P. 107 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 10601-10700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	inaprc 10601	An equivalent to the Tarski-Grothendieck Axiom: there is a proper class of inaccessible cardinals. (Contributed by Mario Carneiro, 9-Jun-2013.)
⊢ Inacc ∉ V
4.2.3  Tarski map function
Syntax	ctskm 10602	Extend class definition to include the map whose value is the smallest Tarski class.
class tarskiMap
Definition	df-tskm 10603*	A function that maps a set 𝑥 to the smallest Tarski class that contains the set. (Contributed by FL, 30-Dec-2010.)
⊢ tarskiMap = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ Tarski ∣ 𝑥 ∈ 𝑦})
Theorem	tskmval 10604*	Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
Theorem	tskmid 10605	The set 𝐴 is an element of the smallest Tarski class that contains 𝐴. CLASSES1 th. 5. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴))
Theorem	tskmcl 10606	A Tarski class that contains 𝐴 is a Tarski class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
⊢ (tarskiMap‘𝐴) ∈ Tarski
Theorem	sstskm 10607*	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 10608*	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 10937). After that, we derive their basic properties, various operations like addition (df-add 10891) and sine (df-sin 15788), and subsets such as the integers (df-z 12329) and natural numbers (df-nn 11983).
5.1  Construction and axiomatization of real and complex numbers
5.1.1  Dedekind-cut construction of real and complex numbers
Syntax	cnpi 10609	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 10909. The actual set of Dedekind cuts is defined by df-np 10746.
class N
Syntax	cpli 10610	Positive integer addition.
class +N
Syntax	cmi 10611	Positive integer multiplication.
class ·N
Syntax	clti 10612	Positive integer ordering relation.
class <N
Syntax	cplpq 10613	Positive pre-fraction addition.
class +pQ
Syntax	cmpq 10614	Positive pre-fraction multiplication.
class ·pQ
Syntax	cltpq 10615	Positive pre-fraction ordering relation.
class <pQ
Syntax	ceq 10616	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 10617	Set of positive fractions.
class Q
Syntax	c1q 10618	The positive fraction constant 1.
class 1Q
Syntax	cerq 10619	Positive fraction equivalence class.
class [Q]
Syntax	cplq 10620	Positive fraction addition.
class +Q
Syntax	cmq 10621	Positive fraction multiplication.
class ·Q
Syntax	crq 10622	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 10623	Positive fraction ordering relation.
class <Q
Syntax	cnp 10624	Set of positive reals.
class P
Syntax	c1p 10625	Positive real constant 1.
class 1P
Syntax	cpp 10626	Positive real addition.
class +P
Syntax	cmp 10627	Positive real multiplication.
class ·P
Syntax	cltp 10628	Positive real ordering relation.
class <P
Syntax	cer 10629	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 10630	Set of signed reals.
class R
Syntax	c0r 10631	The signed real constant 0.
class 0R
Syntax	c1r 10632	The signed real constant 1.
class 1R
Syntax	cm1r 10633	The signed real constant -1.
class -1R
Syntax	cplr 10634	Signed real addition.
class +R
Syntax	cmr 10635	Signed real multiplication.
class ·R
Syntax	cltr 10636	Signed real ordering relation.
class <R
Definition	df-ni 10637	Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10886, 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 10638	Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10886, 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 10639	Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10886, 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 10640	Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 10886, 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 10641	Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
Theorem	elni2 10642	Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) (New usage is discouraged.)
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
Theorem	pinn 10643	A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
Theorem	pion 10644	A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → 𝐴 ∈ On)
Theorem	piord 10645	A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → Ord 𝐴)
Theorem	niex 10646	The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
⊢ N ∈ V
Theorem	0npi 10647	The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
⊢ ¬ ∅ ∈ N
Theorem	1pi 10648	Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.)
⊢ 1o ∈ N
Theorem	addpiord 10649	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 10650	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 10651	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 10652	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 10653	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 10654	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 10655	Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
⊢ dom +N = (N × N)
Theorem	dmmulpi 10656	Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
⊢ dom ·N = (N × N)
Theorem	addclpi 10657	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 10658	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 10659	Addition of positive integers is commutative. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
⊢ (𝐴 +N 𝐵) = (𝐵 +N 𝐴)
Theorem	addasspi 10660	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 10661	Multiplication of positive integers is commutative. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)
⊢ (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴)
Theorem	mulasspi 10662	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 10663	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 10664	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 10665	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 10666	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 10667*	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 10668	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 10669	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 10670	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 10671	No positive integer is less than one. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
⊢ ¬ 𝐴 <N 1o
Theorem	indpi 10672*	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 10673*	Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10886, 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 10679) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 10676). (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 10674*	Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10886, 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 10675*	Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10886, 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 10676*	Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10886, 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 10677*	Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10886, 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 10678	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 10695. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
⊢ [Q] = ( ~Q ∩ ((N × N) × Q))
Definition	df-plq 10679	Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10886, 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 10680	Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10886, 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 10681	Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers df-c 10886, 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 10682	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 10886, 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 10683	Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10886, 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 10684	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 10685	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 10686	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 10687	The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
⊢ ~Q ∈ V
Theorem	nqex 10688	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 10689	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 10690	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 10691	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 10692	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 10693	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 10694*	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 10695	Corollary of nqereu 10694: the function [Q] is actually a function. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
⊢ [Q]:(N × N)⟶Q
Theorem	nqercl 10696	Corollary of nqereu 10694: closure of [Q]. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
Theorem	nqerrel 10697	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 10698	Corollary of nqereu 10694: 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 10699	Corollary of nqereu 10694: 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 10700	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)))