P. 101 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 10001-10100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sstskm 10001*	Being a part of (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥)))
Theorem	eltskm 10002*	Belonging to (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥)))
PART 5  REAL AND COMPLEX NUMBERS This section derives the basics of real and complex numbers. We first construct and axiomatize real and complex numbers (e.g., ax-resscn 10331). After that, we derive their basic properties, various operations like addition (df-add 10285) and sine (df-sin 15211), and subsets such as the integers (df-z 11734) and natural numbers (df-nn 11380).
5.1  Construction and axiomatization of real and complex numbers
5.1.1  Dedekind-cut construction of real and complex numbers
Syntax	cnpi 10003	The set of positive integers, which is the set of natural numbers ω with 0 removed. Note: This is the start of the Dedekind-cut construction of real and complex numbers. The last lemma of the construction is mulcnsrec 10303. The actual set of Dedekind cuts is defined by df-np 10140.
class N
Syntax	cpli 10004	Positive integer addition.
class +N
Syntax	cmi 10005	Positive integer multiplication.
class ·N
Syntax	clti 10006	Positive integer ordering relation.
class <N
Syntax	cplpq 10007	Positive pre-fraction addition.
class +pQ
Syntax	cmpq 10008	Positive pre-fraction multiplication.
class ·pQ
Syntax	cltpq 10009	Positive pre-fraction ordering relation.
class <pQ
Syntax	ceq 10010	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 10011	Set of positive fractions.
class Q
Syntax	c1q 10012	The positive fraction constant 1.
class 1Q
Syntax	cerq 10013	Positive fraction equivalence class.
class [Q]
Syntax	cplq 10014	Positive fraction addition.
class +Q
Syntax	cmq 10015	Positive fraction multiplication.
class ·Q
Syntax	crq 10016	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 10017	Positive fraction ordering relation.
class <Q
Syntax	cnp 10018	Set of positive reals.
class P
Syntax	c1p 10019	Positive real constant 1.
class 1P
Syntax	cpp 10020	Positive real addition.
class +P
Syntax	cmp 10021	Positive real multiplication.
class ·P
Syntax	cltp 10022	Positive real ordering relation.
class <P
Syntax	cer 10023	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 10024	Set of signed reals.
class R
Syntax	c0r 10025	The signed real constant 0.
class 0R
Syntax	c1r 10026	The signed real constant 1.
class 1R
Syntax	cm1r 10027	The signed real constant -1.
class -1R
Syntax	cplr 10028	Signed real addition.
class +R
Syntax	cmr 10029	Signed real multiplication.
class ·R
Syntax	cltr 10030	Signed real ordering relation.
class <R
Definition	df-ni 10031	Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10280, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
⊢ N = (ω ∖ {∅})
Definition	df-pli 10032	Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10280, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
⊢ +N = ( +o ↾ (N × N))
Definition	df-mi 10033	Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10280, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
⊢ ·N = ( ·o ↾ (N × N))
Definition	df-lti 10034	Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10280, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.) (New usage is discouraged.)
⊢ <N = ( E ∩ (N × N))
Theorem	elni 10035	Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
Theorem	elni2 10036	Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) (New usage is discouraged.)
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
Theorem	pinn 10037	A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
Theorem	pion 10038	A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → 𝐴 ∈ On)
Theorem	piord 10039	A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → Ord 𝐴)
Theorem	niex 10040	The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
⊢ N ∈ V
Theorem	0npi 10041	The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
⊢ ¬ ∅ ∈ N
Theorem	1pi 10042	Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.)
⊢ 1o ∈ N
Theorem	addpiord 10043	Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))
Theorem	mulpiord 10044	Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
Theorem	mulidpi 10045	1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴)
Theorem	ltpiord 10046	Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵))
Theorem	ltsopi 10047	Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
⊢ <N Or N
Theorem	ltrelpi 10048	Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.) (New usage is discouraged.)
⊢ <N ⊆ (N × N)
Theorem	dmaddpi 10049	Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
⊢ dom +N = (N × N)
Theorem	dmmulpi 10050	Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
⊢ dom ·N = (N × N)
Theorem	addclpi 10051	Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) ∈ N)
Theorem	mulclpi 10052	Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N)
Theorem	addcompi 10053	Addition of positive integers is commutative. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
⊢ (𝐴 +N 𝐵) = (𝐵 +N 𝐴)
Theorem	addasspi 10054	Addition of positive integers is associative. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
⊢ ((𝐴 +N 𝐵) +N 𝐶) = (𝐴 +N (𝐵 +N 𝐶))
Theorem	mulcompi 10055	Multiplication of positive integers is commutative. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)
⊢ (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴)
Theorem	mulasspi 10056	Multiplication of positive integers is associative. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)
⊢ ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶))
Theorem	distrpi 10057	Multiplication of positive integers is distributive. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)
⊢ (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶))
Theorem	addcanpi 10058	Addition cancellation law for positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 +N 𝐵) = (𝐴 +N 𝐶) ↔ 𝐵 = 𝐶))
Theorem	mulcanpi 10059	Multiplication cancellation law for positive integers. (Contributed by NM, 4-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))
Theorem	addnidpi 10060	There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → ¬ (𝐴 +N 𝐵) = 𝐴)
Theorem	ltexpi 10061*	Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ ∃𝑥 ∈ N (𝐴 +N 𝑥) = 𝐵))
Theorem	ltapi 10062	Ordering property of addition for positive integers. (Contributed by NM, 7-Mar-1996.) (New usage is discouraged.)
⊢ (𝐶 ∈ N → (𝐴 <N 𝐵 ↔ (𝐶 +N 𝐴) <N (𝐶 +N 𝐵)))
Theorem	ltmpi 10063	Ordering property of multiplication for positive integers. (Contributed by NM, 8-Feb-1996.) (New usage is discouraged.)
⊢ (𝐶 ∈ N → (𝐴 <N 𝐵 ↔ (𝐶 ·N 𝐴) <N (𝐶 ·N 𝐵)))
Theorem	1lt2pi 10064	One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
⊢ 1o <N (1o +N 1o)
Theorem	nlt1pi 10065	No positive integer is less than one. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
⊢ ¬ 𝐴 <N 1o
Theorem	indpi 10066*	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 10067*	Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10280, 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 10073) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 10070). (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 10068*	Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10280, 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 10069*	Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10280, 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 10070*	Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10280, 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 10071*	Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10280, 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 10072	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 10089. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
⊢ [Q] = ( ~Q ∩ ((N × N) × Q))
Definition	df-plq 10073	Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10280, 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 10074	Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10280, 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 10075	Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers df-c 10280, 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 10076	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 10280, 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 10077	Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10280, 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 10078	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 10079	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 10080	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 10081	The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
⊢ ~Q ∈ V
Theorem	nqex 10082	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 10083	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 10084	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 10085	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 10086	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 10087	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 10088*	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 10089	Corollary of nqereu 10088: the function [Q] is actually a function. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
⊢ [Q]:(N × N)⟶Q
Theorem	nqercl 10090	Corollary of nqereu 10088: closure of [Q]. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
Theorem	nqerrel 10091	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 10092	Corollary of nqereu 10088: 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 10093	Corollary of nqereu 10088: 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 10094	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 10095	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 10096	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 10097	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 10098	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 10099	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 10100	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)))