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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdjucomen 7101 Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ≈ (𝐵𝐴))

Theoremdjuassen 7102 Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝐵) ⊔ 𝐶) ≈ (𝐴 ⊔ (𝐵𝐶)))

Theoremxpdjuen 7103 Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵𝐶)) ≈ ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)))

Theoremdjudoml 7104 A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
((𝐴𝑉𝐵𝑊) → 𝐴 ≼ (𝐴𝐵))

Theoremdjudomr 7105 A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
((𝐴𝑉𝐵𝑊) → 𝐵 ≼ (𝐴𝐵))

2.6.43  Excluded middle and the power set of a singleton

Theorempw1on 7106 The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
𝒫 1o ∈ On

Theorempw1dom2 7107 The power set of 1o dominates 2o. Also see pwpw0ss 3741 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.)
2o ≼ 𝒫 1o

Theorempw1ne0 7108 The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
𝒫 1o ≠ ∅

Theorempw1ne1 7109 The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
𝒫 1o ≠ 1o

Theorempw1ne3 7110 The power set of 1o is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
𝒫 1o ≠ 3o

Theorempw1nel3 7111 Negated excluded middle implies that the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
EXMID → ¬ 𝒫 1o ∈ 3o)

Theoremsucpw1ne3 7112 Negated excluded middle implies that the successor of the power set of 1o is not three . (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
EXMID → suc 𝒫 1o ≠ 3o)

Theoremsucpw1nel3 7113 The successor of the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
¬ suc 𝒫 1o ∈ 3o

Theorem3nelsucpw1 7114 Three is not an element of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
¬ 3o ∈ suc 𝒫 1o

Theoremsucpw1nss3 7115 Negated excluded middle implies that the successor of the power set of 1o is not a subset of 3o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
EXMID → ¬ suc 𝒫 1o ⊆ 3o)

Theorem3nsssucpw1 7116 Negated excluded middle implies that 3o is not a subset of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
EXMID → ¬ 3o ⊆ suc 𝒫 1o)

Theoremonntri35 7117* Double negated ordinal trichotomy.

There are five equivalent statements: (1) ¬ ¬ ∀𝑥 ∈ On∀𝑦 ∈ On(𝑥𝑦𝑥 = 𝑦𝑦𝑥), (2) ¬ ¬ ∀𝑥 ∈ On∀𝑦 ∈ On(𝑥𝑦𝑦𝑥), (3) 𝑥 ∈ On∀𝑦 ∈ On¬ ¬ (𝑥𝑦𝑥 = 𝑦𝑦𝑥), (4) 𝑥 ∈ On∀𝑦 ∈ On¬ ¬ (𝑥𝑦𝑦𝑥), and (5) ¬ ¬ EXMID. That these are all equivalent is expressed by (1) implies (3) (onntri13 7118), (3) implies (5) (onntri35 7117), (5) implies (1) (onntri51 7119), (2) implies (4) (onntri24 7121), (4) implies (5) (onntri45 7120), and (5) implies (2) (onntri52 7122).

(Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)

(∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ¬ ¬ EXMID)

Theoremonntri13 7118 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))

Theoremonntri51 7119 Double negated ordinal trichotomy. The hypothesis represents the proposition that ordinal trichotomy follows from excluded middle, which is proved in the Metamath Proof Explorer so of course could be proved here, but isn't yet. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥))       (¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥))

Theoremonntri45 7120* Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥) → ¬ ¬ EXMID)

Theoremonntri24 7121 Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑦𝑥))

Theoremonntri52 7122* Double negated ordinal trichotomy. Like onntri51 7119, the hypothesis is only here temporarily until that can be proved. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
(EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥))       (¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥))

PART 3  CHOICE PRINCIPLES

We have already introduced the full Axiom of Choice df-ac 7091 but since it implies excluded middle as shown at exmidac 7094, 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

Syntaxwacc 7123 Formula for an abbreviation of countable choice.
wff CCHOICE

Definitiondf-cc 7124* The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7091 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.)
(CCHOICE ↔ ∀𝑥(dom 𝑥 ≈ ω → ∃𝑓(𝑓𝑥𝑓 Fn dom 𝑥)))

Theoremccfunen 7125* Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)
(𝜑CCHOICE)    &   (𝜑𝐴 ≈ ω)    &   (𝜑 → ∀𝑥𝐴𝑤 𝑤𝑥)       (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))

Theoremcc1 7126* 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 ∧ ∀𝑧𝑥𝑤 𝑤𝑧) → ∃𝑓𝑧𝑥 (𝑓𝑧) ∈ 𝑧))

Theoremcc2lem 7127* Lemma for cc2 7128. (Contributed by Jim Kingdon, 27-Apr-2024.)
(𝜑CCHOICE)    &   (𝜑𝐹 Fn ω)    &   (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹𝑥))    &   𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐹𝑛)))    &   𝐺 = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘(𝐴𝑛))))       (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔𝑛) ∈ (𝐹𝑛)))

Theoremcc2 7128* Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
(𝜑CCHOICE)    &   (𝜑𝐹 Fn ω)    &   (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹𝑥))       (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔𝑛) ∈ (𝐹𝑛)))

Theoremcc3 7129* Countable choice using a sequence F(n) . (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Jim Kingdon, 29-Apr-2024.)
(𝜑CCHOICE)    &   (𝜑 → ∀𝑛𝑁 𝐹 ∈ V)    &   (𝜑 → ∀𝑛𝑁𝑤 𝑤𝐹)    &   (𝜑𝑁 ≈ ω)       (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ 𝐹))

Theoremcc4f 7130* 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.)
(𝜑CCHOICE)    &   (𝜑𝐴𝑉)    &   𝑛𝐴    &   (𝜑𝑁 ≈ ω)    &   (𝑥 = (𝑓𝑛) → (𝜓𝜒))    &   (𝜑 → ∀𝑛𝑁𝑥𝐴 𝜓)       (𝜑 → ∃𝑓(𝑓:𝑁𝐴 ∧ ∀𝑛𝑁 𝜒))

Theoremcc4 7131* 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.)
(𝜑CCHOICE)    &   (𝜑𝐴𝑉)    &   (𝜑𝑁 ≈ ω)    &   (𝑥 = (𝑓𝑛) → (𝜓𝜒))    &   (𝜑 → ∀𝑛𝑁𝑥𝐴 𝜓)       (𝜑 → ∃𝑓(𝑓:𝑁𝐴 ∧ ∀𝑛𝑁 𝜒))

Theoremcc4n 7132* Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7131, 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.)
(𝜑CCHOICE)    &   (𝜑 → ∀𝑛𝑁 {𝑥𝐴𝜓} ∈ 𝑉)    &   (𝜑𝑁 ≈ ω)    &   (𝑥 = (𝑓𝑛) → (𝜓𝜒))    &   (𝜑 → ∀𝑛𝑁𝑥𝐴 𝜓)       (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 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 6382 and similar theorems ), going from there to positive integers (df-ni 7165) and then positive rational numbers (df-nqqs 7209) 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. With excluded middle, it is natural to define the cut as the lower set only (as Metamath Proof Explorer does), but 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 7327. The Cauchy reals (without countable choice) fail to satisfy ax-caucvg 7793 and the MacNeille reals fail to satisfy axltwlin 7885, 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

Syntaxcnpi 7133 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

class +N

Syntaxcmi 7135 Positive integer multiplication.
class ·N

Syntaxclti 7136 Positive integer ordering relation.
class <N

class +pQ

Syntaxcmpq 7138 Positive pre-fraction multiplication.
class ·pQ

Syntaxcltpq 7139 Positive pre-fraction ordering relation.
class <pQ

Syntaxceq 7140 Equivalence class used to construct positive fractions.
class ~Q

Syntaxcnq 7141 Set of positive fractions.
class Q

Syntaxc1q 7142 The positive fraction constant 1.
class 1Q

class +Q

Syntaxcmq 7144 Positive fraction multiplication.
class ·Q

Syntaxcrq 7145 Positive fraction reciprocal operation.
class *Q

Syntaxcltq 7146 Positive fraction ordering relation.
class <Q

Syntaxceq0 7147 Equivalence class used to construct nonnegative fractions.
class ~Q0

Syntaxcnq0 7148 Set of nonnegative fractions.
class Q0

Syntaxc0q0 7149 The nonnegative fraction constant 0.
class 0Q0

class +Q0

Syntaxcmq0 7151 Nonnegative fraction multiplication.
class ·Q0

Syntaxcnp 7152 Set of positive reals.
class P

Syntaxc1p 7153 Positive real constant 1.
class 1P

class +P

Syntaxcmp 7155 Positive real multiplication.
class ·P

Syntaxcltp 7156 Positive real ordering relation.
class <P

Syntaxcer 7157 Equivalence class used to construct signed reals.
class ~R

Syntaxcnr 7158 Set of signed reals.
class R

Syntaxc0r 7159 The signed real constant 0.
class 0R

Syntaxc1r 7160 The signed real constant 1.
class 1R

Syntaxcm1r 7161 The signed real constant -1.
class -1R

class +R

Syntaxcmr 7163 Signed real multiplication.
class ·R

Syntaxcltr 7164 Signed real ordering relation.
class <R

Definitiondf-ni 7165 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 ∖ {∅})

Definitiondf-pli 7166 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))

Definitiondf-mi 7167 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))

Definitiondf-lti 7168 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))

Theoremelni 7169 Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
(𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))

Theorempinn 7170 A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
(𝐴N𝐴 ∈ ω)

Theorempion 7171 A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
(𝐴N𝐴 ∈ On)

Theorempiord 7172 A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
(𝐴N → Ord 𝐴)

Theoremniex 7173 The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
N ∈ V

Theorem0npi 7174 The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
¬ ∅ ∈ N

Theoremelni2 7175 Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
(𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))

Theorem1pi 7176 Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
1oN

((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))

Theoremmulpiord 7178 Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.)
((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))

Theoremmulidpi 7179 1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.)
(𝐴N → (𝐴 ·N 1o) = 𝐴)

Theoremltpiord 7180 Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴𝐵))

Theoremltsopi 7181 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

Theorempitric 7182 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
((𝐴N𝐵N) → (𝐴 <N 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 <N 𝐴)))

Theorempitri3or 7183 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴 = 𝐵𝐵 <N 𝐴))

Theoremltdcpi 7184 Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
((𝐴N𝐵N) → DECID 𝐴 <N 𝐵)

Theoremltrelpi 7185 Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
<N ⊆ (N × N)

Theoremdmaddpi 7186 Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
dom +N = (N × N)

Theoremdmmulpi 7187 Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
dom ·N = (N × N)

Theoremaddclpi 7188 Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
((𝐴N𝐵N) → (𝐴 +N 𝐵) ∈ N)

Theoremmulclpi 7189 Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)

Theoremaddcompig 7190 Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐵 +N 𝐴))

Theoremaddasspig 7191 Addition of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((𝐴N𝐵N𝐶N) → ((𝐴 +N 𝐵) +N 𝐶) = (𝐴 +N (𝐵 +N 𝐶)))

Theoremmulcompig 7192 Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴))

Theoremmulasspig 7193 Multiplication of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶)))

Theoremdistrpig 7194 Multiplication of positive integers is distributive. (Contributed by Jim Kingdon, 26-Aug-2019.)
((𝐴N𝐵N𝐶N) → (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)))

Theoremaddcanpig 7195 Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.)
((𝐴N𝐵N𝐶N) → ((𝐴 +N 𝐵) = (𝐴 +N 𝐶) ↔ 𝐵 = 𝐶))

Theoremmulcanpig 7196 Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.)
((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))

Theoremaddnidpig 7197 There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.)
((𝐴N𝐵N) → ¬ (𝐴 +N 𝐵) = 𝐴)

Theoremltexpi 7198* Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
((𝐴N𝐵N) → (𝐴 <N 𝐵 ↔ ∃𝑥N (𝐴 +N 𝑥) = 𝐵))

Theoremltapig 7199 Ordering property of addition for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.)
((𝐴N𝐵N𝐶N) → (𝐴 <N 𝐵 ↔ (𝐶 +N 𝐴) <N (𝐶 +N 𝐵)))

Theoremltmpig 7200 Ordering property of multiplication for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.)
((𝐴N𝐵N𝐶N) → (𝐴 <N 𝐵 ↔ (𝐶 ·N 𝐴) <N (𝐶 ·N 𝐵)))

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