P. 69 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	c0r 6801	The signed real constant 0.
class 0R
Syntax	c1r 6802	The signed real constant 1.
class 1R
Syntax	cm1r 6803	The signed real constant -1.
class -1R
Syntax	cplr 6804	Signed real addition.
class +R
Syntax	cmr 6805	Signed real multiplication.
class ·R
Syntax	cltr 6806	Signed real ordering relation.
class <R
Definition	df-ni 6807	Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.)
⊢ N = (ω ∖ {∅})
Definition	df-pli 6808	Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.)
⊢ +N = ( +𝑜 ↾ (N × N))
Definition	df-mi 6809	Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.)
⊢ ·N = ( ·𝑜 ↾ (N × N))
Definition	df-lti 6810	Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.)
⊢ <N = ( E ∩ (N × N))
Theorem	elni 6811	Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
Theorem	pinn 6812	A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
Theorem	pion 6813	A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
⊢ (𝐴 ∈ N → 𝐴 ∈ On)
Theorem	piord 6814	A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
⊢ (𝐴 ∈ N → Ord 𝐴)
Theorem	niex 6815	The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
⊢ N ∈ V
Theorem	0npi 6816	The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
⊢ ¬ ∅ ∈ N
Theorem	elni2 6817	Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
Theorem	1pi 6818	Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
⊢ 1𝑜 ∈ N
Theorem	addpiord 6819	Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +𝑜 𝐵))
Theorem	mulpiord 6820	Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵))
Theorem	mulidpi 6821	1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ (𝐴 ∈ N → (𝐴 ·N 1𝑜) = 𝐴)
Theorem	ltpiord 6822	Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵))
Theorem	ltsopi 6823	Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.)
⊢ <N Or N
Theorem	pitric 6824	Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <N 𝐴)))
Theorem	pitri3or 6825	Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <N 𝐴))
Theorem	ltdcpi 6826	Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → DECID 𝐴 <N 𝐵)
Theorem	ltrelpi 6827	Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
⊢ <N ⊆ (N × N)
Theorem	dmaddpi 6828	Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
⊢ dom +N = (N × N)
Theorem	dmmulpi 6829	Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
⊢ dom ·N = (N × N)
Theorem	addclpi 6830	Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) ∈ N)
Theorem	mulclpi 6831	Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N)
Theorem	addcompig 6832	Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐵 +N 𝐴))
Theorem	addasspig 6833	Addition of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 +N 𝐵) +N 𝐶) = (𝐴 +N (𝐵 +N 𝐶)))
Theorem	mulcompig 6834	Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴))
Theorem	mulasspig 6835	Multiplication of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶)))
Theorem	distrpig 6836	Multiplication of positive integers is distributive. (Contributed by Jim Kingdon, 26-Aug-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)))
Theorem	addcanpig 6837	Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 +N 𝐵) = (𝐴 +N 𝐶) ↔ 𝐵 = 𝐶))
Theorem	mulcanpig 6838	Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))
Theorem	addnidpig 6839	There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ¬ (𝐴 +N 𝐵) = 𝐴)
Theorem	ltexpi 6840*	Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ ∃𝑥 ∈ N (𝐴 +N 𝑥) = 𝐵))
Theorem	ltapig 6841	Ordering property of addition for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 <N 𝐵 ↔ (𝐶 +N 𝐴) <N (𝐶 +N 𝐵)))
Theorem	ltmpig 6842	Ordering property of multiplication for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 <N 𝐵 ↔ (𝐶 ·N 𝐴) <N (𝐶 ·N 𝐵)))
Theorem	1lt2pi 6843	One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
⊢ 1𝑜 <N (1𝑜 +N 1𝑜)
Theorem	nlt1pig 6844	No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
⊢ (𝐴 ∈ N → ¬ 𝐴 <N 1𝑜)
Theorem	indpi 6845*	Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.)
⊢ (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)
Theorem	nnppipi 6846	A natural number plus a positive integer is a positive integer. (Contributed by Jim Kingdon, 10-Nov-2019.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → (𝐴 +𝑜 𝐵) ∈ N)
Definition	df-plpq 6847*	Define pre-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. This "pre-addition" operation works directly with ordered pairs of integers. The actual positive fraction addition +Q (df-plqqs 6852) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 6850). (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.)
⊢ +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 6848*	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) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
Definition	df-ltpq 6849*	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)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))}
Definition	df-enq 6850*	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)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))}
Definition	df-nqqs 6851	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 )
Definition	df-plqqs 6852*	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⟩ ∣ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ +pQ ⟨𝑢, 𝑓⟩)] ~Q ))}
Definition	df-mqqs 6853*	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⟩ ∣ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ ⟨𝑢, 𝑓⟩)] ~Q ))}
Definition	df-1nqqs 6854	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
Definition	df-rq 6855*	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⟩ ∣ (𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ (𝑥 ·Q 𝑦) = 1Q)}
Definition	df-ltnqqs 6856*	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⟩ ∣ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~Q ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ~Q ) ∧ (𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣)))}
Theorem	dfplpq2 6857*	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)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·N 𝑓) +N (𝑣 ·N 𝑢)), (𝑣 ·N 𝑓)⟩))}
Theorem	dfmpq2 6858*	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)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 ·N 𝑢), (𝑣 ·N 𝑓)⟩))}
Theorem	enqbreq 6859	Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.)
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (⟨𝐴, 𝐵⟩ ~Q ⟨𝐶, 𝐷⟩ ↔ (𝐴 ·N 𝐷) = (𝐵 ·N 𝐶)))
Theorem	enqbreq2 6860	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 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
Theorem	enqer 6861	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)
Theorem	enqeceq 6862	Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.)
⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q = [⟨𝐶, 𝐷⟩] ~Q ↔ (𝐴 ·N 𝐷) = (𝐵 ·N 𝐶)))
Theorem	enqex 6863	The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
⊢ ~Q ∈ V
Theorem	enqdc 6864	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 6865	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 6866	The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
⊢ Q ∈ V
Theorem	0nnq 6867	The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
⊢ ¬ ∅ ∈ Q
Theorem	ltrelnq 6868	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 6869	The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)
⊢ 1Q ∈ Q
Theorem	addcmpblnq 6870	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 6871	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 6872	Lemma for addpipqqs 6873. (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 6873	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 6874	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 6875	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 6876	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 6877	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 6878	Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) ∈ Q)
Theorem	mulclnq 6879	Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ Q)
Theorem	dmaddpqlem 6880*	Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 6882. (Contributed by Jim Kingdon, 15-Sep-2019.)
⊢ (𝑥 ∈ Q → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
Theorem	nqpi 6881*	Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 6880 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 6882	Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
⊢ dom +Q = (Q × Q)
Theorem	dmmulpq 6883	Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
⊢ dom ·Q = (Q × Q)
Theorem	addcomnqg 6884	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 6885	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 6886	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 6887	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 6888	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 6889	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 6890	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 6891	The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
⊢ (𝐴 ∈ N → 1Q = [⟨𝐴, 𝐴⟩] ~Q )
Theorem	mulidnq 6892	Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴)
Theorem	recexnq 6893*	Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.)
⊢ (𝐴 ∈ Q → ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q))
Theorem	recmulnqg 6894	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 6895	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 6896	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 6897	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 6898	Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
⊢ (*Q‘1Q) = 1Q
Theorem	nqtri3or 6899	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 6900	Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → DECID 𝐴 <Q 𝐵)