P. 109 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 10801-10900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	axgroth6 10801*	The Tarski-Grothendieck axiom using abbreviations. This version is called Tarski's axiom: given a set 𝑥, there exists a set 𝑦 containing 𝑥, the subsets of the members of 𝑦, the power sets of the members of 𝑦, and the subsets of 𝑦 of cardinality less than that of 𝑦. (Contributed by NM, 21-Jun-2009.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦))
Theorem	grothomex 10802	The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9600). Note that our proof depends on neither the Axiom of Infinity nor Regularity. (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.)
⊢ ω ∈ V
Theorem	grothac 10803	The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 10441). This can be put in a more conventional form via ween 10007 and dfac8 10107. Note that the mere existence of strongly inaccessible cardinals doesn't imply AC, but rather the particular form of the Tarski-Grothendieck axiom (see https://fomarchive.ugent.be/2008-March/012783.html 10107). (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.)
⊢ dom card = V
Theorem	axgroth3 10804*	Alternate version of the Tarski-Grothendieck Axiom. ax-cc 10407 is used to derive this version. (Contributed by NM, 26-Mar-2007.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → ((𝑦 ∖ 𝑧) ≼ 𝑧 ∨ 𝑧 ∈ 𝑦)))
Theorem	axgroth4 10805*	Alternate version of the Tarski-Grothendieck Axiom. ax-ac 10431 is used to derive this version. (Contributed by NM, 16-Apr-2007.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 ∃𝑣 ∈ 𝑦 ∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ (𝑦 ∩ 𝑣)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → ((𝑦 ∖ 𝑧) ≼ 𝑧 ∨ 𝑧 ∈ 𝑦)))
Theorem	grothprimlem 10806*	Lemma for grothprim 10807. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.)
⊢ ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣))))
Theorem	grothprim 10807*	The Tarski-Grothendieck Axiom ax-groth 10796 expanded into set theory primitives using 163 symbols (allowing the defined symbols ∧, ∨, ↔, and ∃). An open problem is whether a shorter equivalent exists (when expanded to primitives). (Contributed by NM, 16-Apr-2007.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧((𝑧 ∈ 𝑦 → ∃𝑣(𝑣 ∈ 𝑦 ∧ ∀𝑤(∀𝑢(𝑢 ∈ 𝑤 → 𝑢 ∈ 𝑧) → (𝑤 ∈ 𝑦 ∧ 𝑤 ∈ 𝑣)))) ∧ ∃𝑤((𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦) → (∀𝑣((𝑣 ∈ 𝑧 → ∃𝑡∀𝑢(∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑣 ∨ ℎ = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣 ∈ 𝑦 → (𝑣 ∈ 𝑧 ∨ ∃𝑢(𝑢 ∈ 𝑧 ∧ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣))))))) ∨ 𝑧 ∈ 𝑦))))
Theorem	grothtsk 10808	The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)
⊢ ∪ Tarski = V
Theorem	inaprc 10809	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 10810	Extend class definition to include the map whose value is the smallest Tarski class.
class tarskiMap
Definition	df-tskm 10811*	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 10812*	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 10813	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 10814	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 10815*	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 10816*	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 11145). After that, we derive their basic properties, various operations like addition (df-add 11099) and sine (df-sin 16113), and subsets such as the integers (df-z 12583) and natural numbers (df-nn 12225).
5.1  Construction and axiomatization of real and complex numbers
5.1.1  Dedekind-cut construction of real and complex numbers
Syntax	cnpi 10817	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 11117. The actual set of Dedekind cuts is defined by df-np 10954.
class N
Syntax	cpli 10818	Positive integer addition.
class +N
Syntax	cmi 10819	Positive integer multiplication.
class ·N
Syntax	clti 10820	Positive integer ordering relation.
class <N
Syntax	cplpq 10821	Positive pre-fraction addition.
class +pQ
Syntax	cmpq 10822	Positive pre-fraction multiplication.
class ·pQ
Syntax	cltpq 10823	Positive pre-fraction ordering relation.
class <pQ
Syntax	ceq 10824	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 10825	Set of positive fractions.
class Q
Syntax	c1q 10826	The positive fraction constant 1.
class 1Q
Syntax	cerq 10827	Positive fraction equivalence class.
class [Q]
Syntax	cplq 10828	Positive fraction addition.
class +Q
Syntax	cmq 10829	Positive fraction multiplication.
class ·Q
Syntax	crq 10830	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 10831	Positive fraction ordering relation.
class <Q
Syntax	cnp 10832	Set of positive reals.
class P
Syntax	c1p 10833	Positive real constant 1.
class 1P
Syntax	cpp 10834	Positive real addition.
class +P
Syntax	cmp 10835	Positive real multiplication.
class ·P
Syntax	cltp 10836	Positive real ordering relation.
class <P
Syntax	cer 10837	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 10838	Set of signed reals.
class R
Syntax	c0r 10839	The signed real constant 0.
class 0R
Syntax	c1r 10840	The signed real constant 1.
class 1R
Syntax	cm1r 10841	The signed real constant -1.
class -1R
Syntax	cplr 10842	Signed real addition.
class +R
Syntax	cmr 10843	Signed real multiplication.
class ·R
Syntax	cltr 10844	Signed real ordering relation.
class <R
Definition	df-ni 10845	Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 11094, 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 10846	Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 11094, 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 10847	Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 11094, 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 10848	Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 11094, 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 10849	Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
Theorem	elni2 10850	Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) (New usage is discouraged.)
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
Theorem	pinn 10851	A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
Theorem	pion 10852	A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → 𝐴 ∈ On)
Theorem	piord 10853	A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) (New usage is discouraged.)
⊢ (𝐴 ∈ N → Ord 𝐴)
Theorem	niex 10854	The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
⊢ N ∈ V
Theorem	0npi 10855	The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
⊢ ¬ ∅ ∈ N
Theorem	1pi 10856	Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.)
⊢ 1o ∈ N
Theorem	addpiord 10857	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 10858	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 10859	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 10860	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 10861	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 10862	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 10863	Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
⊢ dom +N = (N × N)
Theorem	dmmulpi 10864	Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
⊢ dom ·N = (N × N)
Theorem	addclpi 10865	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 10866	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 10867	Addition of positive integers is commutative. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
⊢ (𝐴 +N 𝐵) = (𝐵 +N 𝐴)
Theorem	addasspi 10868	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 10869	Multiplication of positive integers is commutative. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)
⊢ (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴)
Theorem	mulasspi 10870	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 10871	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 10872	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 10873	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 10874	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 10875*	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 10876	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 10877	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 10878	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 10879	No positive integer is less than one. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
⊢ ¬ 𝐴 <N 1o
Theorem	indpi 10880*	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 10881*	Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11094, 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 10887) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 10884). (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 10882*	Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11094, 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 10883*	Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11094, 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 10884*	Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11094, 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 10885*	Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11094, 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 10886	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 10903. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
⊢ [Q] = ( ~Q ∩ ((N × N) × Q))
Definition	df-plq 10887	Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11094, 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 10888	Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11094, 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 10889	Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers df-c 11094, 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 10890	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 11094, 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 10891	Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11094, 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 10892	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 10893	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 10894	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 10895	The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
⊢ ~Q ∈ V
Theorem	nqex 10896	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 10897	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 10898	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 10899	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 10900	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)