P. 75 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7401-7500   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	c1q 7401	The positive fraction constant 1.
class 1Q
Syntax	cplq 7402	Positive fraction addition.
class +Q
Syntax	cmq 7403	Positive fraction multiplication.
class ·Q
Syntax	crq 7404	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 7405	Positive fraction ordering relation.
class <Q
Syntax	ceq0 7406	Equivalence class used to construct nonnegative fractions.
class ~Q0
Syntax	cnq0 7407	Set of nonnegative fractions.
class Q0
Syntax	c0q0 7408	The nonnegative fraction constant 0.
class 0Q0
Syntax	cplq0 7409	Nonnegative fraction addition.
class +Q0
Syntax	cmq0 7410	Nonnegative fraction multiplication.
class ·Q0
Syntax	cnp 7411	Set of positive reals.
class P
Syntax	c1p 7412	Positive real constant 1.
class 1P
Syntax	cpp 7413	Positive real addition.
class +P
Syntax	cmp 7414	Positive real multiplication.
class ·P
Syntax	cltp 7415	Positive real ordering relation.
class <P
Syntax	cer 7416	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 7417	Set of signed reals.
class R
Syntax	c0r 7418	The signed real constant 0.
class 0R
Syntax	c1r 7419	The signed real constant 1.
class 1R
Syntax	cm1r 7420	The signed real constant -1.
class -1R
Syntax	cplr 7421	Signed real addition.
class +R
Syntax	cmr 7422	Signed real multiplication.
class ·R
Syntax	cltr 7423	Signed real ordering relation.
class <R
Definition	df-ni 7424	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 7425	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 7426	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 7427	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 7428	Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
Theorem	pinn 7429	A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
Theorem	pion 7430	A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
⊢ (𝐴 ∈ N → 𝐴 ∈ On)
Theorem	piord 7431	A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
⊢ (𝐴 ∈ N → Ord 𝐴)
Theorem	niex 7432	The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
⊢ N ∈ V
Theorem	0npi 7433	The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
⊢ ¬ ∅ ∈ N
Theorem	elni2 7434	Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
Theorem	1pi 7435	Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
⊢ 1o ∈ N
Theorem	addpiord 7436	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 7437	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 7438	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 7439	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 7440	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 7441	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 7442	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 7443	Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → DECID 𝐴 <N 𝐵)
Theorem	ltrelpi 7444	Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
⊢ <N ⊆ (N × N)
Theorem	dmaddpi 7445	Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
⊢ dom +N = (N × N)
Theorem	dmmulpi 7446	Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
⊢ dom ·N = (N × N)
Theorem	addclpi 7447	Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) ∈ N)
Theorem	mulclpi 7448	Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N)
Theorem	addcompig 7449	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 7450	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 7451	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 7452	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 7453	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 7454	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 7455	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 7456	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 7457*	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 7458	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 7459	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 7460	One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
⊢ 1o <N (1o +N 1o)
Theorem	nlt1pig 7461	No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
⊢ (𝐴 ∈ N → ¬ 𝐴 <N 1o)
Theorem	indpi 7462*	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 7463	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 7464*	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 7469) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 7467). (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 7465*	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 7466*	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 7467*	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 7468	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 7469*	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 7470*	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 7471	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 7472*	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 7473*	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 7474*	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 7475*	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 7476	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 7477	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 7478	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 7479	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 7480	The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
⊢ ~Q ∈ V
Theorem	enqdc 7481	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 7482	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 7483	The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
⊢ Q ∈ V
Theorem	0nnq 7484	The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
⊢ ¬ ∅ ∈ Q
Theorem	ltrelnq 7485	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 7486	The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)
⊢ 1Q ∈ Q
Theorem	addcmpblnq 7487	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 7488	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 7489	Lemma for addpipqqs 7490. (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 7490	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 7491	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 7492	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 7493	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 7494	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 7495	Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) ∈ Q)
Theorem	mulclnq 7496	Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ Q)
Theorem	dmaddpqlem 7497*	Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7499. (Contributed by Jim Kingdon, 15-Sep-2019.)
⊢ (𝑥 ∈ Q → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
Theorem	nqpi 7498*	Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7497 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 7499	Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
⊢ dom +Q = (Q × Q)
Theorem	dmmulpq 7500	Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
⊢ dom ·Q = (Q × Q)