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Theorem List for Intuitionistic Logic Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmkvprop 6901* Markov's Principle expressed in terms of propositions (or more precisely, the 𝐴 = ω case is Markov's Principle). (Contributed by Jim Kingdon, 19-Mar-2023.)
((𝐴 ∈ Markov ∧ ∀𝑛𝐴 DECID 𝜑 ∧ ¬ ∀𝑛𝐴 ¬ 𝜑) → ∃𝑛𝐴 𝜑)

Theoremfodjumkvlemres 6902* Lemma for fodjumkv 6903. The final result with 𝑃 expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.)
(𝜑𝑀 ∈ Markov)    &   (𝜑𝐹:𝑀onto→(𝐴𝐵))    &   𝑃 = (𝑦𝑀 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))       (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴))

Theoremfodjumkv 6903* A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.)
(𝜑𝑀 ∈ Markov)    &   (𝜑𝐹:𝑀onto→(𝐴𝐵))       (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴))

2.6.37  Cardinal numbers

Syntaxccrd 6904 Extend class definition to include the cardinal size function.
class card

Definitiondf-card 6905* Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.)
card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})

Theoremcardcl 6906* The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
(∃𝑦 ∈ On 𝑦𝐴 → (card‘𝐴) ∈ On)

Theoremisnumi 6907 A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ dom card)

Theoremfinnum 6908 Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴 ∈ Fin → 𝐴 ∈ dom card)

Theoremonenon 6909 Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
(𝐴 ∈ On → 𝐴 ∈ dom card)

Theoremcardval3ex 6910* The value of (card‘𝐴). (Contributed by Jim Kingdon, 30-Aug-2021.)
(∃𝑥 ∈ On 𝑥𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})

Theoremoncardval 6911* The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
(𝐴 ∈ On → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})

Theoremcardonle 6912 The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.)
(𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)

Theoremcard0 6913 The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
(card‘∅) = ∅

Theoremcarden2bex 6914* If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)
((𝐴𝐵 ∧ ∃𝑥 ∈ On 𝑥𝐴) → (card‘𝐴) = (card‘𝐵))

Theorempm54.43 6915 Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.)
((𝐴 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) = ∅ ↔ (𝐴𝐵) ≈ 2o))

Theorempr2nelem 6916 Lemma for pr2ne 6917. (Contributed by FL, 17-Aug-2008.)
((𝐴𝐶𝐵𝐷𝐴𝐵) → {𝐴, 𝐵} ≈ 2o)

Theorempr2ne 6917 If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)
((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2o𝐴𝐵))

Theoremen2eleq 6918 Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
((𝑋𝑃𝑃 ≈ 2o) → 𝑃 = {𝑋, (𝑃 ∖ {𝑋})})

Theoremen2other2 6919 Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ { (𝑃 ∖ {𝑋})}) = 𝑋)

Theoremdju1p1e2 6920 Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.)
(1o ⊔ 1o) ≈ 2o

Theoreminfpwfidom 6921 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 6922 Lemma for exmidfodomr 6927. A variant of djur 6837. (Contributed by Jim Kingdon, 2-Jul-2022.)
(𝜑𝐴 ⊆ 1o)    &   (𝜑𝐵 ∈ (𝐴 ⊔ 1o))       (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))

Theoremexmidfodomrlemreseldju 6923 Lemma for exmidfodomrlemrALT 6926. A variant of eldju 6839. (Contributed by Jim Kingdon, 9-Jul-2022.)
(𝜑𝐴 ⊆ 1o)    &   (𝜑𝐵 ∈ (𝐴 ⊔ 1o))       (𝜑 → ((∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))

Theoremexmidfodomrlemim 6924* 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 6925* 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 6926* 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 6925. In particular, this proof uses eldju 6839 instead of djur 6837 and avoids djulclb 6827. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Jim Kingdon, 9-Jul-2022.)
(∀𝑥𝑦((∃𝑧 𝑧𝑦𝑦𝑥) → ∃𝑓 𝑓:𝑥onto𝑦) → EXMID)

Theoremexmidfodomr 6927* 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))

PART 3  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 6275 and similar theorems ), going from there to positive integers (df-ni 6960) and then positive rational numbers (df-nqqs 7004) 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".

3.1  Construction and axiomatization of real and complex numbers

3.1.1  Dedekind-cut construction of real and complex numbers

Syntaxcnpi 6928 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 6930 Positive integer multiplication.
class ·N

Syntaxclti 6931 Positive integer ordering relation.
class <N

class +pQ

Syntaxcmpq 6933 Positive pre-fraction multiplication.
class ·pQ

Syntaxcltpq 6934 Positive pre-fraction ordering relation.
class <pQ

Syntaxceq 6935 Equivalence class used to construct positive fractions.
class ~Q

Syntaxcnq 6936 Set of positive fractions.
class Q

Syntaxc1q 6937 The positive fraction constant 1.
class 1Q

class +Q

Syntaxcmq 6939 Positive fraction multiplication.
class ·Q

Syntaxcrq 6940 Positive fraction reciprocal operation.
class *Q

Syntaxcltq 6941 Positive fraction ordering relation.
class <Q

Syntaxceq0 6942 Equivalence class used to construct nonnegative fractions.
class ~Q0

Syntaxcnq0 6943 Set of nonnegative fractions.
class Q0

Syntaxc0q0 6944 The nonnegative fraction constant 0.
class 0Q0

class +Q0

Syntaxcmq0 6946 Nonnegative fraction multiplication.
class ·Q0

Syntaxcnp 6947 Set of positive reals.
class P

Syntaxc1p 6948 Positive real constant 1.
class 1P

class +P

Syntaxcmp 6950 Positive real multiplication.
class ·P

Syntaxcltp 6951 Positive real ordering relation.
class <P

Syntaxcer 6952 Equivalence class used to construct signed reals.
class ~R

Syntaxcnr 6953 Set of signed reals.
class R

Syntaxc0r 6954 The signed real constant 0.
class 0R

Syntaxc1r 6955 The signed real constant 1.
class 1R

Syntaxcm1r 6956 The signed real constant -1.
class -1R

class +R

Syntaxcmr 6958 Signed real multiplication.
class ·R

Syntaxcltr 6959 Signed real ordering relation.
class <R

Definitiondf-ni 6960 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 6961 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 6962 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 6963 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 6964 Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
(𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))

Theorempinn 6965 A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
(𝐴N𝐴 ∈ ω)

Theorempion 6966 A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
(𝐴N𝐴 ∈ On)

Theorempiord 6967 A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
(𝐴N → Ord 𝐴)

Theoremniex 6968 The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
N ∈ V

Theorem0npi 6969 The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
¬ ∅ ∈ N

Theoremelni2 6970 Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
(𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))

Theorem1pi 6971 Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
1oN

((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵))

Theoremmulpiord 6973 Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.)
((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))

Theoremmulidpi 6974 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 6975 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 6976 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 6977 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
((𝐴N𝐵N) → (𝐴 <N 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 <N 𝐴)))

Theorempitri3or 6978 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴 = 𝐵𝐵 <N 𝐴))

Theoremltdcpi 6979 Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
((𝐴N𝐵N) → DECID 𝐴 <N 𝐵)

Theoremltrelpi 6980 Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
<N ⊆ (N × N)

Theoremdmaddpi 6981 Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
dom +N = (N × N)

Theoremdmmulpi 6982 Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
dom ·N = (N × N)

Theoremaddclpi 6983 Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
((𝐴N𝐵N) → (𝐴 +N 𝐵) ∈ N)

Theoremmulclpi 6984 Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)

Theoremaddcompig 6985 Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐵 +N 𝐴))

Theoremaddasspig 6986 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 6987 Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴))

Theoremmulasspig 6988 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 6989 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 6990 Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.)
((𝐴N𝐵N𝐶N) → ((𝐴 +N 𝐵) = (𝐴 +N 𝐶) ↔ 𝐵 = 𝐶))

Theoremmulcanpig 6991 Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.)
((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))

Theoremaddnidpig 6992 There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.)
((𝐴N𝐵N) → ¬ (𝐴 +N 𝐵) = 𝐴)

Theoremltexpi 6993* 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 6994 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 6995 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 6996 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
1o <N (1o +N 1o)

Theoremnlt1pig 6997 No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
(𝐴N → ¬ 𝐴 <N 1o)

Theoremindpi 6998* 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 6999 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 7000* 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 7005) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 7003). (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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