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Theorem List for Intuitionistic Logic Explorer - 6501-6600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoreminfeuti 6501* An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))

Theoreminfsnti 6502* The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑𝐵𝐴)       (𝜑 → inf({𝐵}, 𝐴, 𝑅) = 𝐵)

Theoreminf00 6503 The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
inf(𝐵, ∅, 𝑅) = ∅

Theoreminfisoti 6504* Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐶 𝑧𝑅𝑦)))    &   ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       (𝜑 → inf((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))

2.6.30  Ordinal isomorphism

Theoremordiso2 6505 Generalize ordiso 6506 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵)

Theoremordiso 6506* Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 ↔ ∃𝑓 𝑓 Isom E , E (𝐴, 𝐵)))

2.6.31  Cardinal numbers

Syntaxccrd 6507 Extend class definition to include the cardinal size function.
class card

Definitiondf-card 6508* 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 6509* The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
(∃𝑦 ∈ On 𝑦𝐴 → (card‘𝐴) ∈ On)

Theoremisnumi 6510 A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ dom card)

Theoremfinnum 6511 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 6512 Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
(𝐴 ∈ On → 𝐴 ∈ dom card)

Theoremcardval3ex 6513* The value of (card‘𝐴). (Contributed by Jim Kingdon, 30-Aug-2021.)
(∃𝑥 ∈ On 𝑥𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})

Theoremoncardval 6514* 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 6515 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 6516 The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
(card‘∅) = ∅

Theoremcarden2bex 6517* 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 6518 Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.)
((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ ↔ (𝐴𝐵) ≈ 2𝑜))

Theorempr2nelem 6519 Lemma for pr2ne 6520. (Contributed by FL, 17-Aug-2008.)
((𝐴𝐶𝐵𝐷𝐴𝐵) → {𝐴, 𝐵} ≈ 2𝑜)

Theorempr2ne 6520 If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)
((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2𝑜𝐴𝐵))

Theoremen2eleq 6521 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 6522 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)

Theoreminfpwfidom 6523 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))

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 6118 and similar theorems ), going from there to positive integers (df-ni 6556) and then positive rational numbers (df-nqqs 6600) 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 6524 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 6526 Positive integer multiplication.
class ·N

Syntaxclti 6527 Positive integer ordering relation.
class <N

class +pQ

Syntaxcmpq 6529 Positive pre-fraction multiplication.
class ·pQ

Syntaxcltpq 6530 Positive pre-fraction ordering relation.
class <pQ

Syntaxceq 6531 Equivalence class used to construct positive fractions.
class ~Q

Syntaxcnq 6532 Set of positive fractions.
class Q

Syntaxc1q 6533 The positive fraction constant 1.
class 1Q

class +Q

Syntaxcmq 6535 Positive fraction multiplication.
class ·Q

Syntaxcrq 6536 Positive fraction reciprocal operation.
class *Q

Syntaxcltq 6537 Positive fraction ordering relation.
class <Q

Syntaxceq0 6538 Equivalence class used to construct non-negative fractions.
class ~Q0

Syntaxcnq0 6539 Set of non-negative fractions.
class Q0

Syntaxc0q0 6540 The non-negative fraction constant 0.
class 0Q0

class +Q0

Syntaxcmq0 6542 Non-negative fraction multiplication.
class ·Q0

Syntaxcnp 6543 Set of positive reals.
class P

Syntaxc1p 6544 Positive real constant 1.
class 1P

class +P

Syntaxcmp 6546 Positive real multiplication.
class ·P

Syntaxcltp 6547 Positive real ordering relation.
class <P

Syntaxcer 6548 Equivalence class used to construct signed reals.
class ~R

Syntaxcnr 6549 Set of signed reals.
class R

Syntaxc0r 6550 The signed real constant 0.
class 0R

Syntaxc1r 6551 The signed real constant 1.
class 1R

Syntaxcm1r 6552 The signed real constant -1.
class -1R

class +R

Syntaxcmr 6554 Signed real multiplication.
class ·R

Syntaxcltr 6555 Signed real ordering relation.
class <R

Definitiondf-ni 6556 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 6557 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 6558 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 6559 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 6560 Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
(𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))

Theorempinn 6561 A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
(𝐴N𝐴 ∈ ω)

Theorempion 6562 A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
(𝐴N𝐴 ∈ On)

Theorempiord 6563 A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
(𝐴N → Ord 𝐴)

Theoremniex 6564 The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
N ∈ V

Theorem0npi 6565 The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
¬ ∅ ∈ N

Theoremelni2 6566 Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
(𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))

Theorem1pi 6567 Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
1𝑜N

((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐴 +𝑜 𝐵))

Theoremmulpiord 6569 Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.)
((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵))

Theoremmulidpi 6570 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 6571 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 6572 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 6573 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
((𝐴N𝐵N) → (𝐴 <N 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 <N 𝐴)))

Theorempitri3or 6574 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴 = 𝐵𝐵 <N 𝐴))

Theoremltdcpi 6575 Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
((𝐴N𝐵N) → DECID 𝐴 <N 𝐵)

Theoremltrelpi 6576 Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
<N ⊆ (N × N)

Theoremdmaddpi 6577 Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
dom +N = (N × N)

Theoremdmmulpi 6578 Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
dom ·N = (N × N)

Theoremaddclpi 6579 Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
((𝐴N𝐵N) → (𝐴 +N 𝐵) ∈ N)

Theoremmulclpi 6580 Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)

Theoremaddcompig 6581 Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐵 +N 𝐴))

Theoremaddasspig 6582 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 6583 Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴))

Theoremmulasspig 6584 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 6585 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 6586 Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.)
((𝐴N𝐵N𝐶N) → ((𝐴 +N 𝐵) = (𝐴 +N 𝐶) ↔ 𝐵 = 𝐶))

Theoremmulcanpig 6587 Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.)
((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))

Theoremaddnidpig 6588 There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.)
((𝐴N𝐵N) → ¬ (𝐴 +N 𝐵) = 𝐴)

Theoremltexpi 6589* 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 6590 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 6591 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 6592 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
1𝑜 <N (1𝑜 +N 1𝑜)

Theoremnlt1pig 6593 No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
(𝐴N → ¬ 𝐴 <N 1𝑜)

Theoremindpi 6594* 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 6595 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 6596* 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 6601) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 6599). (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))⟩)

Definitiondf-mpq 6597* 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) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)

Definitiondf-ltpq 6598* 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)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))}

Definitiondf-enq 6599* 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)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))}

Definitiondf-nqqs 6600 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 )

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