P. 74 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	enqex 7301	The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
⊢ ~Q ∈ V
Theorem	enqdc 7302	The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → DECID ⟨𝐴, 𝐵⟩ ~Q ⟨𝐶, 𝐷⟩)
Theorem	enqdc1 7303	The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ (N × N)) → DECID ⟨𝐴, 𝐵⟩ ~Q 𝐶)
Theorem	nqex 7304	The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
⊢ Q ∈ V
Theorem	0nnq 7305	The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
⊢ ¬ ∅ ∈ Q
Theorem	ltrelnq 7306	Positive fraction 'less than' is a relation on positive fractions. (Contributed by NM, 14-Feb-1996.) (Revised by Mario Carneiro, 27-Apr-2013.)
⊢ <Q ⊆ (Q × Q)
Theorem	1nq 7307	The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)
⊢ 1Q ∈ Q
Theorem	addcmpblnq 7308	Lemma showing compatibility of addition. (Contributed by NM, 27-Aug-1995.)
⊢ ((((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ N ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ N ∧ 𝑆 ∈ N))) → (((𝐴 ·N 𝐷) = (𝐵 ·N 𝐶) ∧ (𝐹 ·N 𝑆) = (𝐺 ·N 𝑅)) → ⟨((𝐴 ·N 𝐺) +N (𝐵 ·N 𝐹)), (𝐵 ·N 𝐺)⟩ ~Q ⟨((𝐶 ·N 𝑆) +N (𝐷 ·N 𝑅)), (𝐷 ·N 𝑆)⟩))
Theorem	mulcmpblnq 7309	Lemma showing compatibility of multiplication. (Contributed by NM, 27-Aug-1995.)
⊢ ((((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ N ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ N ∧ 𝑆 ∈ N))) → (((𝐴 ·N 𝐷) = (𝐵 ·N 𝐶) ∧ (𝐹 ·N 𝑆) = (𝐺 ·N 𝑅)) → ⟨(𝐴 ·N 𝐹), (𝐵 ·N 𝐺)⟩ ~Q ⟨(𝐶 ·N 𝑅), (𝐷 ·N 𝑆)⟩))
Theorem	addpipqqslem 7310	Lemma for addpipqqs 7311. (Contributed by Jim Kingdon, 11-Sep-2019.)
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ⟨((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)), (𝐵 ·N 𝐷)⟩ ∈ (N × N))
Theorem	addpipqqs 7311	Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q +Q [⟨𝐶, 𝐷⟩] ~Q ) = [⟨((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)), (𝐵 ·N 𝐷)⟩] ~Q )
Theorem	mulpipq2 7312	Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.)
⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
Theorem	mulpipq 7313	Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.)
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (⟨𝐴, 𝐵⟩ ·pQ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)
Theorem	mulpipqqs 7314	Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q ·Q [⟨𝐶, 𝐷⟩] ~Q ) = [⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩] ~Q )
Theorem	ordpipqqs 7315	Ordering of positive fractions in terms of positive integers. (Contributed by Jim Kingdon, 14-Sep-2019.)
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q <Q [⟨𝐶, 𝐷⟩] ~Q ↔ (𝐴 ·N 𝐷) <N (𝐵 ·N 𝐶)))
Theorem	addclnq 7316	Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) ∈ Q)
Theorem	mulclnq 7317	Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ Q)
Theorem	dmaddpqlem 7318*	Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7320. (Contributed by Jim Kingdon, 15-Sep-2019.)
⊢ (𝑥 ∈ Q → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
Theorem	nqpi 7319*	Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7318 but also shows that the numerator and denominator are positive integers. (Contributed by Jim Kingdon, 20-Sep-2019.)
⊢ (𝐴 ∈ Q → ∃𝑤∃𝑣((𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))
Theorem	dmaddpq 7320	Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
⊢ dom +Q = (Q × Q)
Theorem	dmmulpq 7321	Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
⊢ dom ·Q = (Q × Q)
Theorem	addcomnqg 7322	Addition of positive fractions is commutative. (Contributed by Jim Kingdon, 15-Sep-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = (𝐵 +Q 𝐴))
Theorem	addassnqg 7323	Addition of positive fractions is associative. (Contributed by Jim Kingdon, 16-Sep-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 +Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q 𝐶)))
Theorem	mulcomnqg 7324	Multiplication of positive fractions is commutative. (Contributed by Jim Kingdon, 17-Sep-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴))
Theorem	mulassnqg 7325	Multiplication of positive fractions is associative. (Contributed by Jim Kingdon, 17-Sep-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶)))
Theorem	mulcanenq 7326	Lemma for distributive law: cancellation of common factor. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩)
Theorem	mulcanenqec 7327	Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 17-Sep-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → [⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩] ~Q = [⟨𝐵, 𝐶⟩] ~Q )
Theorem	distrnqg 7328	Multiplication of positive fractions is distributive. (Contributed by Jim Kingdon, 17-Sep-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 ·Q (𝐵 +Q 𝐶)) = ((𝐴 ·Q 𝐵) +Q (𝐴 ·Q 𝐶)))
Theorem	1qec 7329	The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
⊢ (𝐴 ∈ N → 1Q = [⟨𝐴, 𝐴⟩] ~Q )
Theorem	mulidnq 7330	Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴)
Theorem	recexnq 7331*	Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.)
⊢ (𝐴 ∈ Q → ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q))
Theorem	recmulnqg 7332	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 7333	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 7334	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 7335	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 7336	Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
⊢ (*Q‘1Q) = 1Q
Theorem	nqtri3or 7337	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 7338	Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → DECID 𝐴 <Q 𝐵)
Theorem	ltsonq 7339	'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 7340	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 7341	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 7342	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 7343	Ordering property of addition for positive fractions. One direction of ltanqg 7341. (Contributed by Jim Kingdon, 9-Dec-2019.)
⊢ ((𝐴 <Q 𝐵 ∧ 𝐶 ∈ Q) → (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵))
Theorem	ltmnqi 7344	Ordering property of multiplication for positive fractions. One direction of ltmnqg 7342. (Contributed by Jim Kingdon, 9-Dec-2019.)
⊢ ((𝐴 <Q 𝐵 ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵))
Theorem	lt2addnq 7345	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 7346	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 7347	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 7348	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 7349*	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 7350*	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 7351*	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 7352*	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 7353*	There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.)
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q 𝑥 <Q 𝐴)
Theorem	nsmallnq 7354*	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 7355*	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 7351). (Contributed by Jim Kingdon, 25-Nov-2019.)
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝐴)
Theorem	ltbtwnnqq 7356*	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 7357*	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 7358*	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 7359*	A version of the Archimedean property. This variation is "stronger" than archnqq 7358 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 7360*	Like prarloclemarch 7359 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 7444. (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 7361	Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7362. (Contributed by Jim Kingdon, 29-Dec-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴)))
Theorem	ltrnqi 7362	Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7361. (Contributed by Jim Kingdon, 24-Sep-2019.)
⊢ (𝐴 <Q 𝐵 → (*Q‘𝐵) <Q (*Q‘𝐴))
Theorem	nnnq 7363	The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)
⊢ (𝐴 ∈ N → [⟨𝐴, 1o⟩] ~Q ∈ Q)
Theorem	ltnnnq 7364	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 7365*	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 7366	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 7367	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 7368*	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 7369*	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 7370*	Multiplication on nonnegative fractions. This definition is similar to df-mq0 7369 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 7371*	Addition on nonnegative fractions. This definition is similar to df-plq0 7368 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 7372	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 7373	The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7376. (Contributed by Jim Kingdon, 14-Nov-2019.)
⊢ (𝑓 ~Q0 𝑔 → 𝑔 ~Q0 𝑓)
Theorem	enq0ref 7374	The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7376. (Contributed by Jim Kingdon, 14-Nov-2019.)
⊢ (𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓)
Theorem	enq0tr 7375	The equivalence relation for nonnegative fractions is transitive. Lemma for enq0er 7376. (Contributed by Jim Kingdon, 14-Nov-2019.)
⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → 𝑓 ~Q0 ℎ)
Theorem	enq0er 7376	The equivalence relation for nonnegative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
⊢ ~Q0 Er (ω × N)
Theorem	enq0breq 7377	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 7378	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 7379	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 7380	The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
⊢ ~Q0 ∈ V
Theorem	nq0ex 7381	The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
⊢ Q0 ∈ V
Theorem	nqnq0 7382	A positive fraction is a nonnegative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
⊢ Q ⊆ Q0
Theorem	nq0nn 7383*	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 7384	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 7385	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 7386	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 7387*	Decomposing nonnegative fractions into natural numbers. Lemma for addnnnq0 7390 and mulnnnq0 7391. (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 7388*	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 7389*	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 7390	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 7391	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 7392	Closure of addition on nonnegative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.)
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 +Q0 𝐵) ∈ Q0)
Theorem	mulclnq0 7393	Closure of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.)
⊢ ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 ·Q0 𝐵) ∈ Q0)
Theorem	nqpnq0nq 7394	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 7395	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 7396	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 7397	Multiplication with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
⊢ (𝐴 ∈ Q0 → (0Q0 ·Q0 𝐴) = 0Q0)
Theorem	nq0a0 7398	Addition with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
⊢ (𝐴 ∈ Q0 → (𝐴 +Q0 0Q0) = 𝐴)
Theorem	nnanq0 7399	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 7400	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)))