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Theorem List for Intuitionistic Logic Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdju1p1e2 7001 Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.)
(1o ⊔ 1o) ≈ 2o

Theoreminfpwfidom 7002 The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption (𝒫 𝐴 ∩ Fin) ∈ V because this theorem also implies that 𝐴 is a set if 𝒫 𝐴 ∩ Fin is.) (Contributed by Mario Carneiro, 17-May-2015.)
((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))

Theoremexmidfodomrlemeldju 7003 Lemma for exmidfodomr 7008. A variant of djur 6906. (Contributed by Jim Kingdon, 2-Jul-2022.)
(𝜑𝐴 ⊆ 1o)    &   (𝜑𝐵 ∈ (𝐴 ⊔ 1o))       (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))

Theoremexmidfodomrlemreseldju 7004 Lemma for exmidfodomrlemrALT 7007. A variant of eldju 6905. (Contributed by Jim Kingdon, 9-Jul-2022.)
(𝜑𝐴 ⊆ 1o)    &   (𝜑𝐵 ∈ (𝐴 ⊔ 1o))       (𝜑 → ((∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))

Theoremexmidfodomrlemim 7005* Excluded middle implies the existence of a mapping from any set onto any inhabited set that it dominates. Proposition 1.1 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.)
(EXMID → ∀𝑥𝑦((∃𝑧 𝑧𝑦𝑦𝑥) → ∃𝑓 𝑓:𝑥onto𝑦))

Theoremexmidfodomrlemr 7006* The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.)
(∀𝑥𝑦((∃𝑧 𝑧𝑦𝑦𝑥) → ∃𝑓 𝑓:𝑥onto𝑦) → EXMID)

TheoremexmidfodomrlemrALT 7007* The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. An alternative proof of exmidfodomrlemr 7006. In particular, this proof uses eldju 6905 instead of djur 6906 and avoids djulclb 6892. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Jim Kingdon, 9-Jul-2022.)
(∀𝑥𝑦((∃𝑧 𝑧𝑦𝑦𝑥) → ∃𝑓 𝑓:𝑥onto𝑦) → EXMID)

Theoremexmidfodomr 7008* Excluded middle is equivalent to the existence of a mapping from any set onto any inhabited set that it dominates. (Contributed by Jim Kingdon, 1-Jul-2022.)
(EXMID ↔ ∀𝑥𝑦((∃𝑧 𝑧𝑦𝑦𝑥) → ∃𝑓 𝑓:𝑥onto𝑦))

2.6.39  Axiom of Choice equivalents

Syntaxwac 7009 Formula for an abbreviation of the axiom of choice.
wff CHOICE

Definitiondf-ac 7010* The expression CHOICE will be used as a readable shorthand for any form of the axiom of choice; all concrete forms are long, cryptic, have dummy variables, or all three, making it useful to have a short name. Similar to the Axiom of Choice (first form) of [Enderton] p. 49.

There are some decisions about how to write this definition especially around whether ax-setind 4412 is needed to show equivalence to other ways of stating choice, and about whether choice functions are available for nonempty sets or inhabited sets. (Contributed by Mario Carneiro, 22-Feb-2015.)

(CHOICE ↔ ∀𝑥𝑓(𝑓𝑥𝑓 Fn dom 𝑥))

Theoremacfun 7011* A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.)
(𝜑CHOICE)    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑥𝐴𝑤 𝑤𝑥)       (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))

Theoremexmidaclem 7012* Lemma for exmidac 7013. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})}    &   𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})}    &   𝐶 = {𝐴, 𝐵}       (CHOICEEXMID)

Theoremexmidac 7013 The axiom of choice implies excluded middle. See acexmid 5727 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.)
(CHOICEEXMID)

2.6.40  Cardinal number arithmetic

Theoremendjudisj 7014 Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by Jim Kingdon, 30-Jul-2023.)
((𝐴𝑉𝐵𝑊 ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ≈ (𝐴𝐵))

Theoremdjuen 7015 Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.)
((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ≈ (𝐵𝐷))

Theoremdjuenun 7016 Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.)
((𝐴𝐵𝐶𝐷 ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≈ (𝐵𝐷))

Theoremdju1en 7017 Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉 ∧ ¬ 𝐴𝐴) → (𝐴 ⊔ 1o) ≈ suc 𝐴)

Theoremdju0en 7018 Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴𝑉 → (𝐴 ⊔ ∅) ≈ 𝐴)

Theoremxp2dju 7019 Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(2o × 𝐴) = (𝐴𝐴)

Theoremdjucomen 7020 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 7021 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 7022 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 7023 A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
((𝐴𝑉𝐵𝑊) → 𝐴 ≼ (𝐴𝐵))

Theoremdjudomr 7024 A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
((𝐴𝑉𝐵𝑊) → 𝐵 ≼ (𝐴𝐵))

PART 3  CHOICE PRINCIPLES

We have already introduced the full Axiom of Choice df-ac 7010 but since it implies excluded middle as shown at exmidac 7013, 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 7025 Formula for an abbreviation of countable choice.
wff CCHOICE

Definitiondf-cc 7026* The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7010 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.)
(CCHOICE ↔ ∀𝑥(dom 𝑥 ≈ ω → ∃𝑓(𝑓𝑥𝑓 Fn dom 𝑥)))

Theoremccfunen 7027* 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))

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 6324 and similar theorems ), going from there to positive integers (df-ni 7060) and then positive rational numbers (df-nqqs 7104) 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".

4.1  Construction and axiomatization of real and complex numbers

4.1.1  Dedekind-cut construction of real and complex numbers

Syntaxcnpi 7028 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 7030 Positive integer multiplication.
class ·N

Syntaxclti 7031 Positive integer ordering relation.
class <N

class +pQ

Syntaxcmpq 7033 Positive pre-fraction multiplication.
class ·pQ

Syntaxcltpq 7034 Positive pre-fraction ordering relation.
class <pQ

Syntaxceq 7035 Equivalence class used to construct positive fractions.
class ~Q

Syntaxcnq 7036 Set of positive fractions.
class Q

Syntaxc1q 7037 The positive fraction constant 1.
class 1Q

class +Q

Syntaxcmq 7039 Positive fraction multiplication.
class ·Q

Syntaxcrq 7040 Positive fraction reciprocal operation.
class *Q

Syntaxcltq 7041 Positive fraction ordering relation.
class <Q

Syntaxceq0 7042 Equivalence class used to construct nonnegative fractions.
class ~Q0

Syntaxcnq0 7043 Set of nonnegative fractions.
class Q0

Syntaxc0q0 7044 The nonnegative fraction constant 0.
class 0Q0

class +Q0

Syntaxcmq0 7046 Nonnegative fraction multiplication.
class ·Q0

Syntaxcnp 7047 Set of positive reals.
class P

Syntaxc1p 7048 Positive real constant 1.
class 1P

class +P

Syntaxcmp 7050 Positive real multiplication.
class ·P

Syntaxcltp 7051 Positive real ordering relation.
class <P

Syntaxcer 7052 Equivalence class used to construct signed reals.
class ~R

Syntaxcnr 7053 Set of signed reals.
class R

Syntaxc0r 7054 The signed real constant 0.
class 0R

Syntaxc1r 7055 The signed real constant 1.
class 1R

Syntaxcm1r 7056 The signed real constant -1.
class -1R

class +R

Syntaxcmr 7058 Signed real multiplication.
class ·R

Syntaxcltr 7059 Signed real ordering relation.
class <R

Definitiondf-ni 7060 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 7061 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 7062 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 7063 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 7064 Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
(𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))

Theorempinn 7065 A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
(𝐴N𝐴 ∈ ω)

Theorempion 7066 A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
(𝐴N𝐴 ∈ On)

Theorempiord 7067 A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
(𝐴N → Ord 𝐴)

Theoremniex 7068 The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
N ∈ V

Theorem0npi 7069 The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
¬ ∅ ∈ N

Theoremelni2 7070 Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
(𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))

Theorem1pi 7071 Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
1oN

((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))

Theoremmulpiord 7073 Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.)
((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))

Theoremmulidpi 7074 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 7075 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 7076 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 7077 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
((𝐴N𝐵N) → (𝐴 <N 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 <N 𝐴)))

Theorempitri3or 7078 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴 = 𝐵𝐵 <N 𝐴))

Theoremltdcpi 7079 Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
((𝐴N𝐵N) → DECID 𝐴 <N 𝐵)

Theoremltrelpi 7080 Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
<N ⊆ (N × N)

Theoremdmaddpi 7081 Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
dom +N = (N × N)

Theoremdmmulpi 7082 Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
dom ·N = (N × N)

Theoremaddclpi 7083 Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
((𝐴N𝐵N) → (𝐴 +N 𝐵) ∈ N)

Theoremmulclpi 7084 Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)

Theoremaddcompig 7085 Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐵 +N 𝐴))

Theoremaddasspig 7086 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 7087 Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴))

Theoremmulasspig 7088 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 7089 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 7090 Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.)
((𝐴N𝐵N𝐶N) → ((𝐴 +N 𝐵) = (𝐴 +N 𝐶) ↔ 𝐵 = 𝐶))

Theoremmulcanpig 7091 Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.)
((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))

Theoremaddnidpig 7092 There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.)
((𝐴N𝐵N) → ¬ (𝐴 +N 𝐵) = 𝐴)

Theoremltexpi 7093* 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 7094 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 7095 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)))

Theorem1lt2pi 7096 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
1o <N (1o +N 1o)

Theoremnlt1pig 7097 No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
(𝐴N → ¬ 𝐴 <N 1o)

Theoremindpi 7098* Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.)
(𝑥 = 1o → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 +N 1o) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦N → (𝜒𝜃))       (𝐴N𝜏)

Theoremnnppipi 7099 A natural number plus a positive integer is a positive integer. (Contributed by Jim Kingdon, 10-Nov-2019.)
((𝐴 ∈ ω ∧ 𝐵N) → (𝐴 +o 𝐵) ∈ N)

Definitiondf-plpq 7100* 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 7105) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 7103). (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) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)

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