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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-mpq 7101* Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 28-Aug-1995.)
·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)

Definitiondf-ltpq 7102* Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 28-Aug-1995.)
<pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))}

Definitiondf-enq 7103* Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.)
~Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))}

Definitiondf-nqqs 7104 Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 16-Aug-1995.)
Q = ((N × N) / ~Q )

Definitiondf-plqqs 7105* Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 24-Aug-1995.)
+Q = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ𝑢, 𝑓⟩)] ~Q ))}

Definitiondf-mqqs 7106* Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 24-Aug-1995.)
·Q = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q ))}

Definitiondf-1nqqs 7107 Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 29-Oct-1995.)
1Q = [⟨1o, 1o⟩] ~Q

Definitiondf-rq 7108* Define reciprocal on positive fractions. It means the same thing as one divided by the argument (although we don't define full division since we will never need it). This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.5 of [Gleason] p. 119, who uses an asterisk to denote this unary operation. (Contributed by Jim Kingdon, 20-Sep-2019.)
*Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q𝑦Q ∧ (𝑥 ·Q 𝑦) = 1Q)}

Definitiondf-ltnqqs 7109* Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 13-Feb-1996.)
<Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~Q𝑦 = [⟨𝑣, 𝑢⟩] ~Q ) ∧ (𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣)))}

Theoremdfplpq2 7110* Alternate definition of pre-addition on positive fractions. (Contributed by Jim Kingdon, 12-Sep-2019.)
+pQ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩))}

Theoremdfmpq2 7111* Alternate definition of pre-multiplication on positive fractions. (Contributed by Jim Kingdon, 13-Sep-2019.)
·pQ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩))}

Theoremenqbreq 7112 Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.)
(((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (⟨𝐴, 𝐵⟩ ~Q𝐶, 𝐷⟩ ↔ (𝐴 ·N 𝐷) = (𝐵 ·N 𝐶)))

Theoremenqbreq2 7113 Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.)
((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))

Theoremenqer 7114 The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
~Q Er (N × N)

Theoremenqeceq 7115 Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.)
(((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q = [⟨𝐶, 𝐷⟩] ~Q ↔ (𝐴 ·N 𝐷) = (𝐵 ·N 𝐶)))

Theoremenqex 7116 The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
~Q ∈ V

Theoremenqdc 7117 The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
(((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → DECID𝐴, 𝐵⟩ ~Q𝐶, 𝐷⟩)

Theoremenqdc1 7118 The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
(((𝐴N𝐵N) ∧ 𝐶 ∈ (N × N)) → DECID𝐴, 𝐵⟩ ~Q 𝐶)

Theoremnqex 7119 The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
Q ∈ V

Theorem0nnq 7120 The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
¬ ∅ ∈ Q

Theoremltrelnq 7121 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)

Theorem1nq 7122 The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)
1QQ

((((𝐴N𝐵N) ∧ (𝐶N𝐷N)) ∧ ((𝐹N𝐺N) ∧ (𝑅N𝑆N))) → (((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)⟩))

Theoremmulcmpblnq 7124 Lemma showing compatibility of multiplication. (Contributed by NM, 27-Aug-1995.)
((((𝐴N𝐵N) ∧ (𝐶N𝐷N)) ∧ ((𝐹N𝐺N) ∧ (𝑅N𝑆N))) → (((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)⟩))

(((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ⟨((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)), (𝐵 ·N 𝐷)⟩ ∈ (N × N))

Theoremaddpipqqs 7126 Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)
(((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q +Q [⟨𝐶, 𝐷⟩] ~Q ) = [⟨((𝐴 ·N 𝐷) +N (𝐵 ·N 𝐶)), (𝐵 ·N 𝐷)⟩] ~Q )

Theoremmulpipq2 7127 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) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)

Theoremmulpipq 7128 Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.)
(((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (⟨𝐴, 𝐵⟩ ·pQ𝐶, 𝐷⟩) = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)

Theoremmulpipqqs 7129 Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)
(((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q ·Q [⟨𝐶, 𝐷⟩] ~Q ) = [⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩] ~Q )

Theoremordpipqqs 7130 Ordering of positive fractions in terms of positive integers. (Contributed by Jim Kingdon, 14-Sep-2019.)
(((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q <Q [⟨𝐶, 𝐷⟩] ~Q ↔ (𝐴 ·N 𝐷) <N (𝐵 ·N 𝐶)))

Theoremaddclnq 7131 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.)
((𝐴Q𝐵Q) → (𝐴 +Q 𝐵) ∈ Q)

Theoremmulclnq 7132 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.)
((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) ∈ Q)

Theoremdmaddpqlem 7133* Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7135. (Contributed by Jim Kingdon, 15-Sep-2019.)
(𝑥Q → ∃𝑤𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )

Theoremnqpi 7134* Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7133 but also shows that the numerator and denominator are positive integers. (Contributed by Jim Kingdon, 20-Sep-2019.)
(𝐴Q → ∃𝑤𝑣((𝑤N𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q ))

Theoremdmaddpq 7135 Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
dom +Q = (Q × Q)

Theoremdmmulpq 7136 Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
dom ·Q = (Q × Q)

Theoremaddcomnqg 7137 Addition of positive fractions is commutative. (Contributed by Jim Kingdon, 15-Sep-2019.)
((𝐴Q𝐵Q) → (𝐴 +Q 𝐵) = (𝐵 +Q 𝐴))

Theoremaddassnqg 7138 Addition of positive fractions is associative. (Contributed by Jim Kingdon, 16-Sep-2019.)
((𝐴Q𝐵Q𝐶Q) → ((𝐴 +Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q 𝐶)))

Theoremmulcomnqg 7139 Multiplication of positive fractions is commutative. (Contributed by Jim Kingdon, 17-Sep-2019.)
((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴))

Theoremmulassnqg 7140 Multiplication of positive fractions is associative. (Contributed by Jim Kingdon, 17-Sep-2019.)
((𝐴Q𝐵Q𝐶Q) → ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶)))

Theoremmulcanenq 7141 Lemma for distributive law: cancellation of common factor. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.)
((𝐴N𝐵N𝐶N) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩)

Theoremmulcanenqec 7142 Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 17-Sep-2019.)
((𝐴N𝐵N𝐶N) → [⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩] ~Q = [⟨𝐵, 𝐶⟩] ~Q )

Theoremdistrnqg 7143 Multiplication of positive fractions is distributive. (Contributed by Jim Kingdon, 17-Sep-2019.)
((𝐴Q𝐵Q𝐶Q) → (𝐴 ·Q (𝐵 +Q 𝐶)) = ((𝐴 ·Q 𝐵) +Q (𝐴 ·Q 𝐶)))

Theorem1qec 7144 The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
(𝐴N → 1Q = [⟨𝐴, 𝐴⟩] ~Q )

Theoremmulidnq 7145 Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
(𝐴Q → (𝐴 ·Q 1Q) = 𝐴)

Theoremrecexnq 7146* Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.)
(𝐴Q → ∃𝑦(𝑦Q ∧ (𝐴 ·Q 𝑦) = 1Q))

Theoremrecmulnqg 7147 Relationship between reciprocal and multiplication on positive fractions. (Contributed by Jim Kingdon, 19-Sep-2019.)
((𝐴Q𝐵Q) → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))

Theoremrecclnq 7148 Closure law for positive fraction reciprocal. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.)
(𝐴Q → (*Q𝐴) ∈ Q)

Theoremrecidnq 7149 A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.)
(𝐴Q → (𝐴 ·Q (*Q𝐴)) = 1Q)

Theoremrecrecnq 7150 Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.)
(𝐴Q → (*Q‘(*Q𝐴)) = 𝐴)

Theoremrec1nq 7151 Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
(*Q‘1Q) = 1Q

Theoremnqtri3or 7152 Trichotomy for positive fractions. (Contributed by Jim Kingdon, 21-Sep-2019.)
((𝐴Q𝐵Q) → (𝐴 <Q 𝐵𝐴 = 𝐵𝐵 <Q 𝐴))

Theoremltdcnq 7153 Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
((𝐴Q𝐵Q) → DECID 𝐴 <Q 𝐵)

Theoremltsonq 7154 '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

Theoremnqtric 7155 Trichotomy for positive fractions. (Contributed by Jim Kingdon, 21-Sep-2019.)
((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 <Q 𝐴)))

Theoremltanqg 7156 Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120. (Contributed by Jim Kingdon, 22-Sep-2019.)
((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))

Theoremltmnqg 7157 Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by Jim Kingdon, 22-Sep-2019.)
((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))

Theoremltanqi 7158 Ordering property of addition for positive fractions. One direction of ltanqg 7156. (Contributed by Jim Kingdon, 9-Dec-2019.)
((𝐴 <Q 𝐵𝐶Q) → (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵))

Theoremltmnqi 7159 Ordering property of multiplication for positive fractions. One direction of ltmnqg 7157. (Contributed by Jim Kingdon, 9-Dec-2019.)
((𝐴 <Q 𝐵𝐶Q) → (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵))

Theoremlt2addnq 7160 Ordering property of addition for positive fractions. (Contributed by Jim Kingdon, 7-Dec-2019.)
(((𝐴Q𝐵Q) ∧ (𝐶Q𝐷Q)) → ((𝐴 <Q 𝐵𝐶 <Q 𝐷) → (𝐴 +Q 𝐶) <Q (𝐵 +Q 𝐷)))

Theoremlt2mulnq 7161 Ordering property of multiplication for positive fractions. (Contributed by Jim Kingdon, 18-Jul-2021.)
(((𝐴Q𝐵Q) ∧ (𝐶Q𝐷Q)) → ((𝐴 <Q 𝐵𝐶 <Q 𝐷) → (𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐷)))

Theorem1lt2nq 7162 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)

Theoremltaddnq 7163 The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
((𝐴Q𝐵Q) → 𝐴 <Q (𝐴 +Q 𝐵))

Theoremltexnqq 7164* 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.)
((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵))

Theoremltexnqi 7165* Ordering on positive fractions in terms of existence of sum. (Contributed by Jim Kingdon, 30-Apr-2020.)
(𝐴 <Q 𝐵 → ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵)

Theoremhalfnqq 7166* One-half of any positive fraction is a fraction. (Contributed by Jim Kingdon, 23-Sep-2019.)
(𝐴Q → ∃𝑥Q (𝑥 +Q 𝑥) = 𝐴)

Theoremhalfnq 7167* 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.)
(𝐴Q → ∃𝑥(𝑥 +Q 𝑥) = 𝐴)

Theoremnsmallnqq 7168* There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.)
(𝐴Q → ∃𝑥Q 𝑥 <Q 𝐴)

Theoremnsmallnq 7169* There is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.)
(𝐴Q → ∃𝑥 𝑥 <Q 𝐴)

Theoremsubhalfnqq 7170* 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 7166). (Contributed by Jim Kingdon, 25-Nov-2019.)
(𝐴Q → ∃𝑥Q (𝑥 +Q 𝑥) <Q 𝐴)

Theoremltbtwnnqq 7171* 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 ↔ ∃𝑥Q (𝐴 <Q 𝑥𝑥 <Q 𝐵))

Theoremltbtwnnq 7172* 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 𝑥𝑥 <Q 𝐵))

Theoremarchnqq 7173* 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.)
(𝐴Q → ∃𝑥N 𝐴 <Q [⟨𝑥, 1o⟩] ~Q )

Theoremprarloclemarch 7174* A version of the Archimedean property. This variation is "stronger" than archnqq 7173 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.)
((𝐴Q𝐵Q) → ∃𝑥N 𝐴 <Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐵))

Theoremprarloclemarch2 7175* Like prarloclemarch 7174 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 7259. (Contributed by Jim Kingdon, 25-Nov-2019.)
((𝐴Q𝐵Q𝐶Q) → ∃𝑥N (1o <N 𝑥𝐴 <Q (𝐵 +Q ([⟨𝑥, 1o⟩] ~Q ·Q 𝐶))))

Theoremltrnqg 7176 Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7177. (Contributed by Jim Kingdon, 29-Dec-2019.)
((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴)))

Theoremltrnqi 7177 Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7176. (Contributed by Jim Kingdon, 24-Sep-2019.)
(𝐴 <Q 𝐵 → (*Q𝐵) <Q (*Q𝐴))

Theoremnnnq 7178 The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)
(𝐴N → [⟨𝐴, 1o⟩] ~QQ)

Theoremltnnnq 7179 Ordering of positive integers via <N or <Q is equivalent. (Contributed by Jim Kingdon, 3-Oct-2020.)
((𝐴N𝐵N) → (𝐴 <N 𝐵 ↔ [⟨𝐴, 1o⟩] ~Q <Q [⟨𝐵, 1o⟩] ~Q ))

Definitiondf-enq0 7180* 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)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))}

Definitiondf-nq0 7181 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 )

Definitiondf-0nq0 7182 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

Definitiondf-plq0 7183* 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⟩ ∣ ((𝑥Q0𝑦Q0) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·o 𝑓) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑓)⟩] ~Q0 ))}

Definitiondf-mq0 7184* 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⟩ ∣ ((𝑥Q0𝑦Q0) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 ))}

Theoremdfmq0qs 7185* Multiplication on nonnegative fractions. This definition is similar to df-mq0 7184 but expands Q0 (Contributed by Jim Kingdon, 22-Nov-2019.)
·Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑓)⟩] ~Q0 ))}

Theoremdfplq0qs 7186* Addition on nonnegative fractions. This definition is similar to df-plq0 7183 but expands Q0 (Contributed by Jim Kingdon, 24-Nov-2019.)
+Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·o 𝑓) +o (𝑣 ·o 𝑢)), (𝑣 ·o 𝑓)⟩] ~Q0 ))}

Theoremenq0enq 7187 Equivalence on positive fractions in terms of equivalence on nonnegative fractions. (Contributed by Jim Kingdon, 12-Nov-2019.)
~Q = ( ~Q0 ∩ ((N × N) × (N × N)))

Theoremenq0sym 7188 The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7191. (Contributed by Jim Kingdon, 14-Nov-2019.)
(𝑓 ~Q0 𝑔𝑔 ~Q0 𝑓)

Theoremenq0ref 7189 The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7191. (Contributed by Jim Kingdon, 14-Nov-2019.)
(𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓)

Theoremenq0tr 7190 The equivalence relation for nonnegative fractions is transitive. Lemma for enq0er 7191. (Contributed by Jim Kingdon, 14-Nov-2019.)
((𝑓 ~Q0 𝑔𝑔 ~Q0 ) → 𝑓 ~Q0 )

Theoremenq0er 7191 The equivalence relation for nonnegative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
~Q0 Er (ω × N)

Theoremenq0breq 7192 Equivalence relation for nonnegative fractions in terms of natural numbers. (Contributed by NM, 27-Aug-1995.)
(((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → (⟨𝐴, 𝐵⟩ ~Q0𝐶, 𝐷⟩ ↔ (𝐴 ·o 𝐷) = (𝐵 ·o 𝐶)))

Theoremenq0eceq 7193 Equivalence class equality of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 24-Nov-2019.)
(((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) → ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ↔ (𝐴 ·o 𝐷) = (𝐵 ·o 𝐶)))

Theoremnqnq0pi 7194 A nonnegative fraction is a positive fraction if its numerator and denominator are positive integers. (Contributed by Jim Kingdon, 10-Nov-2019.)
((𝐴N𝐵N) → [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q )

Theoremenq0ex 7195 The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
~Q0 ∈ V

Theoremnq0ex 7196 The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
Q0 ∈ V

Theoremnqnq0 7197 A positive fraction is a nonnegative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
QQ0

Theoremnq0nn 7198* Decomposition of a nonnegative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.)
(𝐴Q0 → ∃𝑤𝑣((𝑤 ∈ ω ∧ 𝑣N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))

Theoremaddcmpblnq0 7199 Lemma showing compatibility of addition on nonnegative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((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)⟩))

Theoremmulcmpblnq0 7200 Lemma showing compatibility of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
((((𝐴 ∈ ω ∧ 𝐵N) ∧ (𝐶 ∈ ω ∧ 𝐷N)) ∧ ((𝐹 ∈ ω ∧ 𝐺N) ∧ (𝑅 ∈ ω ∧ 𝑆N))) → (((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)⟩))

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