Home Metamath Proof ExplorerTheorem List (p. 331 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27745) Hilbert Space Explorer (27746-29270) Users' Mathboxes (29271-42316)

Theorem List for Metamath Proof Explorer - 33001-33100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-evalfn 33001 The evaluation at a class is a function on the universal class. (General form of slotfn 15856). (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by BJ, 27-Dec-2021.)
Slot 𝐴 Fn V

Theorembj-evalval 33002 Value of the evaluation at a class. (Closed form of strfvnd 15857 and strfvn 15860). (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 15-Nov-2014.) (Revised by BJ, 27-Dec-2021.)
(𝐹𝑉 → (Slot 𝐴𝐹) = (𝐹𝐴))

Theorembj-evalid 33003 The evaluation at a set of the identity function is that set. (General form of ndxarg 15863.) The restriction to a set 𝑉 is necessary since the argument of the function Slot 𝐴 (like that of any function) has to be a set for the evaluation to be meaningful. (Contributed by BJ, 27-Dec-2021.)
((𝑉𝑊𝐴𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴)

Theorembj-ndxarg 33004 Proof of ndxarg 15863 from bj-evalid 33003. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸‘ndx) = 𝑁

Theorembj-ndxid 33005 Proof of ndxid 15864 from ndxarg 15863. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       𝐸 = Slot (𝐸‘ndx)

Theorembj-evalidval 33006 Closed general form of strndxid 15866. Both sides are equal to (𝐹𝐴) by bj-evalid 33003 and bj-evalval 33002 respectively, but bj-evalidval 33006 adds something to bj-evalid 33003 and bj-evalval 33002 in that Slot 𝐴 appears on both sides. (Contributed by BJ, 27-Dec-2021.)
((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (Slot 𝐴𝐹))

20.14.5.16  Elementwise operations

Syntaxcelmtwise 33007 Syntax for elementwise operations.
class elmtwise

Definitiondf-elmtwise 33008* Define the elementwise operation associated with a given operation. For instance, + is the addition of complex numbers (axaddf 9951), so if 𝐴 and 𝐵 are sets of complex numbers, then (𝐴(elmtwise‘ + )𝐵) is the set of numbers of the form (𝑥 + 𝑦) with 𝑥𝐴 and 𝑦𝐵. The set of odd natural numbers is (({2}(elmtwise‘ · )ℕ0)(elmtwise‘ + ){1}), or less formally 2ℕ0 + 1. (Contributed by BJ, 22-Dec-2021.)
elmtwise = (𝑜 ∈ V ↦ (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∣ ∃𝑢𝑥𝑣𝑦 𝑧 = (𝑢𝑜𝑣)}))

20.14.5.17  Elementwise intersection (families of sets induced on a subset)

Many kinds of structures are given by families of subsets of a given set: Moore collections (df-mre 16227), topologies (df-top 20680), pi-systems, rings of sets, delta-rings, lambda-systems/Dynkin systems, algebras/fields of sets, sigma-algebras/sigma-fields/tribes (df-siga 30145), sigma rings, monotone classes, matroids/independent sets, bornologies, filters.

There is a natural notion of structure induced on a subset. It is often given by an elementwise intersection, namely, the family of intersections of sets in the original family with the given subset. In this subsection, we define this notion and prove its main properties. Classical conditions on families of subsets include being nonempty, containing the whole set, containing the empty set, being stable under unions, intersections, subsets, supersets, (relative) complements. Therefore, we prove related properties for the elementwise intersection.

We will call (𝑋t 𝐴) the elementwise intersection on the family 𝑋 by the class 𝐴.

REMARK: many theorems are already in set.mm ; MM>search *rest* /J

Theorembj-rest00 33009 An elementwise intersection on the empty family is the empty set. TODO: this is 0rest 16071. (Contributed by BJ, 27-Apr-2021.)
(∅ ↾t 𝐴) = ∅

Theorembj-restsn 33010 An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 33013 and bj-restsnid 33015. (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌𝐴)})

Theorembj-restsnss 33011 Special case of bj-restsn 33010. (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑌) → ({𝑌} ↾t 𝐴) = {𝐴})

Theorembj-restsnss2 33012 Special case of bj-restsn 33010. (Contributed by BJ, 27-Apr-2021.)
((𝐴𝑉𝑌𝐴) → ({𝑌} ↾t 𝐴) = {𝑌})

Theorembj-restsn0 33013 An elementwise intersection on the singleton on the empty set is the singleton on the empty set. Special case of bj-restsn 33010 and bj-restsnss2 33012. TODO: this is restsn 20955. (Contributed by BJ, 27-Apr-2021.)
(𝐴𝑉 → ({∅} ↾t 𝐴) = {∅})

Theorembj-restsn10 33014 Special case of bj-restsn 33010, bj-restsnss 33011, and bj-rest10 33016. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → ({𝑋} ↾t ∅) = {∅})

Theorembj-restsnid 33015 The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 33010 and bj-restsnss 33011. (Contributed by BJ, 27-Apr-2021.)
({𝐴} ↾t 𝐴) = {𝐴}

Theorembj-rest10 33016 An elementwise intersection on a nonempty family by the empty set is the singleton on the empty set. TODO: this generalizes rest0 20954 and could replace it. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → (𝑋 ≠ ∅ → (𝑋t ∅) = {∅}))

Theorembj-rest10b 33017 Alternate version of bj-rest10 33016. (Contributed by BJ, 27-Apr-2021.)
(𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋t ∅) = {∅})

Theorembj-restn0 33018 An elementwise intersection on a nonempty family is nonempty. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (𝑋 ≠ ∅ → (𝑋t 𝐴) ≠ ∅))

Theorembj-restn0b 33019 Alternate version of bj-restn0 33018. (Contributed by BJ, 27-Apr-2021.)
((𝑋 ∈ (𝑉 ∖ {∅}) ∧ 𝐴𝑊) → (𝑋t 𝐴) ≠ ∅)

Theorembj-restpw 33020 The elementwise intersection on a powerset is the powerset of the intersection. This allows to prove for instance that the topology induced on a subset by the discrete topology is the discrete topology on that subset. See also restdis 20963 (which uses distop 20780 and restopn2 20962). (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑊) → (𝒫 𝑌t 𝐴) = 𝒫 (𝑌𝐴))

Theorembj-rest0 33021 An elementwise intersection on a family containing the empty set contains the empty set. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋t 𝐴)))

Theorembj-restb 33022 An elementwise intersection by a set on a family containing a superset of that set contains that set. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → ((𝐴𝐵𝐵𝑋) → 𝐴 ∈ (𝑋t 𝐴)))

Theorembj-restv 33023 An elementwise intersection by a subset on a family containing the whole set contains the whole subset. (Contributed by BJ, 27-Apr-2021.)
((𝐴 𝑋 𝑋𝑋) → 𝐴 ∈ (𝑋t 𝐴))

Theorembj-resta 33024 An elementwise intersection by a set on a family containing that set contains that set. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → (𝐴𝑋𝐴 ∈ (𝑋t 𝐴)))

Theorembj-restuni 33025 The union of an elementwise intersection by a set is equal to the intersection with that set of the union of the family. See also restuni 20947 and restuni2 20952. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (𝑋t 𝐴) = ( 𝑋𝐴))

Theorembj-restuni2 33026 The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 20947 and restuni2 20952. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴 𝑋) → (𝑋t 𝐴) = 𝐴)

Theorembj-restreg 33027 A reformulation of the axiom of regularity using elementwise intersection. (RK: might have to be placed later since theorems in this section are to be moved early (in the section related to the algebra of sets).) (Contributed by BJ, 27-Apr-2021.)
((𝐴𝑉𝐴 ≠ ∅) → ∅ ∈ (𝐴t 𝐴))

20.14.5.18  Moore collections (complements)

Theorembj-intss 33028 A nonempty intersection of a family of subsets of a class is included in that class. (Contributed by BJ, 7-Dec-2021.)
(𝐴 ⊆ 𝒫 𝑋 → (𝐴 ≠ ∅ → 𝐴𝑋))

Theorembj-raldifsn 33029* All elements in a set satisfy a given property if and only if all but one satisfy that property and that one also does. Typically, this can be used for characterizations that are proved using different methods for a given element and for all others, for instance zero and nonzero numbers, or the empty set and nonempty sets. (Contributed by BJ, 7-Dec-2021.)
(𝑥 = 𝐵 → (𝜑𝜓))       (𝐵𝐴 → (∀𝑥𝐴 𝜑 ↔ (∀𝑥 ∈ (𝐴 ∖ {𝐵})𝜑𝜓)))

Theorembj-0int 33030* If 𝐴 is a collection of subsets of 𝑋, like a topology, two equivalent ways to say that arbitrary intersections of elements of 𝐴 relative to 𝑋 belong to some class 𝐵 (in typical applications, 𝐴 itself). (Contributed by BJ, 7-Dec-2021.)
(𝐴 ⊆ 𝒫 𝑋 → ((𝑋𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅}) 𝑥𝐵) ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑋 𝑥) ∈ 𝐵))

Theorembj-mooreset 33031* A Moore collection is a set. That is, if we define a "Moore predicate" by (Moore𝐴 ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴), then any class satisfying that predicate is actually a set. Therefore, the definition df-bj-moore 33033 is sufficient. Note that the closed sets of a topology form a Moore collection, so this remark also applies to topologies and many other families of sets (namely, as soon as the whole set is required to be a closed set, as can be seen from the proof, which relies crucially on uniexr 6957).

Note: if, in the above predicate, we substitute 𝒫 𝑋 for 𝐴, then the last ∈ 𝒫 𝑋 could be weakened to ⊆ 𝒫 𝑋, and then the predicate would be obviously satisfied since 𝒫 𝑋 = 𝑋, making 𝒫 𝑋 a Moore collection in this weaker sense, even if 𝑋 is a proper class, but the addition of this single case does not add anything interesting. (Contributed by BJ, 8-Dec-2021.)

(∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴𝐴 ∈ V)

Syntaxcmoore 33032 Syntax for the class of Moore collections.
class Moore

Definitiondf-bj-moore 33033* Define the class of Moore collections. This is to df-mre 16227 what df-top 20680 is to df-topon 20697. For the sake of consistency, the function defined at df-mre 16227 should be denoted by "MooreOn".

Note: df-mre 16227 singles out the empty intersection. This is not necessary. It could be written instead Moore = (𝑥 ∈ V ↦ {𝑦 ∈ 𝒫 𝒫 𝑥 ∣ ∀𝑧 ∈ 𝒫 𝑦(𝑥 𝑧) ∈ 𝑦}) and the equivalence of both definitions is proved by bj-0int 33030.

There is no added generality in defining a "Moore predicate" for arbitrary classes, since a Moore class satisfying such a predicate is automatically a set (see bj-mooreset 33031). (Contributed by BJ, 27-Apr-2021.)

Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥( 𝑥 𝑦) ∈ 𝑥}

Theorembj-ismoore 33034* Characterization of Moore collections among sets. (Contributed by BJ, 9-Dec-2021.)
(𝐴𝑉 → (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))

Theorembj-ismoorec 33035* Characterization of Moore collections. (Contributed by BJ, 9-Dec-2021.)
(𝐴Moore ↔ (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))

Theorembj-ismoored0 33036 Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝐴Moore 𝐴𝐴)

Theorembj-ismoored 33037 Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝜑𝐴Moore)    &   (𝜑𝐵𝐴)       (𝜑 → ( 𝐴 𝐵) ∈ 𝐴)

Theorembj-ismoored2 33038 Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝜑𝐴Moore)    &   (𝜑𝐵𝐴)    &   (𝜑𝐵 ≠ ∅)       (𝜑 𝐵𝐴)

Theorembj-ismooredr 33039* Sufficient condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → ( 𝐴 𝑥) ∈ 𝐴)       (𝜑𝐴Moore)

Theorembj-ismooredr2 33040* Sufficient condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝜑𝐴𝑉)    &   (𝜑 𝐴𝐴)    &   (((𝜑𝑥𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐴)       (𝜑𝐴Moore)

Theorembj-discrmoore 33041 The discrete Moore collection on a set. (Contributed by BJ, 9-Dec-2021.)
(𝐴 ∈ V ↔ 𝒫 𝐴Moore)

Theorembj-0nmoore 33042 The empty set is not a Moore collection. (Contributed by BJ, 9-Dec-2021.)
¬ ∅ ∈ Moore

Theorembj-snmoore 33043 A singleton is a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝐴 ∈ V ↔ {𝐴} ∈ Moore)

20.14.5.19  Maps-to notation for functions with three arguments

Theorembj-0nelmpt 33044 The empty set is not an element of a function (given in maps-to notation). (Contributed by BJ, 30-Dec-2020.)
¬ ∅ ∈ (𝑥𝐴𝐵)

Theorembj-mptval 33045 Value of a function given in maps-to notation. (Contributed by BJ, 30-Dec-2020.)
𝑥𝐴       (∀𝑥𝐴 𝐵𝑉 → (𝑋𝐴 → (((𝑥𝐴𝐵)‘𝑋) = 𝑌𝑋(𝑥𝐴𝐵)𝑌)))

Theorembj-dfmpt2a 33046* An equivalent definition of df-mpt2 6640. (Contributed by BJ, 30-Dec-2020.)
(𝑥𝐴, 𝑦𝐵𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}

Theorembj-mpt2mptALT 33047* Alternate proof of mpt2mpt 6737. (Contributed by BJ, 30-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)       (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)

Syntaxcmpt3 33048 Extend the definition of a class to include maps-to notation for functions with three arguments.
class (𝑥𝐴, 𝑦𝐵, 𝑧𝐶𝐷)

Definitiondf-bj-mpt3 33049* Define maps-to notation for functions with three arguments. See df-mpt 4721 and df-mpt2 6640 for functions with one and two arguments respectively. This definition is analogous to bj-dfmpt2a 33046. (Contributed by BJ, 11-Apr-2020.)
(𝑥𝐴, 𝑦𝐵, 𝑧𝐶𝐷) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝑠 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷)}

20.14.5.20  Currying

Currying and uncurrying. See also df-cur and df-unc 7379. Contrary to these, the definitions in this section are parameterized.

Syntaxcfset 33050 Notation for the set of functions between two sets.
class Set

Definitiondf-bj-fset 33051* Define the set of functions between two sets. Same as df-map 7844 with arguments swapped. TODO: prove the same staple lemmas as for 𝑚. (Contributed by BJ, 11-Apr-2020.)
Set⟶ = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑥𝑦})

Syntaxccur- 33052 Extend class notation to include the parameterized currying function.
class curry_

Definitiondf-bj-cur 33053* Define currying. See also df-cur 7378. (Contributed by BJ, 11-Apr-2020.)
curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set𝑧) ↦ (𝑎𝑥 ↦ (𝑏𝑦 ↦ (𝑓‘⟨𝑎, 𝑏⟩)))))

Syntaxcunc- 33054 Extend class notation to include the parameterized uncurrying function.
class uncurry_

Definitiondf-bj-unc 33055* Define uncurrying. See also df-unc 7379. (Contributed by BJ, 11-Apr-2020.)
uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set𝑧)) ↦ (𝑎𝑥, 𝑏𝑦 ↦ ((𝑓𝑎)‘𝑏))))

20.14.6  Extended real and complex numbers, real and complex projective lines

In this section, we indroduce several supersets of the set of real numbers and the set of complex numbers.

Once they are given their usual topologies, which are locally compact, both topological spaces have a one-point compactification. They are denoted by ℝ̂ and ℂ̂ respectively, defined in df-bj-cchat 33091 and df-bj-rrhat 33093, and the point at infinity is denoted by , defined in df-bj-infty 33089.

Both and also have "directional compactifications", denoted respectively by ℝ̅, defined in df-bj-rrbar 33087 (already defined as *, see df-xr 10063) and ℂ̅, defined in df-bj-ccbar 33074.

Since ℂ̅ does not seem to be standard, we describe it in some detail. It is obtained by adding to a "point at infinity at the end of each ray with origin at 0". Although ℂ̅ is not an important object in itself, the motivation for introducing it is to provide a common superset to both ℝ̅ and and to define algebraic operations (addition, opposite, multiplication, inverse) as widely as reasonably possible.

Mathematically, ℂ̅ is the quotient of ((ℂ × ℝ≥0) ∖ {⟨0, 0⟩}) by the diagonal multiplicative action of >0 (think of the closed "northern hemisphere" in ^3 identified with (ℂ × ℝ), that each open ray from 0 included in the closed northern half-space intersects exactly once).

Since in set.mm, we want to have a genuine inclusion ℂ ⊆ ℂ̅, we instead define ℂ̅ as the (disjoint) union of with a circle at infinity denoted by . To have a genuine inclusion ℝ̅ ⊆ ℂ̅, we define +∞ and -∞ as certain points in .

Thanks to this framework, one has the genuine inclusions ℝ ⊆ ℝ̅ and ℝ ⊆ ℝ̂ and similarly ℂ ⊆ ℂ̅ and ℂ ⊆ ℂ̂. Furthermore, one has ℝ ⊆ ℂ as well as ℝ̅ ⊆ ℂ̅ and ℝ̂ ⊆ ℂ̂.

Furthermore, we define the main algebraic operations on (ℂ̅ ∪ ℂ̂), which is not very mathematical, but "overloads" the operations, so that one can use the same notation in all cases.

20.14.6.1  Diagonal in a Cartesian square

Complements on the idendity relation and definition of the diagonal in the Cartesian square of a set.

Theorembj-elid 33056 Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
(𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))

Theorembj-elid2 33057 Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
(𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))

Theorembj-elid3 33058 Characterization of the elements of I. (Contributed by BJ, 29-Mar-2020.)
(⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Syntaxcdiag2 33059 Syntax for the diagonal of the Cartesian square of a set.
class Diag

Definitiondf-bj-diag 33060 Define the diagonal of the Cartesian square of a set. (Contributed by BJ, 22-Jun-2019.)
Diag = (𝑥 ∈ V ↦ ( I ∩ (𝑥 × 𝑥)))

Theorembj-diagval 33061 Value of the diagonal. (Contributed by BJ, 22-Jun-2019.)
(𝐴𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))

Theorembj-eldiag 33062 Characterization of the elements of the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
(𝐴𝑉 → (𝐵 ∈ (Diag‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵))))

Theorembj-eldiag2 33063 Characterization of the elements of the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
(𝐴𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Diag‘𝐴) ↔ (𝐵𝐴𝐵 = 𝐶)))

20.14.6.2  Extended numbers and projective lines as sets

TODO(?): replace df-bj-inftyexpi 33065 with a function inftyexpi2pi defined on (0[,)1) since we plan to put this section as early as possible, before the definition of π. It would be best to use df-0r 9867 and df-1r 9868 but intervals are defined for real numbers, and not these temporary reals.

It looks like to define the sets, the addition and the opposite, one only needs some basic results about addition, opposite and ordering, which could use df-plr 9864, df-ltr 9866, df-0r 9867, df-1r 9868, df-ltr 9866. The idea is then to define the order relation directly on ℝ̅, skipping .

Syntaxcinftyexpi 33064 Syntax for the function inftyexpi parameterizing .
class inftyexpi

Definitiondf-bj-inftyexpi 33065 Definition of the auxiliary function inftyexpi parameterizing the circle at infinity in ℂ̅. We use coupling with to simplify the proof of bj-ccinftydisj 33071. It could seem more natural to define inftyexpi on all of using prcpal but we want to use only basic functions in the definition of ℂ̅. (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)

Theorembj-inftyexpiinv 33066 Utility theorem for the inverse of inftyexpi. (Contributed by BJ, 22-Jun-2019.) This utility theorem is irrelevant and should generally not be used. (New usage is discouraged.)
(𝐴 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐴)) = 𝐴)

Theorembj-inftyexpiinj 33067 Injectivity of the parameterization inftyexpi. Remark: a more conceptual proof would use bj-inftyexpiinv 33066 and the fact that a function with a retraction is injective. (Contributed by BJ, 22-Jun-2019.)
((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵)))

Theorembj-inftyexpidisj 33068 An element of the circle at infinity is not a complex number. (Contributed by BJ, 22-Jun-2019.) This utility theorem is irrelevant and should generally not be used. (New usage is discouraged.)
¬ (inftyexpi ‘𝐴) ∈ ℂ

Syntaxcccinfty 33069 Syntax for the circle at infinity .
class

Definitiondf-bj-ccinfty 33070 Definition of the circle at infinity . (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
= ran inftyexpi

Theorembj-ccinftydisj 33071 The circle at infinity is disjoint from the set of complex numbers. (Contributed by BJ, 22-Jun-2019.)
(ℂ ∩ ℂ) = ∅

Theorembj-elccinfty 33072 A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
(𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) ∈ ℂ)

Syntaxcccbar 33073 Syntax for the set of extended complex numbers ℂ̅.
class ℂ̅

Definitiondf-bj-ccbar 33074 Definition of the set of extended complex numbers ℂ̅. (Contributed by BJ, 22-Jun-2019.)
ℂ̅ = (ℂ ∪ ℂ)

Theorembj-ccssccbar 33075 Complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
ℂ ⊆ ℂ̅

Theorembj-ccinftyssccbar 33076 Infinite extended complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
⊆ ℂ̅

Syntaxcpinfty 33077 Syntax for +∞.
class +∞

Definitiondf-bj-pinfty 33078 Definition of +∞. (Contributed by BJ, 27-Jun-2019.)
+∞ = (inftyexpi ‘0)

Theorembj-pinftyccb 33079 The class +∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
+∞ ∈ ℂ̅

Theorembj-pinftynrr 33080 The extended complex number +∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
¬ +∞ ∈ ℂ

Syntaxcminfty 33081 Syntax for -∞.
class -∞

Definitiondf-bj-minfty 33082 Definition of -∞. (Contributed by BJ, 27-Jun-2019.)
-∞ = (inftyexpi ‘π)

Theorembj-minftyccb 33083 The class -∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
-∞ ∈ ℂ̅

Theorembj-minftynrr 33084 The extended complex number -∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
¬ -∞ ∈ ℂ

Theorembj-pinftynminfty 33085 The extended complex numbers +∞ and -∞ are different. (Contributed by BJ, 27-Jun-2019.)
+∞ ≠ -∞

Syntaxcrrbar 33086 Syntax for the set of extended real numbers ℝ̅.
class ℝ̅

Definitiondf-bj-rrbar 33087 Definition of the set of extended real numbers ℝ̅. See df-xr 10063. (Contributed by BJ, 29-Jun-2019.)
ℝ̅ = (ℝ ∪ {-∞, +∞})

Syntaxcinfty 33088 Syntax for .
class

Definitiondf-bj-infty 33089 Definition of , the point at infinity of the real or complex projective line. (Contributed by BJ, 27-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
∞ = 𝒫

Syntaxccchat 33090 Syntax for ℂ̂.
class ℂ̂

Definitiondf-bj-cchat 33091 Define the complex projective line, or Riemann sphere. (Contributed by BJ, 27-Jun-2019.)
ℂ̂ = (ℂ ∪ {∞})

Syntaxcrrhat 33092 Syntax for ℝ̂.
class ℝ̂

Definitiondf-bj-rrhat 33093 Define the real projective line. (Contributed by BJ, 27-Jun-2019.)
ℝ̂ = (ℝ ∪ {∞})

Theorembj-rrhatsscchat 33094 The real projective line is included in the complex projective line. (Contributed by BJ, 27-Jun-2019.)
ℝ̂ ⊆ ℂ̂

We define the operations on the extended real and complex numbers and on the real and complex projective lines simultaneously, thus "overloading" the operations.

class +ℂ̅

Definitiondf-bj-addc 33096 Define the additions on the extended complex numbers (on the subset of (ℂ̅ × ℂ̅) where it makes sense) and on the complex projective line (Riemann sphere). (Contributed by BJ, 22-Jun-2019.)
+ℂ̅ = (𝑥 ∈ (((ℂ × ℂ̅) ∪ (ℂ̅ × ℂ)) ∪ ((ℂ̂ × ℂ̂) ∪ (Diag‘ℂ))) ↦ if(((1st𝑥) = ∞ ∨ (2nd𝑥) = ∞), ∞, if((1st𝑥) ∈ ℂ, if((2nd𝑥) ∈ ℂ, ((1st𝑥) + (2nd𝑥)), (2nd𝑥)), (1st𝑥))))

Syntaxcoppcc 33097 Syntax for the opposite of extended complex numbers.
class -ℂ̅

Definitiondf-bj-oppc 33098 Define the negation (operation givin the opposite) the set of extended complex numbers and the complex projective line (Riemann sphere). One could use the prcpal function in the infinite case, but we want to use only basic functions at this point. (Contributed by BJ, 22-Jun-2019.)
-ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = ∞, ∞, if(𝑥 ∈ ℂ, -𝑥, (inftyexpi ‘if(0 < (1st𝑥), ((1st𝑥) − π), ((1st𝑥) + π))))))

20.14.6.4  Argument, multiplication and inverse

Since one needs arguments in order to define multiplication in ℂ̅, it seems harder to put this at the very beginning of the introduction of complex numbers.

Syntaxcprcpal 33099 Syntax for the function prcpal.
class prcpal

Definitiondf-bj-prcpal 33100 Define the function prcpal. (Contributed by BJ, 22-Jun-2019.)
prcpal = (𝑥 ∈ ℝ ↦ ((𝑥 mod (2 · π)) − if((𝑥 mod (2 · π)) ≤ π, 0, (2 · π))))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
 Copyright terms: Public domain < Previous  Next >