P. 73 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	onntri3or 7201*	Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
⊢ (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
Theorem	onntri2or 7202*	Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
⊢ (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
PART 3  CHOICE PRINCIPLES We have already introduced the full Axiom of Choice df-ac 7162 but since it implies excluded middle as shown at exmidac 7165, it is not especially relevant to us. In this section we define countable choice and dependent choice, which are not as strong as thus often considered in mathematics which seeks to avoid full excluded middle.
3.1  Countable Choice and Dependent Choice
3.1.1  Introduce Countable Choice
Syntax	wacc 7203	Formula for an abbreviation of countable choice.
wff CCHOICE
Definition	df-cc 7204*	The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7162 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.)
⊢ (CCHOICE ↔ ∀𝑥(dom 𝑥 ≈ ω → ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)))
Theorem	ccfunen 7205*	Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)
⊢ (𝜑 → CCHOICE)    &   ⊢ (𝜑 → 𝐴 ≈ ω)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))
Theorem	cc1 7206*	Countable choice in terms of a choice function on a countably infinite set of inhabited sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
⊢ (CCHOICE → ∀𝑥((𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑓‘𝑧) ∈ 𝑧))
Theorem	cc2lem 7207*	Lemma for cc2 7208. (Contributed by Jim Kingdon, 27-Apr-2024.)
⊢ (𝜑 → CCHOICE)    &   ⊢ (𝜑 → 𝐹 Fn ω)    &   ⊢ (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑥))    &   ⊢ 𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐹‘𝑛)))    &   ⊢ 𝐺 = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘(𝐴‘𝑛))))    ⇒   ⊢ (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔‘𝑛) ∈ (𝐹‘𝑛)))
Theorem	cc2 7208*	Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
⊢ (𝜑 → CCHOICE)    &   ⊢ (𝜑 → 𝐹 Fn ω)    &   ⊢ (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔‘𝑛) ∈ (𝐹‘𝑛)))
Theorem	cc3 7209*	Countable choice using a sequence F(n) . (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Jim Kingdon, 29-Apr-2024.)
⊢ (𝜑 → CCHOICE)    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 𝐹 ∈ V)    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑤 𝑤 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑁 ≈ ω)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹))
Theorem	cc4f 7210*	Countable choice by showing the existence of a function 𝑓 which can choose a value at each index 𝑛 such that 𝜒 holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
⊢ (𝜑 → CCHOICE)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ Ⅎ𝑛𝐴    &   ⊢ (𝜑 → 𝑁 ≈ ω)    &   ⊢ (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒))
Theorem	cc4 7211*	Countable choice by showing the existence of a function 𝑓 which can choose a value at each index 𝑛 such that 𝜒 holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 1-May-2024.)
⊢ (𝜑 → CCHOICE)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ≈ ω)    &   ⊢ (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒))
Theorem	cc4n 7212*	Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7211, the hypotheses only require an A(n) for each value of 𝑛, not a single set 𝐴 which suffices for every 𝑛 ∈ ω. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
⊢ (𝜑 → CCHOICE)    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ≈ ω)    &   ⊢ (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 𝜒))
PART 4  REAL AND COMPLEX NUMBERS This section derives the basics of real and complex numbers. To construct the real numbers constructively, we follow two main sources. The first is Metamath Proof Explorer, which has the advantage of being already formalized in metamath. Its disadvantage, for our purposes, is that it assumes the law of the excluded middle throughout. Since we have already developed natural numbers ( for example, nna0 6442 and similar theorems ), going from there to positive integers (df-ni 7245) and then positive rational numbers (df-nqqs 7289) does not involve a major change in approach compared with the Metamath Proof Explorer. It is when we proceed to Dedekind cuts that we bring in more material from Section 11.2 of [HoTT], which focuses on the aspects of Dedekind cuts which are different without excluded middle or choice principles. With excluded middle, it is natural to define a cut as the lower set only (as Metamath Proof Explorer does), but here we define the cut as a pair of both the lower and upper sets, as [HoTT] does. There are also differences in how we handle order and replacing "not equal to zero" with "apart from zero". When working constructively, there are several possible definitions of real numbers. Here we adopt the most common definition, as two-sided Dedekind cuts with the properties described at df-inp 7407. The Cauchy reals (without countable choice) fail to satisfy ax-caucvg 7873 and the MacNeille reals fail to satisfy axltwlin 7966, and we do not develop them here. For more on differing definitions of the reals, see the introduction to Chapter 11 in [HoTT] or Section 1.2 of [BauerHanson].
4.1  Construction and axiomatization of real and complex numbers
4.1.1  Dedekind-cut construction of real and complex numbers
Syntax	cnpi 7213	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.
class N
Syntax	cpli 7214	Positive integer addition.
class +N
Syntax	cmi 7215	Positive integer multiplication.
class ·N
Syntax	clti 7216	Positive integer ordering relation.
class <N
Syntax	cplpq 7217	Positive pre-fraction addition.
class +pQ
Syntax	cmpq 7218	Positive pre-fraction multiplication.
class ·pQ
Syntax	cltpq 7219	Positive pre-fraction ordering relation.
class <pQ
Syntax	ceq 7220	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 7221	Set of positive fractions.
class Q
Syntax	c1q 7222	The positive fraction constant 1.
class 1Q
Syntax	cplq 7223	Positive fraction addition.
class +Q
Syntax	cmq 7224	Positive fraction multiplication.
class ·Q
Syntax	crq 7225	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 7226	Positive fraction ordering relation.
class <Q
Syntax	ceq0 7227	Equivalence class used to construct nonnegative fractions.
class ~Q0
Syntax	cnq0 7228	Set of nonnegative fractions.
class Q0
Syntax	c0q0 7229	The nonnegative fraction constant 0.
class 0Q0
Syntax	cplq0 7230	Nonnegative fraction addition.
class +Q0
Syntax	cmq0 7231	Nonnegative fraction multiplication.
class ·Q0
Syntax	cnp 7232	Set of positive reals.
class P
Syntax	c1p 7233	Positive real constant 1.
class 1P
Syntax	cpp 7234	Positive real addition.
class +P
Syntax	cmp 7235	Positive real multiplication.
class ·P
Syntax	cltp 7236	Positive real ordering relation.
class <P
Syntax	cer 7237	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 7238	Set of signed reals.
class R
Syntax	c0r 7239	The signed real constant 0.
class 0R
Syntax	c1r 7240	The signed real constant 1.
class 1R
Syntax	cm1r 7241	The signed real constant -1.
class -1R
Syntax	cplr 7242	Signed real addition.
class +R
Syntax	cmr 7243	Signed real multiplication.
class ·R
Syntax	cltr 7244	Signed real ordering relation.
class <R
Definition	df-ni 7245	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 7246	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 7247	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 7248	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 7249	Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
Theorem	pinn 7250	A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
Theorem	pion 7251	A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
⊢ (𝐴 ∈ N → 𝐴 ∈ On)
Theorem	piord 7252	A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
⊢ (𝐴 ∈ N → Ord 𝐴)
Theorem	niex 7253	The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
⊢ N ∈ V
Theorem	0npi 7254	The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
⊢ ¬ ∅ ∈ N
Theorem	elni2 7255	Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
Theorem	1pi 7256	Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
⊢ 1o ∈ N
Theorem	addpiord 7257	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 7258	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 7259	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 7260	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 7261	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 7262	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 7263	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 7264	Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → DECID 𝐴 <N 𝐵)
Theorem	ltrelpi 7265	Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
⊢ <N ⊆ (N × N)
Theorem	dmaddpi 7266	Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
⊢ dom +N = (N × N)
Theorem	dmmulpi 7267	Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
⊢ dom ·N = (N × N)
Theorem	addclpi 7268	Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) ∈ N)
Theorem	mulclpi 7269	Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N)
Theorem	addcompig 7270	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 7271	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 7272	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 7273	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 7274	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 7275	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 7276	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 7277	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 7278*	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 7279	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 7280	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 7281	One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
⊢ 1o <N (1o +N 1o)
Theorem	nlt1pig 7282	No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
⊢ (𝐴 ∈ N → ¬ 𝐴 <N 1o)
Theorem	indpi 7283*	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 7284	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 7285*	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 7290) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 7288). (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 7286*	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 7287*	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 7288*	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 7289	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 7290*	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 7291*	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 7292	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 7293*	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 7294*	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 7295*	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 7296*	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 7297	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 7298	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 7299	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 7300	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)))