P. 73 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	recidnq 7201	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 7202	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 7203	Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
⊢ (*Q‘1Q) = 1Q
Theorem	nqtri3or 7204	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 7205	Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → DECID 𝐴 <Q 𝐵)
Theorem	ltsonq 7206	'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 7207	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 7208	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 7209	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 7210	Ordering property of addition for positive fractions. One direction of ltanqg 7208. (Contributed by Jim Kingdon, 9-Dec-2019.)
⊢ ((𝐴 <Q 𝐵 ∧ 𝐶 ∈ Q) → (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵))
Theorem	ltmnqi 7211	Ordering property of multiplication for positive fractions. One direction of ltmnqg 7209. (Contributed by Jim Kingdon, 9-Dec-2019.)
⊢ ((𝐴 <Q 𝐵 ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵))
Theorem	lt2addnq 7212	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 7213	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 7214	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 7215	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 7216*	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 7217*	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 7218*	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 7219*	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 7220*	There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.)
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q 𝑥 <Q 𝐴)
Theorem	nsmallnq 7221*	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 7222*	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 7218). (Contributed by Jim Kingdon, 25-Nov-2019.)
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝐴)
Theorem	ltbtwnnqq 7223*	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 7224*	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 7225*	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 7226*	A version of the Archimedean property. This variation is "stronger" than archnqq 7225 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 7227*	Like prarloclemarch 7226 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 7311. (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 7228	Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7229. (Contributed by Jim Kingdon, 29-Dec-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴)))
Theorem	ltrnqi 7229	Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7228. (Contributed by Jim Kingdon, 24-Sep-2019.)
⊢ (𝐴 <Q 𝐵 → (*Q‘𝐵) <Q (*Q‘𝐴))
Theorem	nnnq 7230	The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)
⊢ (𝐴 ∈ N → [⟨𝐴, 1o⟩] ~Q ∈ Q)
Theorem	ltnnnq 7231	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 7232*	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 7233	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 7234	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 7235*	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 7236*	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 7237*	Multiplication on nonnegative fractions. This definition is similar to df-mq0 7236 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 7238*	Addition on nonnegative fractions. This definition is similar to df-plq0 7235 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 7239	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 7240	The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7243. (Contributed by Jim Kingdon, 14-Nov-2019.)
⊢ (𝑓 ~Q0 𝑔 → 𝑔 ~Q0 𝑓)
Theorem	enq0ref 7241	The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7243. (Contributed by Jim Kingdon, 14-Nov-2019.)
⊢ (𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓)
Theorem	enq0tr 7242	The equivalence relation for nonnegative fractions is transitive. Lemma for enq0er 7243. (Contributed by Jim Kingdon, 14-Nov-2019.)
⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → 𝑓 ~Q0 ℎ)
Theorem	enq0er 7243	The equivalence relation for nonnegative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
⊢ ~Q0 Er (ω × N)
Theorem	enq0breq 7244	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 7245	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 7246	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 7247	The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
⊢ ~Q0 ∈ V
Theorem	nq0ex 7248	The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
⊢ Q0 ∈ V
Theorem	nqnq0 7249	A positive fraction is a nonnegative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
⊢ Q ⊆ Q0
Theorem	nq0nn 7250*	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 7251	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 7252	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 7253	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 7254*	Decomposing nonnegative fractions into natural numbers. Lemma for addnnnq0 7257 and mulnnnq0 7258. (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 7255*	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 7256*	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 7257	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 7258	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 7259	Closure of addition on nonnegative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.)
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 +Q0 𝐵) ∈ Q0)
Theorem	mulclnq0 7260	Closure of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.)
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 ·Q0 𝐵) ∈ Q0)
Theorem	nqpnq0nq 7261	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 7262	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 7263	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 7264	Multiplication with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
⊢ (𝐴 ∈ Q0 → (0Q0 ·Q0 𝐴) = 0Q0)
Theorem	nq0a0 7265	Addition with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
⊢ (𝐴 ∈ Q0 → (𝐴 +Q0 0Q0) = 𝐴)
Theorem	nnanq0 7266	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 7267	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 7268	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 7269	A natural number closure law. Lemma for addassnq0 7270. (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 7270	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 7271	Multiplication of nonnegative fractions is distributive. Version of distrnq0 7267 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 7272	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 7273	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 7274*	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 7275*	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 7276*	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 7316. 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 7277*	Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 7276 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 7278*	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 7279	Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ P ⊆ (𝒫 Q × 𝒫 Q)
Theorem	preqlu 7280	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 7281	The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.)
⊢ P ∈ V
Theorem	elinp 7282*	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 7283	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 7284*	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 7285*	A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ 𝐿)
Theorem	prmu 7286*	A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ 𝑈)
Theorem	prssnql 7287	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 7288	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 7289	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 7290	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 7291	The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
⊢ ¬ ∅ ∈ P
Theorem	prcdnql 7292	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 7293	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 7294	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 7295	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 7296*	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 7297*	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 7298*	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 7299	A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 <Q 𝐵) → (𝐴 ∈ 𝐿 ∨ 𝐵 ∈ 𝑈))
Theorem	prdisj 7300	A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.)
⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ Q) → ¬ (𝐴 ∈ 𝐿 ∧ 𝐴 ∈ 𝑈))