝑥) = 𝐴)
Theorem	nsmallnqq 7723*	There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.)
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q 𝑥 <Q 𝐴)
Theorem	nsmallnq 7724*	There is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.)
⊢ (𝐴 ∈ Q → ∃𝑥 𝑥 <Q 𝐴)
Theorem	subhalfnqq 7725*	There is a number which is less than half of any positive fraction. The case where 𝐴 is one is Lemma 11.4 of [BauerTaylor], p. 50, and they use the word "approximate half" for such a number (since there may be constructions, for some structures other than the rationals themselves, which rely on such an approximate half but do not require division by two as seen at halfnqq 7721). (Contributed by Jim Kingdon, 25-Nov-2019.)
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝐴)
Theorem	ltbtwnnqq 7726*	There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by Jim Kingdon, 24-Sep-2019.)
⊢ (𝐴 <Q 𝐵 ↔ ∃𝑥 ∈ Q (𝐴 <Q 𝑥 ∧ 𝑥 <Q 𝐵))
Theorem	ltbtwnnq 7727*	There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
⊢ (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 <Q 𝑥 ∧ 𝑥 <Q 𝐵))
Theorem	archnqq 7728*	For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Jim Kingdon, 1-Dec-2019.)
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ N 𝐴 <Q [⟨𝑥, 1o⟩] ~Q )
Theorem	prarloclemarch 7729*	A version of the Archimedean property. This variation is "stronger" than archnqq 7728 in the sense that we provide an integer which is larger than a given rational 𝐴 even after being multiplied by a second rational 𝐵. (Contributed by Jim Kingdon, 30-Nov-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ∃𝑥 ∈ N 𝐴 <Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵))
Theorem	prarloclemarch2 7730*	Like prarloclemarch 7729 but the integer must be at least two, and there is also 𝐵 added to the right hand side. These details follow straightforwardly but are chosen to be helpful in the proof of prarloc 7814. (Contributed by Jim Kingdon, 25-Nov-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ∃𝑥 ∈ N (1o <N 𝑥 ∧ 𝐴 <Q (𝐵 +Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐶))))
Theorem	ltrnqg 7731	Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7732. (Contributed by Jim Kingdon, 29-Dec-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴)))
Theorem	ltrnqi 7732	Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7731. (Contributed by Jim Kingdon, 24-Sep-2019.)
⊢ (𝐴 <Q 𝐵 → (*Q‘𝐵) <Q (*Q‘𝐴))
Theorem	nnnq 7733	The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)
⊢ (𝐴 ∈ N → [⟨𝐴, 1o⟩] ~Q ∈ Q)
Theorem	ltnnnq 7734	Ordering of positive integers via <N or <Q is equivalent. (Contributed by Jim Kingdon, 3-Oct-2020.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ [⟨𝐴, 1o⟩] ~Q <Q [⟨𝐵, 1o⟩] ~Q ))
Definition	df-enq0 7735*	Define equivalence relation for nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
⊢ ~Q0 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))}
Definition	df-nq0 7736	Define class of nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
⊢ Q0 = ((ω × N) / ~Q0 )
Definition	df-0nq0 7737	Define nonnegative fraction constant 0. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 5-Nov-2019.)
⊢ 0Q0 = [⟨∅, 1o⟩] ~Q0
Definition	df-plq0 7738*	Define addition on nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
⊢ +Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ Q0 ∧ 𝑦 ∈ Q0) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·o 𝑓) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑓)⟩] ~Q0 ))}
Definition	df-mq0 7739*	Define multiplication on nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
⊢ ·Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ Q0 ∧ 𝑦 ∈ Q0) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 ))}
Theorem	dfmq0qs 7740*	Multiplication on nonnegative fractions. This definition is similar to df-mq0 7739 but expands Q0. (Contributed by Jim Kingdon, 22-Nov-2019.)
⊢ ·Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 ))}
Theorem	dfplq0qs 7741*	Addition on nonnegative fractions. This definition is similar to df-plq0 7738 but expands Q0. (Contributed by Jim Kingdon, 24-Nov-2019.)
⊢ +Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·o 𝑓) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑓)⟩] ~Q0 ))}
Theorem	enq0enq 7742	Equivalence on positive fractions in terms of equivalence on nonnegative fractions. (Contributed by Jim Kingdon, 12-Nov-2019.)
⊢ ~Q = ( ~Q0 ∩ ((N × N) × (N × N)))
Theorem	enq0sym 7743	The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7746. (Contributed by Jim Kingdon, 14-Nov-2019.)
⊢ (𝑓 ~Q0 𝑔 → 𝑔 ~Q0 𝑓)
Theorem	enq0ref 7744	The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7746. (Contributed by Jim Kingdon, 14-Nov-2019.)
⊢ (𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓)
Theorem	enq0tr 7745	The equivalence relation for nonnegative fractions is transitive. Lemma for enq0er 7746. (Contributed by Jim Kingdon, 14-Nov-2019.)
⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → 𝑓 ~Q0 ℎ)
Theorem	enq0er 7746	The equivalence relation for nonnegative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
⊢ ~Q0 Er (ω × N)
Theorem	enq0breq 7747	Equivalence relation for nonnegative fractions in terms of natural numbers. (Contributed by NM, 27-Aug-1995.)
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → (⟨𝐴, 𝐵⟩ ~Q0 ⟨𝐶, 𝐷⟩ ↔ (𝐴 ·o 𝐷) = (𝐵 ·o 𝐶)))
Theorem	enq0eceq 7748	Equivalence class equality of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 24-Nov-2019.)
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ↔ (𝐴 ·o 𝐷) = (𝐵 ·o 𝐶)))
Theorem	nqnq0pi 7749	A nonnegative fraction is a positive fraction if its numerator and denominator are positive integers. (Contributed by Jim Kingdon, 10-Nov-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q )
Theorem	enq0ex 7750	The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
⊢ ~Q0 ∈ V
Theorem	nq0ex 7751	The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
⊢ Q0 ∈ V
Theorem	nqnq0 7752	A positive fraction is a nonnegative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
⊢ Q ⊆ Q0
Theorem	nq0nn 7753*	Decomposition of a nonnegative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.)
⊢ (𝐴 ∈ Q0 → ∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
Theorem	addcmpblnq0 7754	Lemma showing compatibility of addition on nonnegative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅)) → ⟨((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)), (𝐵 ·o 𝐺)⟩ ~Q0 ⟨((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)), (𝐷 ·o 𝑆)⟩))
Theorem	mulcmpblnq0 7755	Lemma showing compatibility of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅)) → ⟨(𝐴 ·o 𝐹), (𝐵 ·o 𝐺)⟩ ~Q0 ⟨(𝐶 ·o 𝑅), (𝐷 ·o 𝑆)⟩))
Theorem	mulcanenq0ec 7756	Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 29-Nov-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → [⟨(𝐴 ·o 𝐵), (𝐴 ·o 𝐶)⟩] ~Q0 = [⟨𝐵, 𝐶⟩] ~Q0 )
Theorem	nnnq0lem1 7757*	Decomposing nonnegative fractions into natural numbers. Lemma for addnnnq0 7760 and mulnnnq0 7761. (Contributed by Jim Kingdon, 23-Nov-2019.)
⊢ (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ((((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ (𝑠 ∈ ω ∧ 𝑓 ∈ N)) ∧ ((𝑢 ∈ ω ∧ 𝑡 ∈ N) ∧ (𝑔 ∈ ω ∧ ℎ ∈ N))) ∧ ((𝑤 ·o 𝑓) = (𝑣 ·o 𝑠) ∧ (𝑢 ·o ℎ) = (𝑡 ·o 𝑔))))
Theorem	addnq0mo 7758*	There is at most one result from adding nonnegative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
⊢ ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·o 𝑡) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑡)⟩] ~Q0 ))
Theorem	mulnq0mo 7759*	There is at most one result from multiplying nonnegative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
⊢ ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 ))
Theorem	addnnnq0 7760	Addition of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.)
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q0 +Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨((𝐴 ·o 𝐷) +o (𝐵 ·o 𝐶)), (𝐵 ·o 𝐷)⟩] ~Q0 )
Theorem	mulnnnq0 7761	Multiplication of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 19-Nov-2019.)
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q0 ·Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩] ~Q0 )
Theorem	addclnq0 7762	Closure of addition on nonnegative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.)
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 +Q0 𝐵) ∈ Q0)
Theorem	mulclnq0 7763	Closure of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.)
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 ·Q0 𝐵) ∈ Q0)
Theorem	nqpnq0nq 7764	A positive fraction plus a nonnegative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q0) → (𝐴 +Q0 𝐵) ∈ Q)
Theorem	nqnq0a 7765	Addition of positive fractions is equal with +Q or +Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = (𝐴 +Q0 𝐵))
Theorem	nqnq0m 7766	Multiplication of positive fractions is equal with ·Q or ·Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = (𝐴 ·Q0 𝐵))
Theorem	nq0m0r 7767	Multiplication with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
⊢ (𝐴 ∈ Q0 → (0Q0 ·Q0 𝐴) = 0Q0)
Theorem	nq0a0 7768	Addition with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
⊢ (𝐴 ∈ Q0 → (𝐴 +Q0 0Q0) = 𝐴)
Theorem	nnanq0 7769	Addition of nonnegative fractions with a common denominator. You can add two fractions with the same denominator by adding their numerators and keeping the same denominator. (Contributed by Jim Kingdon, 1-Dec-2019.)
⊢ ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → [⟨(𝑁 +o 𝑀), 𝐴⟩] ~Q0 = ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ))
Theorem	distrnq0 7770	Multiplication of nonnegative fractions is distributive. (Contributed by Jim Kingdon, 27-Nov-2019.)
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0 ∧ 𝐶 ∈ Q0) → (𝐴 ·Q0 (𝐵 +Q0 𝐶)) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶)))
Theorem	mulcomnq0 7771	Multiplication of nonnegative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.)
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 ·Q0 𝐵) = (𝐵 ·Q0 𝐴))
Theorem	addassnq0lemcl 7772	A natural number closure law. Lemma for addassnq0 7773. (Contributed by Jim Kingdon, 3-Dec-2019.)
⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ N) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ N)) → (((𝐼 ·o 𝐿) +o (𝐽 ·o 𝐾)) ∈ ω ∧ (𝐽 ·o 𝐿) ∈ N))
Theorem	addassnq0 7773	Addition of nonnegative fractions is associative. (Contributed by Jim Kingdon, 29-Nov-2019.)
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0 ∧ 𝐶 ∈ Q0) → ((𝐴 +Q0 𝐵) +Q0 𝐶) = (𝐴 +Q0 (𝐵 +Q0 𝐶)))
Theorem	distnq0r 7774	Multiplication of nonnegative fractions is distributive. Version of distrnq0 7770 with the multiplications commuted. (Contributed by Jim Kingdon, 29-Nov-2019.)
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0 ∧ 𝐶 ∈ Q0) → ((𝐵 +Q0 𝐶) ·Q0 𝐴) = ((𝐵 ·Q0 𝐴) +Q0 (𝐶 ·Q0 𝐴)))
Theorem	addpinq1 7775	Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.)
⊢ (𝐴 ∈ N → [⟨(𝐴 +N 1o), 1o⟩] ~Q = ([⟨𝐴, 1o⟩] ~Q +Q 1Q))
Theorem	nq02m 7776	Multiply a nonnegative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.)
⊢ (𝐴 ∈ Q0 → ([⟨2o, 1o⟩] ~Q0 ·Q0 𝐴) = (𝐴 +Q0 𝐴))
Definition	df-inp 7777*	Define the set of positive reals. A "Dedekind cut" is a partition of the positive rational numbers into two classes such that all the numbers of one class are less than all the numbers of the other. Here we follow the definition of a Dedekind cut from Definition 11.2.1 of [HoTT], p. (varies) with the one exception that we define it over positive rational numbers rather than all rational numbers. A Dedekind cut is an ordered pair of a lower set 𝑙 and an upper set 𝑢 which is inhabited (∃𝑞 ∈ Q𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q𝑟 ∈ 𝑢), rounded (∀𝑞 ∈ Q(𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q(𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)) and likewise for 𝑢), disjoint (∀𝑞 ∈ Q¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢)) and located (∀𝑞 ∈ Q∀𝑟 ∈ Q(𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))). See HoTT for more discussion of those terms and different ways of defining Dedekind cuts. (Note: This is a "temporary" definition used in the construction of complex numbers, and is intended to be used only by the construction.) (Contributed by Jim Kingdon, 25-Sep-2019.)
⊢ P = {⟨𝑙, 𝑢⟩ ∣ (((𝑙 ⊆ Q ∧ 𝑢 ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q 𝑟 ∈ 𝑢)) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)) ∧ ∀𝑟 ∈ Q (𝑟 ∈ 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))))}
Definition	df-i1p 7778*	Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers and is intended to be used only by the construction. (Contributed by Jim Kingdon, 25-Sep-2019.)
⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
Definition	df-iplp 7779*	Define addition on positive reals. From Section 11.2.1 of [HoTT], p. (varies). We write this definition to closely resemble the definition in HoTT although some of the conditions are redundant (for example, 𝑟 ∈ (1st ‘𝑥) implies 𝑟 ∈ Q) and can be simplified as shown at genpdf 7819. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 26-Sep-2019.)
⊢ +P = (𝑥 ∈ P, 𝑦 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}⟩)
Definition	df-imp 7780*	Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 7779 or the definition of multiplication on positive reals in Metamath Proof Explorer. This is as opposed to the more complicated definition of multiplication given in Section 11.2.1 of [HoTT], p. (varies), which appears to be motivated by handling negative numbers or handling modified Dedekind cuts in which locatedness is omitted. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 29-Sep-2019.)
⊢ ·P = (𝑥 ∈ P, 𝑦 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 ·Q 𝑠))}⟩)
Definition	df-iltp 7781*	Define ordering on positive reals. We define 𝑥<P 𝑦 if there is a positive fraction 𝑞 which is an element of the upper cut of 𝑥 and the lower cut of 𝑦. From the definition of < in Section 11.2.1 of [HoTT], p. (varies). This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 29-Sep-2019.)
⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
Theorem	npsspw 7782	Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ P ⊆ (𝒫 Q × 𝒫 Q)
Theorem	preqlu 7783	Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵))))
Theorem	npex 7784	The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.)
⊢ P ∈ V
Theorem	elinp 7785*	Membership in positive reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ (⟨𝐿, 𝑈⟩ ∈ P ↔ (((𝐿 ⊆ Q ∧ 𝑈 ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ 𝐿 ∧ ∃𝑟 ∈ Q 𝑟 ∈ 𝑈)) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝐿)) ∧ ∀𝑟 ∈ Q (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑈))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝐿 ∧ 𝑞 ∈ 𝑈) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))))
Theorem	prop 7786	A positive real is an ordered pair of a lower cut and an upper cut. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
Theorem	elnp1st2nd 7787*	Membership in positive reals, using 1st and 2nd to refer to the lower and upper cut. (Contributed by Jim Kingdon, 3-Oct-2019.)
⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐴) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐴)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐴) ∨ 𝑟 ∈ (2nd ‘𝐴))))))
Theorem	prml 7788*	A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ 𝐿)
Theorem	prmu 7789*	A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ 𝑈)
Theorem	prssnql 7790	The lower cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
⊢ (⟨𝐿, 𝑈⟩ ∈ P → 𝐿 ⊆ Q)
Theorem	prssnqu 7791	The upper cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
⊢ (⟨𝐿, 𝑈⟩ ∈ P → 𝑈 ⊆ Q)
Theorem	elprnql 7792	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 7793	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 7794	The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
⊢ ¬ ∅ ∈ P
Theorem	prcdnql 7795	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 7796	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 7797	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 7798	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 7799*	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 7800*	An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐵 ∈ 𝑈) → ∃𝑥 ∈ 𝑈 𝑥 <Q 𝐵)