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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 -∞
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