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Theorem List for Metamath Proof Explorer - 32201-32300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-pr2ex 32201 Sethood of the second projection. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → pr2 𝐴 ∈ V)

Theorembj-2uplth 32202 The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 4871). (Contributed by BJ, 6-Oct-2018.)
(⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theorembj-2uplex 32203 A couple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Oct-2018.)
(⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))

Theorembj-2upln0 32204 A couple is nonempty. (Contributed by BJ, 21-Apr-2019.)
𝐴, 𝐵⦆ ≠ ∅

Theorembj-2upln1upl 32205 A couple is never equal to a monuple. It is in order to have this "non-clashing" result that tagging was used. Without tagging, we would have 𝐴, ∅⦆ = ⦅𝐴. Note that in the context of Morse tuples, it is natural to define the 0-tuple as the empty set. Therefore, the present theorem together with bj-1upln0 32190 and bj-2upln0 32204 tell us that an m-tuple may equal an n-tuple only when m = n, at least for m, n <= 2, but this result would extend as soon as we define n-tuples for higher values of n. (Contributed by BJ, 21-Apr-2019.)
𝐴, 𝐵⦆ ≠ ⦅𝐶

21.14.5.14  Set theory: miscellaneous

Miscellaneous theorems of set theory.

Theorembj-vjust2 32206 Justification theorem for bj-df-v 32207. See also vjust 3174 and bj-vjust 31974. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
{𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}

Theorembj-df-v 32207 Alternate definition of the universal class. Actually, the current definition df-v 3175 should be proved from this one, and vex 3176 should be proved from this proposed definition together with bj-vexwv 32051, which would remove from vex 3176 dependency on ax-13 2234 (see also comment of bj-vexw 32049). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
V = {𝑥 ∣ ⊤}

Theorembj-df-nul 32208 Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
∅ = {𝑥 ∣ ⊥}

Theorembj-nul 32209* Two formulations of the axiom of the empty set ax-nul 4717. Proposal: place it right before ax-nul 4717. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(∅ ∈ V ↔ ∃𝑥𝑦 ¬ 𝑦𝑥)

Theorembj-nuliota 32210* Definition of the empty set using the definite description binder. See also bj-nuliotaALT 32211. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)

Theorembj-nuliotaALT 32211* Alternate proof of bj-nuliota 32210. Note that this alternate proof uses the fact that 𝑥𝜑 evaluates to when there is no 𝑥 satisfying 𝜑 (iotanul 5783). This is an implementation detail of the encoding currently used in set.mm and should be avoided. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)

Theorembj-vtoclgfALT 32212 Alternate proof of vtoclgf 3237. Proof from vtoclgft 3227. (This may have been the original proof before shortening.) (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)

Theorembj-pwcfsdom 32213 Remove hypothesis from pwcfsdom 9284. Illustration of how to remove a "proof-facilitating hypothesis". (Can use it to shorten theorems using pwcfsdom 9284.) (Contributed by BJ, 14-Sep-2019.)
(ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))

Theorembj-grur1 32214 Remove hypothesis from grur1 9521. Illustration of how to remove a "definitional hypothesis". This makes its uses longer, but the theorem feels more self-contained. Looks preferable when the defined term appears only once in the conclusion. (Contributed by BJ, 14-Sep-2019.)
((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → 𝑈 = (𝑅1‘(𝑈 ∩ On)))

21.14.5.15  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 16069), topologies (df-top 20521), pi-systems, rings of sets, delta-rings, lambda-systems/Dynkin systems, algebras/fields of sets, sigma-algebras/sigma-fields/tribes (df-siga 29498), 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 32215 An elementwise intersection on the empty family is the empty set. TODO: this is 0rest 15913. (Contributed by BJ, 27-Apr-2021.)
(∅ ↾t 𝐴) = ∅

Theorembj-restsn 32216 An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 32219 and bj-restsnid 32221. (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌𝐴)})

Theorembj-restsnss 32217 Special case of bj-restsn 32216. (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑌) → ({𝑌} ↾t 𝐴) = {𝐴})

Theorembj-restsnss2 32218 Special case of bj-restsn 32216. (Contributed by BJ, 27-Apr-2021.)
((𝐴𝑉𝑌𝐴) → ({𝑌} ↾t 𝐴) = {𝑌})

Theorembj-restsn0 32219 An elementwise intersection on the singleton on the empty set is the singleton on the empty set. Special case of bj-restsn 32216 and bj-restsnss2 32218. TODO: this is restsn 20784. (Contributed by BJ, 27-Apr-2021.)
(𝐴𝑉 → ({∅} ↾t 𝐴) = {∅})

Theorembj-restsn10 32220 Special case of bj-restsn 32216, bj-restsnss 32217, and bj-rest10 32222. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → ({𝑋} ↾t ∅) = {∅})

Theorembj-restsnid 32221 The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 32216 and bj-restsnss 32217. (Contributed by BJ, 27-Apr-2021.)
({𝐴} ↾t 𝐴) = {𝐴}

Theorembj-rest10 32222 An elementwise intersection on a nonempty family by the empty set is the singleton on the empty set. TODO: this generalizes rest0 20783 and could replace it. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → (𝑋 ≠ ∅ → (𝑋t ∅) = {∅}))

Theorembj-rest10b 32223 Alternate version of bj-rest10 32222. (Contributed by BJ, 27-Apr-2021.)
(𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋t ∅) = {∅})

Theorembj-restn0 32224 An elementwise intersection on a nonempty family is nonempty. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (𝑋 ≠ ∅ → (𝑋t 𝐴) ≠ ∅))

Theorembj-restn0b 32225 Alternate version of bj-restn0 32224. (Contributed by BJ, 27-Apr-2021.)
((𝑋 ∈ (𝑉 ∖ {∅}) ∧ 𝐴𝑊) → (𝑋t 𝐴) ≠ ∅)

Theorembj-restpw 32226 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 20792 (which uses distop 20610 and restopn2 20791). (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑊) → (𝒫 𝑌t 𝐴) = 𝒫 (𝑌𝐴))

Theorembj-rest0 32227 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 32228 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 32229 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 32230 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 32231 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 20776 and restuni2 20781. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (𝑋t 𝐴) = ( 𝑋𝐴))

Theorembj-restuni2 32232 The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 20776 and restuni2 20781. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴 𝑋) → (𝑋t 𝐴) = 𝐴)

Theorembj-restreg 32233 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))

21.14.5.16  Topology (complements)

Theorembj-toptopon2 32234 A topology is the same thing as a topology on the union of its open sets. space. (Contributed by BJ, 27-Apr-2021.)
(𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))

Theorembj-topontopon 32235 A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
(𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘ 𝐽))

Theorembj-funtopon 32236 TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
Fun TopOn

Theorembj-elpw3 32237 A variant of elpwg 4116. (Contributed by BJ, 29-Apr-2021.)
((𝐴 ∈ V ∧ 𝐴𝐵) ↔ 𝐴 ∈ 𝒫 𝐵)

Theorembj-sspwpw 32238 The union of a set is included in a given class if and only if that set is an element of the double power class of that class. (Contributed by BJ, 29-Apr-2021.)
((𝐴 ∈ V ∧ 𝐴𝐵) ↔ 𝐴 ∈ 𝒫 𝒫 𝐵)

Theorembj-sspwpwab 32239* The class of families whose union is included in a given class is equal to the double power class of that class. (Contributed by BJ, 29-Apr-2021.)
{𝑥 𝑥𝐴} = 𝒫 𝒫 𝐴

Theorembj-sspwpweq 32240* The class of families whose union is equal to a given class is included in the double power class of that class. (Contributed by BJ, 29-Apr-2021.)
{𝑥 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴

Theorembj-toponss 32241 The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.)
(TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴

Theorembj-dmtopon 32242 The domain of TopOn is V. (Contributed by BJ, 29-Apr-2021.)
dom TopOn = V

Theorembj-fntopon 32243 TopOn is a function with domain V. Analogue of fnmre 16074. (Contributed by BJ, 29-Apr-2021.)
TopOn Fn V

Theorembj-toprntopon 32244 A topology is the same thing as a topology on a set (variable-free version). (Contributed by BJ, 27-Apr-2021.)
Top = ran TopOn

Theorembj-xnex 32245* Lemma for snnex 6862 and bj-pwnex 32246. (Contributed by BJ, 2-May-2021.)
(∀𝑦(𝐴𝑉𝑦𝐴) → {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∉ V)

Theorembj-pwnex 32246* The class of all power sets is a proper class. See also snnex 6862. (Contributed by BJ, 2-May-2021.)
{𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V

Theorembj-topnex 32247 The class of all topologies is a proper class. (Contributed by BJ, 2-May-2021.)
Top ∉ V

Syntaxcmoo 32248 Syntax for the class of Moore collections.
class Moore

Definitiondf-bj-mre 32249* Define the class of Moore collections. This is to df-mre 16069 what df-top 20521 is to df-topon 20523.

Note: df-mre 16069 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})

Remark: as usual, if one wanted a more general definition, one could define a new syntax and (Moore𝐴 ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴). (Contributed by BJ, 27-Apr-2021.)

Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥( 𝑥 𝑦) ∈ 𝑥}

21.14.5.17  Maps-to notation for functions with three arguments

Theorembj-0nelmpt 32250 The empty set is not an element of a function (given in maps-to notation). (Contributed by BJ, 30-Dec-2020.)
¬ ∅ ∈ (𝑥𝐴𝐵)

Theorembj-mptval 32251 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 32252* An equivalent definition of df-mpt2 6554. (Contributed by BJ, 30-Dec-2020.)
(𝑥𝐴, 𝑦𝐵𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}

Theorembj-mpt2mptALT 32253* Alternate proof of mpt2mpt 6650. (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 32254 Extend the definition of a class to include maps-to notation for functions with three arguments.
class (𝑥𝐴, 𝑦𝐵, 𝑧𝐶𝐷)

Definitiondf-bj-mpt3 32255* Define maps-to notation for functions with three arguments. See df-mpt 4645 and df-mpt2 6554 for functions with one and two arguments respectively. This definition is analogous to bj-dfmpt2a 32252. (Contributed by BJ, 11-Apr-2020.)
(𝑥𝐴, 𝑦𝐵, 𝑧𝐶𝐷) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝑠 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷)}

21.14.5.18  Currying

Currying and uncurrying. See also df-cur and df-unc 7281. Contrary to these, the definitions in this section are parameterized.

Syntaxcfset 32256 Notation for the set of functions between two sets.
class Set

Definitiondf-bj-fset 32257* Define the set of functions between two sets. Same as df-map 7746 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- 32258 Extend class notation to include the parameterized currying function.
class curry_

Syntaxcunc- 32259 Extend class notation to include the parameterized uncurrying function.
class uncurry_

Definitiondf-bj-cur 32260* Define currying. See also df-cur 7280. (Contributed by BJ, 11-Apr-2020.)
curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set𝑧) ↦ (𝑎𝑥 ↦ (𝑏𝑦 ↦ (𝑓‘⟨𝑎, 𝑏⟩)))))

Definitiondf-bj-unc 32261* Define uncurrying. See also df-unc 7281. (Contributed by BJ, 11-Apr-2020.)
uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set𝑧)) ↦ (𝑎𝑥, 𝑏𝑦 ↦ ((𝑓𝑎)‘𝑏))))

21.14.6  Extended real and complex numbers, real and complex projectives 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 32297 and df-bj-rrhat 32299, and the point at infinity is denoted by , defined in df-bj-infty 32295.

Both and also have "directional compactifications", denoted respectively by ℝ̅, defined in df-bj-rrbar 32293 (already defined as *, see df-xr 9957) and ℂ̅, defined in df-bj-ccbar 32280.

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.

21.14.6.1  Diagonal in a Cartesian square

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

Theorembj-elid 32262 Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
(𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))

Theorembj-elid2 32263 Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
(𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))

Theorembj-elid3 32264 Characterization of the elements of I. (Contributed by BJ, 29-Mar-2020.)
(⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Syntaxcdiag2 32265 Syntax for the diagonal of the Cartesian square of a set.
class Diag

Definitiondf-bj-diag 32266 Define the diagonal of the Cartesian square of a set. (Contributed by BJ, 22-Jun-2019.)
Diag = (𝑥 ∈ V ↦ ( I ∩ (𝑥 × 𝑥)))

Theorembj-diagval 32267 Value of the diagonal. (Contributed by BJ, 22-Jun-2019.)
(𝐴𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))

Theorembj-eldiag 32268 Characterization of the elements the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
(𝐴𝑉 → (𝐵 ∈ (Diag‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵))))

Theorembj-eldiag2 32269 Characterization of the elements the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
(𝐴𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Diag‘𝐴) ↔ (𝐵𝐴𝐵 = 𝐶)))

21.14.6.2  Extended numbers and projective lines as sets

TODO(?): replace df-bj-inftyexpi 32271 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 9761 and df-1r 9762 but intervals are defined for real numers, 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 9758, df-ltr 9760, df-0r 9761, df-1r 9762, df-ltr 9760. The idea is then to define the order relation directly on ℝ̅, skipping .

Syntaxcinftyexpi 32270 Syntax for the function inftyexpi parameterizing .
class inftyexpi

Definitiondf-bj-inftyexpi 32271 Definition of the auxiliary function inftyexpi parameterizing the circle at infinity in ℂ̅. We use coupling with to simplify the proof of bj-ccinftydisj 32277. 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 32272 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 32273 Injectivity of the parameterization inftyexpi. Remark: a more conceptual proof would use bj-inftyexpiinv 32272 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 32274 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 32275 Syntax for the circle at infinity .
class

Definitiondf-bj-ccinfty 32276 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 32277 The circle at infinity is disjoint from the set of complex numbers. (Contributed by BJ, 22-Jun-2019.)
(ℂ ∩ ℂ) = ∅

Theorembj-elccinfty 32278 A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
(𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) ∈ ℂ)

Syntaxcccbar 32279 Syntax for the set of extended complex numbers ℂ̅.
class ℂ̅

Definitiondf-bj-ccbar 32280 Definition of the set of extended complex numbers ℂ̅. (Contributed by BJ, 22-Jun-2019.)
ℂ̅ = (ℂ ∪ ℂ)

Theorembj-ccssccbar 32281 Complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
ℂ ⊆ ℂ̅

Theorembj-ccinftyssccbar 32282 Infinite extended complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
⊆ ℂ̅

Syntaxcpinfty 32283 Syntax for +∞.
class +∞

Definitiondf-bj-pinfty 32284 Definition of +∞. (Contributed by BJ, 27-Jun-2019.)
+∞ = (inftyexpi ‘0)

Theorembj-pinftyccb 32285 The class +∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
+∞ ∈ ℂ̅

Theorembj-pinftynrr 32286 The extended complex number +∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
¬ +∞ ∈ ℂ

Syntaxcminfty 32287 Syntax for -∞.
class -∞

Definitiondf-bj-minfty 32288 Definition of -∞. (Contributed by BJ, 27-Jun-2019.)
-∞ = (inftyexpi ‘π)

Theorembj-minftyccb 32289 The class -∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
-∞ ∈ ℂ̅

Theorembj-minftynrr 32290 The extended complex number -∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
¬ -∞ ∈ ℂ

Theorembj-pinftynminfty 32291 The extended complex numbers +∞ and -∞ are different. (Contributed by BJ, 27-Jun-2019.)
+∞ ≠ -∞

Syntaxcrrbar 32292 Syntax for the set of extended real numbers ℝ̅.
class ℝ̅

Definitiondf-bj-rrbar 32293 Definition of the set of extended real numbers ℝ̅. See df-xr 9957. (Contributed by BJ, 29-Jun-2019.)
ℝ̅ = (ℝ ∪ {-∞, +∞})

Syntaxcinfty 32294 Syntax for .
class

Definitiondf-bj-infty 32295 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 32296 Syntax for ℂ̂.
class ℂ̂

Definitiondf-bj-cchat 32297 Define the complex projective line, or Riemann sphere. (Contributed by BJ, 27-Jun-2019.)
ℂ̂ = (ℂ ∪ {∞})

Syntaxcrrhat 32298 Syntax for ℝ̂.
class ℝ̂

Definitiondf-bj-rrhat 32299 Define the real projective line. (Contributed by BJ, 27-Jun-2019.)
ℝ̂ = (ℝ ∪ {∞})

Theorembj-rrhatsscchat 32300 The real projective line is included in the complex projective line. (Contributed by BJ, 27-Jun-2019.)
ℝ̂ ⊆ ℂ̂

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