P. 110 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 10901-11000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nlt1pi 10901	No positive integer is less than one. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
⊢ ¬ 𝐴 <N 1o
Theorem	indpi 10902*	Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
⊢ (𝑥 = 1o → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 +N 1o) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ N → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ N → 𝜏)
Definition	df-plpq 10903*	Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. This "pre-addition" operation works directly with ordered pairs of integers. The actual positive fraction addition +Q (df-plq 10909) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 10906). (Analogous remarks apply to the other "pre-" operations in the complex number construction that follows.) From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)
⊢ +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
Definition	df-mpq 10904*	Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11116, 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.) (New usage is discouraged.)
⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
Definition	df-ltpq 10905*	Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)
⊢ <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))}
Definition	df-enq 10906*	Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11116, 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.) (New usage is discouraged.)
⊢ ~Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))}
Definition	df-nq 10907*	Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11116, 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.) (New usage is discouraged.)
⊢ Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))}
Definition	df-erq 10908	Define a convenience function that "reduces" a fraction to lowest terms. Note that in this form, it is not obviously a function; we prove this in nqerf 10925. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
⊢ [Q] = ( ~Q ∩ ((N × N) × Q))
Definition	df-plq 10909	Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11116, 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.) (New usage is discouraged.)
⊢ +Q = (([Q] ∘ +pQ ) ↾ (Q × Q))
Definition	df-mq 10910	Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11116, 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.) (New usage is discouraged.)
⊢ ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q))
Definition	df-1nq 10911	Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers df-c 11116, 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.) (New usage is discouraged.)
⊢ 1Q = ⟨1o, 1o⟩
Definition	df-rq 10912	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 df-c 11116, 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 NM, 6-Mar-1996.) (New usage is discouraged.)
⊢ *Q = (◡ ·Q “ {1Q})
Definition	df-ltnq 10913	Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 13-Feb-1996.) (New usage is discouraged.)
⊢ <Q = ( <pQ ∩ (Q × Q))
Theorem	enqbreq 10914	Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (⟨𝐴, 𝐵⟩ ~Q ⟨𝐶, 𝐷⟩ ↔ (𝐴 ·N 𝐷) = (𝐵 ·N 𝐶)))
Theorem	enqbreq2 10915	Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
Theorem	enqer 10916	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.) (New usage is discouraged.)
⊢ ~Q Er (N × N)
Theorem	enqex 10917	The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
⊢ ~Q ∈ V
Theorem	nqex 10918	The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)
⊢ Q ∈ V
Theorem	0nnq 10919	The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)
⊢ ¬ ∅ ∈ Q
Theorem	elpqn 10920	Each positive fraction is an ordered pair of positive integers (the numerator and denominator, in "lowest terms". (Contributed by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
Theorem	ltrelnq 10921	Positive fraction 'less than' is a relation on positive fractions. (Contributed by NM, 14-Feb-1996.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)
⊢ <Q ⊆ (Q × Q)
Theorem	pinq 10922	The representatives of positive integers as positive fractions. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ Q)
Theorem	1nq 10923	The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
⊢ 1Q ∈ Q
Theorem	nqereu 10924*	There is a unique element of Q equivalent to each element of N × N. (Contributed by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ (N × N) → ∃!𝑥 ∈ Q 𝑥 ~Q 𝐴)
Theorem	nqerf 10925	Corollary of nqereu 10924: the function [Q] is actually a function. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
⊢ [Q]:(N × N)⟶Q
Theorem	nqercl 10926	Corollary of nqereu 10924: closure of [Q]. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
Theorem	nqerrel 10927	Any member of (N × N) relates to the representative of its equivalence class. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
Theorem	nqerid 10928	Corollary of nqereu 10924: the function [Q] acts as the identity on members of Q. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
Theorem	enqeq 10929	Corollary of nqereu 10924: if two fractions are both reduced and equivalent, then they are equal. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~Q 𝐵) → 𝐴 = 𝐵)
Theorem	nqereq 10930	The function [Q] acts as a substitute for equivalence classes, and it satisfies the fundamental requirement for equivalence representatives: the representatives are equal iff the members are equivalent. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ([Q]‘𝐴) = ([Q]‘𝐵)))
Theorem	addpipq2 10931	Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
Theorem	addpipq 10932	Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (⟨𝐴, 𝐵⟩ +pQ ⟨𝐶, 𝐷⟩) = ⟨((𝐴 ·N 𝐷) +N (𝐶 ·N 𝐵)), (𝐵 ·N 𝐷)⟩)
Theorem	addpqnq 10933	Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))
Theorem	mulpipq2 10934	Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
Theorem	mulpipq 10935	Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (⟨𝐴, 𝐵⟩ ·pQ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)
Theorem	mulpqnq 10936	Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
Theorem	ordpipq 10937	Ordering of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ (⟨𝐴, 𝐵⟩ <pQ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))
Theorem	ordpinq 10938	Ordering of positive fractions in terms of positive integers. (Contributed by NM, 13-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
Theorem	addpqf 10939	Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ +pQ :((N × N) × (N × N))⟶(N × N)
Theorem	addclnq 10940	Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) ∈ Q)
Theorem	mulpqf 10941	Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ ·pQ :((N × N) × (N × N))⟶(N × N)
Theorem	mulclnq 10942	Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ Q)
Theorem	addnqf 10943	Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
⊢ +Q :(Q × Q)⟶Q
Theorem	mulnqf 10944	Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
⊢ ·Q :(Q × Q)⟶Q
Theorem	addcompq 10945	Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
⊢ (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴)
Theorem	addcomnq 10946	Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
⊢ (𝐴 +Q 𝐵) = (𝐵 +Q 𝐴)
Theorem	mulcompq 10947	Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
⊢ (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴)
Theorem	mulcomnq 10948	Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
⊢ (𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴)
Theorem	adderpqlem 10949	Lemma for adderpq 10951. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ (𝐴 +pQ 𝐶) ~Q (𝐵 +pQ 𝐶)))
Theorem	mulerpqlem 10950	Lemma for mulerpq 10952. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ (𝐴 ·pQ 𝐶) ~Q (𝐵 ·pQ 𝐶)))
Theorem	adderpq 10951	Addition is compatible with the equivalence relation. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ (([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(𝐴 +pQ 𝐵))
Theorem	mulerpq 10952	Multiplication is compatible with the equivalence relation. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ (([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵))
Theorem	addassnq 10953	Addition of positive fractions is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
⊢ ((𝐴 +Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q 𝐶))
Theorem	mulassnq 10954	Multiplication of positive fractions is associative. (Contributed by NM, 1-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶))
Theorem	mulcanenq 10955	Lemma for distributive law: cancellation of common factor. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q ⟨𝐵, 𝐶⟩)
Theorem	distrnq 10956	Multiplication of positive fractions is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
⊢ (𝐴 ·Q (𝐵 +Q 𝐶)) = ((𝐴 ·Q 𝐵) +Q (𝐴 ·Q 𝐶))
Theorem	1nqenq 10957	The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → 1Q ~Q ⟨𝐴, 𝐴⟩)
Theorem	mulidnq 10958	Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴)
Theorem	recmulnq 10959	Relationship between reciprocal and multiplication on positive fractions. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
⊢ (𝐴 ∈ Q → ((*Q‘𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))
Theorem	recidnq 10960	A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ Q → (𝐴 ·Q (*Q‘𝐴)) = 1Q)
Theorem	recclnq 10961	Closure law for positive fraction reciprocal. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ Q → (*Q‘𝐴) ∈ Q)
Theorem	recrecnq 10962	Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ Q → (*Q‘(*Q‘𝐴)) = 𝐴)
Theorem	dmrecnq 10963	Domain of reciprocal on positive fractions. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
⊢ dom *Q = Q
Theorem	ltsonq 10964	'Less than' is a strict ordering on positive fractions. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.) (New usage is discouraged.)
⊢ <Q Or Q
Theorem	lterpq 10965	Compatibility of ordering on equivalent fractions. (Contributed by Mario Carneiro, 9-May-2013.) (New usage is discouraged.)
⊢ (𝐴 <pQ 𝐵 ↔ ([Q]‘𝐴) <Q ([Q]‘𝐵))
Theorem	ltanq 10966	Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
⊢ (𝐶 ∈ Q → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))
Theorem	ltmnq 10967	Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
⊢ (𝐶 ∈ Q → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
Theorem	1lt2nq 10968	One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
⊢ 1Q <Q (1Q +Q 1Q)
Theorem	ltaddnq 10969	The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝐵))
Theorem	ltexnq 10970*	Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by NM, 24-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
⊢ (𝐵 ∈ Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
Theorem	halfnq 10971*	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.) (New usage is discouraged.)
⊢ (𝐴 ∈ Q → ∃𝑥(𝑥 +Q 𝑥) = 𝐴)
Theorem	nsmallnq 10972*	The is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ Q → ∃𝑥 𝑥 <Q 𝐴)
Theorem	ltbtwnnq 10973*	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.) (New usage is discouraged.)
⊢ (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 <Q 𝑥 ∧ 𝑥 <Q 𝐵))
Theorem	ltrnq 10974	Ordering property of reciprocal for positive fractions. Proposition 9-2.6(iv) of [Gleason] p. 120. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
⊢ (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴))
Theorem	archnq 10975*	For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ N 𝐴 <Q ⟨𝑥, 1o⟩)
Definition	df-np 10976*	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. A positive real is defined as the lower class of a Dedekind cut. Definition 9-3.1 of [Gleason] p. 121. (Note: This is a "temporary" definition used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction.) (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
⊢ P = {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧))}
Definition	df-1p 10977	Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. Definition of [Gleason] p. 122. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
⊢ 1P = {𝑥 ∣ 𝑥 <Q 1Q}
Definition	df-plp 10978*	Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. From Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
⊢ +P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 +Q 𝑢)})
Definition	df-mp 10979*	Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. From Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
⊢ ·P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 ·Q 𝑢)})
Definition	df-ltp 10980*	Define ordering on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 11116, and is intended to be used only by the construction. From Proposition 9-3.2 of [Gleason] p. 122. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
Theorem	npex 10981	The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
⊢ P ∈ V
Theorem	elnp 10982*	Membership in positive reals. (Contributed by NM, 16-Feb-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ P ↔ ((∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
Theorem	elnpi 10983*	Membership in positive reals. (Contributed by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
Theorem	prn0 10984	A positive real is not empty. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ P → 𝐴 ≠ ∅)
Theorem	prpssnq 10985	A positive real is a subset of the positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ P → 𝐴 ⊊ Q)
Theorem	elprnq 10986	A positive real is a set of positive fractions. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q)
Theorem	0npr 10987	The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.)
⊢ ¬ ∅ ∈ P
Theorem	prcdnq 10988	A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121. (Contributed by NM, 25-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → (𝐶 <Q 𝐵 → 𝐶 ∈ 𝐴))
Theorem	prub 10989	A positive fraction not in a positive real is an upper bound. Remark (1) of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ Q) → (¬ 𝐶 ∈ 𝐴 → 𝐵 <Q 𝐶))
Theorem	prnmax 10990*	A positive real has no largest member. Definition 9-3.1(iii) of [Gleason] p. 121. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 <Q 𝑥)
Theorem	npomex 10991	A simplifying observation, and an indication of why any attempt to develop a theory of the real numbers without the Axiom of Infinity is doomed to failure: since every member of P is an infinite set, the negation of Infinity implies that P, and hence ℝ, is empty. (Note that this proof, which used the fact that Dedekind cuts have no maximum, could just as well have used that they have no minimum, since they are downward-closed by prcdnq 10988 and nsmallnq 10972). (Contributed by Mario Carneiro, 11-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) (New usage is discouraged.)
⊢ (𝐴 ∈ P → ω ∈ V)
Theorem	prnmadd 10992*	A positive real has no largest member. Addition version. (Contributed by NM, 7-Apr-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥(𝐵 +Q 𝑥) ∈ 𝐴)
Theorem	ltrelpr 10993	Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
⊢ <P ⊆ (P × P)
Theorem	genpv 10994*	Value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)})
Theorem	genpelv 10995*	Membership in value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝐶 = (𝑔𝐺ℎ)))
Theorem	genpprecl 10996*	Pre-closure law for general operation on positive reals. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶𝐺𝐷) ∈ (𝐴𝐹𝐵)))
Theorem	genpdm 10997*	Domain of general operation on positive reals. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ dom 𝐹 = (P × P)
Theorem	genpn0 10998*	The result of an operation on positive reals is not empty. (Contributed by NM, 28-Feb-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∅ ⊊ (𝐴𝐹𝐵))
Theorem	genpss 10999*	The result of an operation on positive reals is a subset of the positive fractions. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ⊆ Q)
Theorem	genpnnp 11000*	The result of an operation on positive reals is different from the set of positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   ⊢ (𝑧 ∈ Q → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥)    ⇒   ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ¬ (𝐴𝐹𝐵) = Q)