Home Metamath Proof ExplorerTheorem List (p. 337 of 437) < 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 (1-28347) Hilbert Space Explorer (28348-29872) Users' Mathboxes (29873-43639)

Theorem List for Metamath Proof Explorer - 33601-33700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-evalid 33601 The evaluation at a set of the identity function is that set. (General form of ndxarg 16280.) 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 33602 Proof of ndxarg 16280 from bj-evalid 33601. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸‘ndx) = 𝑁

Theorembj-ndxid 33603 Proof of ndxid 16281 from ndxarg 16280. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       𝐸 = Slot (𝐸‘ndx)

Theorembj-evalidval 33604 Closed general form of strndxid 16282. Both sides are equal to (𝐹𝐴) by bj-evalid 33601 and bj-evalval 33600 respectively, but bj-evalidval 33604 adds something to bj-evalid 33601 and bj-evalval 33600 in that Slot 𝐴 appears on both sides. (Contributed by BJ, 27-Dec-2021.)
((𝑉𝑊𝐴𝑉𝐹𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (Slot 𝐴𝐹))

20.14.5.17  Elementwise operations

Syntaxcelwise 33605 Syntax for elementwise operations.
class elwise

Definitiondf-elwise 33606* Define the elementwise operation associated with a given operation. For instance, + is the addition of complex numbers (axaddf 10302), so if 𝐴 and 𝐵 are sets of complex numbers, then (𝐴(elwise‘ + )𝐵) is the set of numbers of the form (𝑥 + 𝑦) with 𝑥𝐴 and 𝑦𝐵. The set of odd natural numbers is (({2}(elwise‘ · )ℕ0)(elwise‘ + ){1}), or less formally 2ℕ0 + 1. (Contributed by BJ, 22-Dec-2021.)
elwise = (𝑜 ∈ V ↦ (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∣ ∃𝑢𝑥𝑣𝑦 𝑧 = (𝑢𝑜𝑣)}))

20.14.5.18  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 16632), topologies (df-top 21106), pi-systems, rings of sets, delta-rings, lambda-systems/Dynkin systems, algebras/fields of sets, sigma-algebras/sigma-fields/tribes (df-siga 30769), 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* / JOIN

Theorembj-rest00 33607 An elementwise intersection on the empty family is the empty set. TODO: this is 0rest 16476. (Contributed by BJ, 27-Apr-2021.)
(∅ ↾t 𝐴) = ∅

Theorembj-restsn 33608 An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 33611 and bj-restsnid 33613. (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌𝐴)})

Theorembj-restsnss 33609 Special case of bj-restsn 33608. (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑌) → ({𝑌} ↾t 𝐴) = {𝐴})

Theorembj-restsnss2 33610 Special case of bj-restsn 33608. (Contributed by BJ, 27-Apr-2021.)
((𝐴𝑉𝑌𝐴) → ({𝑌} ↾t 𝐴) = {𝑌})

Theorembj-restsn0 33611 An elementwise intersection on the singleton on the empty set is the singleton on the empty set. Special case of bj-restsn 33608 and bj-restsnss2 33610. TODO: this is restsn 21382. (Contributed by BJ, 27-Apr-2021.)
(𝐴𝑉 → ({∅} ↾t 𝐴) = {∅})

Theorembj-restsn10 33612 Special case of bj-restsn 33608, bj-restsnss 33609, and bj-rest10 33614. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → ({𝑋} ↾t ∅) = {∅})

Theorembj-restsnid 33613 The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 33608 and bj-restsnss 33609. (Contributed by BJ, 27-Apr-2021.)
({𝐴} ↾t 𝐴) = {𝐴}

Theorembj-rest10 33614 An elementwise intersection on a nonempty family by the empty set is the singleton on the empty set. TODO: this generalizes rest0 21381 and could replace it. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → (𝑋 ≠ ∅ → (𝑋t ∅) = {∅}))

Theorembj-rest10b 33615 Alternate version of bj-rest10 33614. (Contributed by BJ, 27-Apr-2021.)
(𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋t ∅) = {∅})

Theorembj-restn0 33616 An elementwise intersection on a nonempty family is nonempty. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (𝑋 ≠ ∅ → (𝑋t 𝐴) ≠ ∅))

Theorembj-restn0b 33617 Alternate version of bj-restn0 33616. (Contributed by BJ, 27-Apr-2021.)
((𝑋 ∈ (𝑉 ∖ {∅}) ∧ 𝐴𝑊) → (𝑋t 𝐴) ≠ ∅)

Theorembj-restpw 33618 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 21390 (which uses distop 21207 and restopn2 21389). (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑊) → (𝒫 𝑌t 𝐴) = 𝒫 (𝑌𝐴))

Theorembj-rest0 33619 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 33620 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 33621 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 33622 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 33623 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 21374 and restuni2 21379. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (𝑋t 𝐴) = ( 𝑋𝐴))

Theorembj-restuni2 33624 The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 21374 and restuni2 21379. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴 𝑋) → (𝑋t 𝐴) = 𝐴)

Theorembj-restreg 33625 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))

20.14.5.19  Moore collections (complements)

Theorembj-intss 33626 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 33627* 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 33628* 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 33629* A Moore collection is a set. That is, if we define a "Moore class-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 33631 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 7249).

Note: if, in the above predicate, we substitute 𝒫 𝑋 for 𝐴, then the last ∈ 𝒫 𝑋 could be weakened to 𝑋, and then the predicate would be obviously satisfied since 𝒫 𝑋 = 𝑋 (unipw 5150) , 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. Instead, we have bj-discrmoore 33639. (Contributed by BJ, 8-Dec-2021.)

(∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴𝐴 ∈ V)

Syntaxcmoore 33630 Syntax for the class of Moore collections.
class Moore

Definitiondf-bj-moore 33631* Define the class of Moore collections. This is to df-mre 16632 what df-top 21106 is to df-topon 21123. For the sake of consistency, the function defined at df-mre 16632 should be denoted by "MooreOn".

Note: df-mre 16632 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 33628.

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 33629). (Contributed by BJ, 27-Apr-2021.)

Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥( 𝑥 𝑦) ∈ 𝑥}

Theorembj-ismoore 33632* Characterization of Moore collections among sets. (Contributed by BJ, 9-Dec-2021.)
(𝐴𝑉 → (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))

Theorembj-ismoorec 33633* Characterization of Moore collections. (Contributed by BJ, 9-Dec-2021.)
(𝐴Moore ↔ (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))

Theorembj-ismoored0 33634 Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝐴Moore 𝐴𝐴)

Theorembj-ismoored 33635 Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝜑𝐴Moore)    &   (𝜑𝐵𝐴)       (𝜑 → ( 𝐴 𝐵) ∈ 𝐴)

Theorembj-ismoored2 33636 Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝜑𝐴Moore)    &   (𝜑𝐵𝐴)    &   (𝜑𝐵 ≠ ∅)       (𝜑 𝐵𝐴)

Theorembj-ismooredr 33637* Sufficient condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → ( 𝐴 𝑥) ∈ 𝐴)       (𝜑𝐴Moore)

Theorembj-ismooredr2 33638* Sufficient condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝜑𝐴𝑉)    &   (𝜑 𝐴𝐴)    &   (((𝜑𝑥𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐴)       (𝜑𝐴Moore)

Theorembj-discrmoore 33639 The discrete Moore collection on a set. (Contributed by BJ, 9-Dec-2021.)
(𝐴 ∈ V ↔ 𝒫 𝐴Moore)

Theorembj-0nmoore 33640 The empty set is not a Moore collection. (Contributed by BJ, 9-Dec-2021.)
¬ ∅ ∈ Moore

Theorembj-snmoore 33641 A singleton is a Moore collection. (Contributed by BJ, 9-Dec-2021.)
(𝐴 ∈ V ↔ {𝐴} ∈ Moore)

20.14.5.20  Maps-to notation for functions with three arguments

Theorembj-0nelmpt 33642 The empty set is not an element of a function (given in maps-to notation). (Contributed by BJ, 30-Dec-2020.)
¬ ∅ ∈ (𝑥𝐴𝐵)

Theorembj-mptval 33643 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 33644* An equivalent definition of df-mpt2 6927. (Contributed by BJ, 30-Dec-2020.)
(𝑥𝐴, 𝑦𝐵𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}

Theorembj-mpt2mptALT 33645* Alternate proof of mpt2mpt 7029. (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 33646 Extend the definition of a class to include maps-to notation for functions with three arguments.
class (𝑥𝐴, 𝑦𝐵, 𝑧𝐶𝐷)

Definitiondf-bj-mpt3 33647* Define maps-to notation for functions with three arguments. See df-mpt 4966 and df-mpt2 6927 for functions with one and two arguments respectively. This definition is analogous to bj-dfmpt2a 33644. (Contributed by BJ, 11-Apr-2020.)
(𝑥𝐴, 𝑦𝐵, 𝑧𝐶𝐷) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝑠 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷)}

20.14.5.21  Currying

Currying and uncurrying. See also df-cur and df-unc 7676. Contrary to these, the definitions in this section are parameterized.

Syntaxcsethom 33648 Syntax for the set of set morphisms.
class Set

Definitiondf-bj-sethom 33649* Define the set of functions (morphisms of sets) between two sets. Same as df-map 8142 with arguments swapped. TODO: prove the same staple lemmas as for 𝑚.

Remark: one may define Set⟶ = (𝑥 ∈ dom Struct , 𝑦 ∈ dom Struct ↦ {𝑓𝑓:(Base‘𝑥)⟶(Base‘𝑦)}) so that for morphisms between other structures, one could write ... = {𝑓 ∈ (𝑥 Set𝑦) ∣ ...}.

(Contributed by BJ, 11-Apr-2020.)

Set⟶ = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑥𝑦})

Syntaxctophom 33650 Syntax for the set of topological morphisms.
class Top

Definitiondf-bj-tophom 33651* Define the set of continuous functions (morphisms of topological spaces) between two topological spaces. Similar to df-cn 21439 (which is in terms of topologies instead of topological spaces). (Contributed by BJ, 10-Feb-2022.)
Top⟶ = (𝑥 ∈ TopSp, 𝑦 ∈ TopSp ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (TopOpen‘𝑦)(𝑓𝑢) ∈ (TopOpen‘𝑥)})

Syntaxcmgmhom 33652 Syntax for the set of magma morphisms.
class Mgm

Definitiondf-bj-mgmhom 33653* Define the set of magma morphisms between two magmas. If domain and codomain are semigroups, monoids, or groups, then one obtains the set of morphisms of these structures. (Contributed by BJ, 10-Feb-2022.)
Mgm⟶ = (𝑥 ∈ Mgm, 𝑦 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (Base‘𝑥)∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+g𝑥)𝑣)) = ((𝑓𝑢)(+g𝑦)(𝑓𝑣))})

Syntaxctopmgmhom 33654 Syntax for the set of topological magma morphisms.
class TopMgm

Definitiondf-bj-topmgmhom 33655* Define the set of topological magma morphisms (continuous magma morphisms) between two topological magmas. If domain and codomain are topological semigroups, monoids, or groups, then one obtains the set of morphisms of these structures. This definition is currently stated with topological monoid domain and codomain, since topological magmas are currently not defined in set.mm. (Contributed by BJ, 10-Feb-2022.)
TopMgm⟶ = (𝑥 ∈ TopMnd, 𝑦 ∈ TopMnd ↦ ((𝑥 Top𝑦) ∩ (𝑥 Mgm𝑦)))

Syntaxccur- 33656 Syntax for the parameterized currying function.
class curry_

Definitiondf-bj-cur 33657* Define currying. See also df-cur 7675. (Contributed by BJ, 11-Apr-2020.)
curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set𝑧) ↦ (𝑎𝑥 ↦ (𝑏𝑦 ↦ (𝑓‘⟨𝑎, 𝑏⟩)))))

Syntaxcunc- 33658 Notation for the parameterized uncurrying function.
class uncurry_

Definitiondf-bj-unc 33659* Define uncurrying. See also df-unc 7676. (Contributed by BJ, 11-Apr-2020.)
uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set𝑧)) ↦ (𝑎𝑥, 𝑏𝑦 ↦ ((𝑓𝑎)‘𝑏))))

20.14.5.22  Setting components of extensible structures

Groundwork for changing the definition, syntax and token for component-setting in extensible structures. See https://github.com/metamath/set.mm/issues/2401

Syntaxcstrset 33660 Syntax for component-setting in extensible structures.
class [𝐵 / 𝐴]struct𝑆

Definitiondf-strset 33661 Component-setting in extensible structures. Define the extensible structure [𝐵 / 𝐴]struct𝑆, which is like the extensible structure 𝑆 except that the value 𝐵 has been put in the slot 𝐴 (replacing the current value if there was already one). In such expressions, 𝐴 is generally substituted for slot mnemonics like Base or +g or dist. (Contributed by BJ, 13-Feb-2022.)
[𝐵 / 𝐴]struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {⟨(𝐴‘ndx), 𝐵⟩})

Theoremsetsstrset 33662 Relation between df-sets 16262 and df-strset 33661. Temporary theorem kept during the transition from the former to the latter. (Contributed by BJ, 13-Feb-2022.)
((𝑆𝑉𝐵𝑊) → [𝐵 / 𝐴]struct𝑆 = (𝑆 sSet ⟨(𝐴‘ndx), 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 33710 and df-bj-rrhat 33712, and the point at infinity is denoted by , defined in df-bj-infty 33708.

Both and also have "directional compactifications", denoted respectively by ℝ̅, defined in df-bj-rrbar 33706 (already defined as *, see df-xr 10415) and ℂ̅, defined in df-bj-ccbar 33693.

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  Identity relation (complements)

Complements on the identity relation.

Theorembj-elid 33663 Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
(𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))

Theorembj-elid2 33664 Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
(𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))

Theorembj-elid3 33665 Characterization of the couples in I. (Contributed by BJ, 29-Mar-2020.)
(⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Theorembj-elid4 33666 Characterization of the couples in I whose first component is a setvar. (Contributed by BJ, 29-Mar-2020.)
(⟨𝑥, 𝐴⟩ ∈ I ↔ 𝑥 = 𝐴)

20.14.6.2  Diagonal in a Cartesian square

Definition of the diagonal in the Cartesian square of a set.

Syntaxcdiag2 33667 Syntax for the diagonal of the Cartesian square of a set.
class Diag

Definitiondf-bj-diag 33668 Define the diagonal of the Cartesian square of a set. (Contributed by BJ, 22-Jun-2019.)
Diag = (𝑥 ∈ V ↦ ( I ∩ (𝑥 × 𝑥)))

Theorembj-diagval 33669 Value of the diagonal. (Contributed by BJ, 22-Jun-2019.)
(𝐴𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))

Theorembj-eldiag 33670 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 33671 Characterization of the elements of the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
(𝐴𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Diag‘𝐴) ↔ (𝐵𝐴𝐵 = 𝐶)))

20.14.6.3  Extended numbers and projective lines as sets

We parameterize the set of infinite extended complex numbers (df-bj-ccinfty 33689) using the reals (df-r 10282) via the function +∞e. Since at that point, we have only defined the set of reals but no operations on it, we define a temporary "fractional part" function, which is more convenient to define on the temporary reals (df-nr 10213) since we use operations on the latter. We also define a temporary "one-half" in order to define minus infinity (df-bj-minfty 33701) and then we can define the sets of extended real and complex numbers and the projective real and complex line, as well as addition and negation and also the order on the extended reals (which bypasses the intermediate definition of a temporary order on the reals and then a superseding one on the extended reals).

Syntaxcfractemp 33672 Syntax for the fractional part of a tempopary real.
class {R

Definitiondf-bj-fractemp 33673* Temporary definition: fractional part of a temporary real.

To understand this definition, recall the canonical injection ω⟶R, 𝑛 ↦ [{𝑥Q𝑥 <Q ⟨suc 𝑛, 1o⟩}, 1P] ~R where we successively take the successor of 𝑛 to land in positive integers, then take the couple with 1o as second component to land in positive rationals, then take the Dedekind cut that positive rational forms, and finally take the equivalence class of the couple with 1P as second component. Adding one at the beginning and subtracting it at the end is necessary since the constructions used in set.mm use the positive integers, positive rationals, and positive reals as intermediate number systems. (Contributed by BJ, 22-Jan-2023.) The precise definition is irrelevant and should generally not be used. One could even inline it. The definitive fractional part of an extended or projective complex number will be defined later. (New usage is discouraged.)

{R = (𝑥R ↦ (𝑦R ((𝑦 = 0R ∨ (0R <R 𝑦𝑦 <R 1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧Q𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥)))

Syntaxcinftyexpitau 33674 Syntax for the function +∞e parameterizing .
class +∞e

Definitiondf-bj-inftyexpitau 33675 Definition of the auxiliary function +∞e parameterizing the circle at infinity in ℂ̅. We use coupling with {R} to simplify the proof of bj-inftyexpitaudisj 33682. (Contributed by BJ, 22-Jan-2023.) The precise definition is irrelevant and should generally not be used. TODO: prove only the necessary lemmas to prove (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((+∞e𝐴) = (+∞e𝐵) ↔ (𝐴𝐵) ∈ ℤ)). (New usage is discouraged.)
+∞e = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st𝑥)), {R}⟩)

SyntaxcccinftyN 33676 Syntax for the circle at infinity ∞N.
class ∞N

Definitiondf-bj-ccinftyN 33677 Definition of the circle at infinity ∞N. (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
∞N = ran +∞e

Theorembj-inftyexpitaufo 33678 The function +∞e written as a surjection with domain and range. (Contributed by BJ, 4-Feb-2023.)
+∞e:ℝ–onto→ℂ∞N

Syntaxchalf 33679 Syntax for the temporary one-half.
class 1/2

Definitiondf-bj-onehalf 33680 Define temporarily the real number "one-half". (Contributed by BJ, 4-Feb-2023.) Once the machinery is developed, the number "one-half" can be denoted by (1 / 2). (New usage is discouraged.)
1/2 = ⟨(𝑥R (𝑥 +R 𝑥) = 1R), 0R

Theorembj-nsnid 33681 A set does not contain the singleton formed on it. More precisely, one can prove that a class contains the singleton formed on it if and only if it is proper and contains the "singleton formed on it", which is the empty set: ¬ ({𝐴} ∈ 𝐴 ↔ (∅ ∈ 𝐴𝐴 ∈ V)). (Contributed by BJ, 4-Feb-2023.)
(𝐴𝑉 → ¬ {𝐴} ∈ 𝐴)

Theorembj-inftyexpitaudisj 33682 An element of the circle at infinity is not a complex number. (Contributed by BJ, 4-Feb-2023.)
¬ (+∞e𝐴) ∈ ℂ

Syntaxcinftyexpi 33683 Syntax for the function +∞ei parameterizing .
class +∞ei

Definitiondf-bj-inftyexpi 33684 Definition of the auxiliary function +∞ei parameterizing the circle at infinity in ℂ̅. We use coupling with to simplify the proof of bj-ccinftydisj 33690. It could seem more natural to define +∞ei on all of , but we want to use only basic functions in the definition of ℂ̅. TODO: transition to df-bj-inftyexpitau 33675 instead. (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
+∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)

Theorembj-inftyexpiinv 33685 Utility theorem for the inverse of +∞ei. (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 ‘(+∞ei𝐴)) = 𝐴)

Theorembj-inftyexpiinj 33686 Injectivity of the parameterization +∞ei. Remark: a more conceptual proof would use bj-inftyexpiinv 33685 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 ↔ (+∞ei𝐴) = (+∞ei𝐵)))

Theorembj-inftyexpidisj 33687 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.)
¬ (+∞ei𝐴) ∈ ℂ

Syntaxcccinfty 33688 Syntax for the circle at infinity .
class

Definitiondf-bj-ccinfty 33689 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 +∞ei

Theorembj-ccinftydisj 33690 The circle at infinity is disjoint from the set of complex numbers. (Contributed by BJ, 22-Jun-2019.)
(ℂ ∩ ℂ) = ∅

Theorembj-elccinfty 33691 A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
(𝐴 ∈ (-π(,]π) → (+∞ei𝐴) ∈ ℂ)

Syntaxcccbar 33692 Syntax for the set of extended complex numbers ℂ̅.
class ℂ̅

Definitiondf-bj-ccbar 33693 Definition of the set of extended complex numbers ℂ̅. (Contributed by BJ, 22-Jun-2019.)
ℂ̅ = (ℂ ∪ ℂ)

Theorembj-ccssccbar 33694 Complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
ℂ ⊆ ℂ̅

Theorembj-ccinftyssccbar 33695 Infinite extended complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
⊆ ℂ̅

Syntaxcpinfty 33696 Syntax for "plus infinity".
class +∞

Definitiondf-bj-pinfty 33697 Definition of "plus infinity". (Contributed by BJ, 27-Jun-2019.)
+∞ = (+∞ei‘0)

Theorembj-pinftyccb 33698 The class +∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
+∞ ∈ ℂ̅

Theorembj-pinftynrr 33699 The extended complex number +∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
¬ +∞ ∈ ℂ

Syntaxcminfty 33700 Syntax for "minus infinity".
class -∞

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 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43639
 Copyright terms: Public domain < Previous  Next >