P. 78 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	addclpi 7701	Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) ∈ N)
Theorem	mulclpi 7702	Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N)
Theorem	addcompig 7703	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 7704	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 7705	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 7706	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 7707	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 7708	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 7709	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 7710	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 7711*	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 7712	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 7713	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 7714	One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
⊢ 1o <N (1o +N 1o)
Theorem	nlt1pig 7715	No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
⊢ (𝐴 ∈ N → ¬ 𝐴 <N 1o)
Theorem	indpi 7716*	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 7717	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 7718*	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 7723) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 7721). (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 7719*	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 7720*	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 7721*	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 7722	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 7723*	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 7724*	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 7725	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 7726*	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 7727*	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 7728*	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 7729*	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 7730	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 7731	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 7732	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 7733	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 7734	The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
⊢ ~Q ∈ V
Theorem	enqdc 7735	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 7736	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 7737	The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
⊢ Q ∈ V
Theorem	0nnq 7738	The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
⊢ ¬ ∅ ∈ Q
Theorem	ltrelnq 7739	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 7740	The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)
⊢ 1Q ∈ Q
Theorem	addcmpblnq 7741	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 7742	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 7743	Lemma for addpipqqs 7744. (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 7744	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 7745	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 7746	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 7747	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 7748	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 7749	Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) ∈ Q)
Theorem	mulclnq 7750	Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) ∈ Q)
Theorem	dmaddpqlem 7751*	Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7753. (Contributed by Jim Kingdon, 15-Sep-2019.)
⊢ (𝑥 ∈ Q → ∃𝑤∃𝑣 𝑥 = [⟨𝑤, 𝑣⟩] ~Q )
Theorem	nqpi 7752*	Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7751 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 7753	Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
⊢ dom +Q = (Q × Q)
Theorem	dmmulpq 7754	Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
⊢ dom ·Q = (Q × Q)
Theorem	addcomnqg 7755	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 7756	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 7757	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 7758	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 7759	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 7760	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 7761	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 7762	The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
⊢ (𝐴 ∈ N → 1Q = [⟨𝐴, 𝐴⟩] ~Q )
Theorem	mulidnq 7763	Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴)
Theorem	recexnq 7764*	Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.)
⊢ (𝐴 ∈ Q → ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q))
Theorem	recmulnqg 7765	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 7766	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 7767	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 7768	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 7769	Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
⊢ (*Q‘1Q) = 1Q
Theorem	nqtri3or 7770	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 7771	Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → DECID 𝐴 <Q 𝐵)
Theorem	ltsonq 7772	'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 7773	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 7774	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 7775	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 7776	Ordering property of addition for positive fractions. One direction of ltanqg 7774. (Contributed by Jim Kingdon, 9-Dec-2019.)
⊢ ((𝐴 <Q 𝐵 ∧ 𝐶 ∈ Q) → (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵))
Theorem	ltmnqi 7777	Ordering property of multiplication for positive fractions. One direction of ltmnqg 7775. (Contributed by Jim Kingdon, 9-Dec-2019.)
⊢ ((𝐴 <Q 𝐵 ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵))
Theorem	lt2addnq 7778	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 7779	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 7780	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 7781	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 7782*	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 7783*	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 7784*	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 7785*	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 7786*	There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.)
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q 𝑥 <Q 𝐴)
Theorem	nsmallnq 7787*	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 7788*	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 7784). (Contributed by Jim Kingdon, 25-Nov-2019.)
⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝐴)
Theorem	ltbtwnnqq 7789*	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 7790*	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 7791*	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 7792*	A version of the Archimedean property. This variation is "stronger" than archnqq 7791 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 7793*	Like prarloclemarch 7792 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 7877. (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 7794	Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7795. (Contributed by Jim Kingdon, 29-Dec-2019.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴)))
Theorem	ltrnqi 7795	Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7794. (Contributed by Jim Kingdon, 24-Sep-2019.)
⊢ (𝐴 <Q 𝐵 → (*Q‘𝐵) <Q (*Q‘𝐴))
Theorem	nnnq 7796	The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)
⊢ (𝐴 ∈ N → [⟨𝐴, 1o⟩] ~Q ∈ Q)
Theorem	ltnnnq 7797	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 7798*	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 7799	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 7800	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