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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
 
20.14.6.3  Addition and opposite

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

 
Syntaxcaddcc 33095 Syntax for the addition of extended complex numbers.
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))))
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