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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | ctopmgmhom 36601 | Syntax for the set of topological magma morphisms. |
class TopMgm⟶ | ||
Definition | df-bj-topmgmhom 36602* | 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 Top⟶ 𝑦) ∩ (𝑥 Mgm⟶ 𝑦))) | ||
Syntax | ccur- 36603 | Syntax for the parameterized currying function. |
class curry_ | ||
Definition | df-bj-cur 36604* | Define currying. See also df-cur 8266. (Contributed by BJ, 11-Apr-2020.) |
⊢ curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set⟶ 𝑧) ↦ (𝑎 ∈ 𝑥 ↦ (𝑏 ∈ 𝑦 ↦ (𝑓‘〈𝑎, 𝑏〉))))) | ||
Syntax | cunc- 36605 | Notation for the parameterized uncurrying function. |
class uncurry_ | ||
Definition | df-bj-unc 36606* | Define uncurrying. See also df-unc 8267. (Contributed by BJ, 11-Apr-2020.) |
⊢ uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set⟶ 𝑧)) ↦ (𝑎 ∈ 𝑥, 𝑏 ∈ 𝑦 ↦ ((𝑓‘𝑎)‘𝑏)))) | ||
Groundwork for changing the definition, syntax and token for component-setting in extensible structures. See https://github.com/metamath/set.mm/issues/2401 | ||
Syntax | cstrset 36607 | Syntax for component-setting in extensible structures. |
class [𝐵 / 𝐴]struct𝑆 | ||
Definition | df-strset 36608 | 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. The V in this definition was chosen to be closer to df-sets 17126, but since extensible structures are functions on ℕ, it will be more natural to replace it with ℕ when df-strset 36608 becomes the main definition. (Contributed by BJ, 13-Feb-2022.) |
⊢ [𝐵 / 𝐴]struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {〈(𝐴‘ndx), 𝐵〉}) | ||
Theorem | setsstrset 36609 | Relation between df-sets 17126 and df-strset 36608. Temporary theorem kept during the transition from the former to the latter. (Contributed by BJ, 13-Feb-2022.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → [𝐵 / 𝐴]struct𝑆 = (𝑆 sSet 〈(𝐴‘ndx), 𝐵〉)) | ||
In this section, we indroduce several supersets of the set ℝ of real numbers and the set ℂ 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 36706 and df-bj-rrhat 36708, and the point at infinity is denoted by ∞, defined in df-bj-infty 36704. Both ℝ and ℂ also have "directional compactifications", denoted respectively by ℝ̅, defined in df-bj-rrbar 36702 (already defined as ℝ*, see df-xr 11276) and ℂ̅, defined in df-bj-ccbar 36689. Since ℂ̅ does not seem to be standard, we describe it in some detail. It is obtained by adding to ℂ 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 ℂ 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 ℝ>0 (think of the closed "northern hemisphere" in ℝ^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 ℂ with a circle at infinity denoted by ℂ∞. To have a genuine inclusion ℝ̅ ⊆ ℂ̅, we define +∞ and -∞ as certain points in ℂ∞. 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. | ||
Theorem | bj-nfald 36610 | Variant of nfald 2317. (Contributed by BJ, 25-Dec-2023.) |
⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) | ||
Theorem | bj-nfexd 36611 | Variant of nfexd 2318. (Contributed by BJ, 25-Dec-2023.) |
⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) | ||
Theorem | copsex2d 36612* | Implicit substitution deduction for ordered pairs. (Contributed by BJ, 25-Dec-2023.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → Ⅎ𝑦𝜒) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ 𝜒)) | ||
Theorem | copsex2b 36613* | Biconditional form of copsex2d 36612. TODO: prove a relative version, that is, with ∃𝑥 ∈ 𝑉∃𝑦 ∈ 𝑊...(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊). (Contributed by BJ, 27-Dec-2023.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → Ⅎ𝑦𝜒) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))) | ||
Theorem | opelopabd 36614* | Membership of an ordere pair in a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → Ⅎ𝑦𝜒) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ 𝜒)) | ||
Theorem | opelopabb 36615* | Membership of an ordered pair in a class abstraction of ordered pairs, biconditional form. (Contributed by BJ, 17-Dec-2023.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → Ⅎ𝑦𝜒) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))) | ||
Theorem | opelopabbv 36616* | Membership of an ordered pair in a class abstraction of ordered pairs, biconditional form. (Contributed by BJ, 17-Dec-2023.) |
⊢ (𝜑 → 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓}) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))) | ||
Theorem | bj-opelrelex 36617 | The coordinates of an ordered pair that belongs to a relation are sets. TODO: Slightly shorter than brrelex12 5724, which could be proved from it. (Contributed by BJ, 27-Dec-2023.) |
⊢ ((Rel 𝑅 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | ||
Theorem | bj-opelresdm 36618 | If an ordered pair is in a restricted binary relation, then its first component is an element of the restricting class. See also opelres 5985. (Contributed by BJ, 25-Dec-2023.) |
⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋) | ||
Theorem | bj-brresdm 36619 |
If two classes are related by a restricted binary relation, then the first
class is an element of the restricting class. See also brres 5986 and
brrelex1 5725.
Remark: there are many pairs like bj-opelresdm 36618 / bj-brresdm 36619, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 36618 / brrelex12 5724 or the opelopabg 5534 / brabg 5535 family). They are straightforwardly equivalent by df-br 5143. The latter is indeed a very direct definition, introducing a "shorthand", and barely necessary, were it not for the frequency of the expression 𝐴𝑅𝐵. Therefore, in the spirit of "definitions are here to be used", most theorems, apart from the most elementary ones, should only have the "br" version, not the "opel" one. (Contributed by BJ, 25-Dec-2023.) |
⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 → 𝐴 ∈ 𝑋) | ||
Theorem | brabd0 36620* | Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → Ⅎ𝑦𝜒) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓}) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒)) | ||
Theorem | brabd 36621* | Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓}) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒)) | ||
Theorem | bj-brab2a1 36622* | "Unbounded" version of brab2a 5765. (Contributed by BJ, 25-Dec-2023.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜓)) | ||
Complements on the identity relation. | ||
Theorem | bj-opabssvv 36623* | A variant of relopabiv 5816 (which could be proved from it, similarly to relxp 5690 from xpss 5688). (Contributed by BJ, 28-Dec-2023.) |
⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ (V × V) | ||
Theorem | bj-funidres 36624 |
The restricted identity relation is a function. (Contributed by BJ,
27-Dec-2023.)
TODO: relabel funi 6579 to funid. |
⊢ Fun ( I ↾ 𝑉) | ||
Theorem | bj-opelidb 36625 |
Characterization of the ordered pair elements of the identity relation.
Remark: in deduction-style proofs, one could save a few syntactic steps by using another antecedent than ⊤ which already appears in the proof. Here for instance this could be the definition I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} but this would make the proof less easy to read. (Contributed by BJ, 27-Dec-2023.) |
⊢ (〈𝐴, 𝐵〉 ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)) | ||
Theorem | bj-opelidb1 36626 | Characterization of the ordered pair elements of the identity relation. Variant of bj-opelidb 36625 where only the sethood of the first component is expressed. (Contributed by BJ, 27-Dec-2023.) |
⊢ (〈𝐴, 𝐵〉 ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) | ||
Theorem | bj-inexeqex 36627 | Lemma for bj-opelid 36629 (but not specific to the identity relation): if the intersection of two classes is a set and the two classes are equal, then both are sets (all three classes are equal, so they all belong to 𝑉, but it is more convenient to have V in the consequent for theorems using it). (Contributed by BJ, 27-Dec-2023.) |
⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | ||
Theorem | bj-elsn0 36628 | If the intersection of two classes is a set, then these classes are equal if and only if one is an element of the singleton formed on the other. Stronger form of elsng 4638 and elsn2g 4662 (which could be proved from it). (Contributed by BJ, 20-Jan-2024.) |
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) | ||
Theorem | bj-opelid 36629 | Characterization of the ordered pair elements of the identity relation when the intersection of their components are sets. Note that the antecedent is more general than either component being a set. (Contributed by BJ, 29-Mar-2020.) |
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ I ↔ 𝐴 = 𝐵)) | ||
Theorem | bj-ideqg 36630 |
Characterization of the classes related by the identity relation when
their intersection is a set. Note that the antecedent is more general
than either class being a set. (Contributed by NM, 30-Apr-2004.) Weaken
the antecedent to sethood of the intersection. (Revised by BJ,
24-Dec-2023.)
TODO: replace ideqg 5848, or at least prove ideqg 5848 from it. |
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | bj-ideqgALT 36631 | Alternate proof of bj-ideqg 36630 from brabga 5530 instead of bj-opelid 36629 itself proved from bj-opelidb 36625. (Contributed by BJ, 27-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | bj-ideqb 36632 | Characterization of classes related by the identity relation. (Contributed by BJ, 24-Dec-2023.) |
⊢ (𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) | ||
Theorem | bj-idres 36633 |
Alternate expression for the restricted identity relation. The
advantage of that expression is to expose it as a "bounded"
class, being
included in the Cartesian square of the restricting class. (Contributed
by BJ, 27-Dec-2023.)
This is an alternate of idinxpresid 6045 (see idinxpres 6044). See also elrid 6043 and elidinxp 6041. (Proof modification is discouraged.) |
⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴)) | ||
Theorem | bj-opelidres 36634 | Characterization of the ordered pairs in the restricted identity relation when the intersection of their component belongs to the restricting class. TODO: prove bj-idreseq 36635 from it. (Contributed by BJ, 29-Mar-2020.) |
⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵)) | ||
Theorem | bj-idreseq 36635 | Sufficient condition for the restricted identity relation to agree with equality. Note that the instance of bj-ideqg 36630 with V substituted for 𝑉 is a direct consequence of bj-idreseq 36635. This is a strengthening of resieq 5990 which should be proved from it (note that currently, resieq 5990 relies on ideq 5849). Note that the intersection in the antecedent is not very meaningful, but is a device to prove versions with either class assumed to be a set. It could be enough to prove the version with a disjunctive antecedent: ⊢ ((𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶) → ...). (Contributed by BJ, 25-Dec-2023.) |
⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴( I ↾ 𝐶)𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | bj-idreseqb 36636 | Characterization for two classes to be related under the restricted identity relation. (Contributed by BJ, 24-Dec-2023.) |
⊢ (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵)) | ||
Theorem | bj-ideqg1 36637 |
For sets, the identity relation is the same thing as equality.
(Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon,
27-Aug-2011.) Generalize to a disjunctive antecedent. (Revised by BJ,
24-Dec-2023.)
TODO: delete once bj-ideqg 36630 is in the main section. |
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | bj-ideqg1ALT 36638 |
Alternate proof of bj-ideqg1 using brabga 5530 instead of the "unbounded"
version bj-brab2a1 36622 or brab2a 5765. (Contributed by BJ, 25-Dec-2023.)
(Proof modification is discouraged.) (New usage is discouraged.)
TODO: delete once bj-ideqg 36630 is in the main section. |
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | bj-opelidb1ALT 36639 | Characterization of the couples in I. (Contributed by BJ, 29-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (〈𝐴, 𝐵〉 ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) | ||
Theorem | bj-elid3 36640 | Characterization of the couples in I whose first component is a setvar. (Contributed by BJ, 29-Mar-2020.) |
⊢ (〈𝑥, 𝐴〉 ∈ I ↔ 𝑥 = 𝐴) | ||
Theorem | bj-elid4 36641 | Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.) |
⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴))) | ||
Theorem | bj-elid5 36642 | Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.) |
⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴))) | ||
Theorem | bj-elid6 36643 | Characterization of the elements of the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.) |
⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵))) | ||
Theorem | bj-elid7 36644 | Characterization of the elements of the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.) |
⊢ (〈𝐵, 𝐶〉 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) | ||
This subsection defines a functionalized version of the identity relation, that can also be seen as the diagonal in a Cartesian square). As explained in df-bj-diag 36646, it will probably be deleted. | ||
Syntax | cdiag2 36645 | Syntax for the diagonal of the Cartesian square of a set. |
class Id | ||
Definition | df-bj-diag 36646 |
Define the functionalized identity, which can also be seen as the diagonal
function. Its value is given in bj-diagval 36647 when it is viewed as the
functionalized identity, and in bj-diagval2 36648 when it is viewed as the
diagonal function.
Indeed, Definition df-br 5143 identifies a binary relation with the class of couples that are related by that binary relation (see eqrel2 37765 for the extensionality property of binary relations). As a consequence, the identity relation, or identity function (see funi 6579), on any class, can alternatively be seen as the diagonal of the cartesian square of that class. The identity relation on the universal class, I, is an "identity relation generator", since its restriction to any class is the identity relation on that class. It may be useful to consider a functionalized version of that fact, and that is the purpose of df-bj-diag 36646. Note: most proofs will only use its values (Id‘𝐴), in which case it may be enough to use ( I ↾ 𝐴) everywhere and dispense with this definition. (Contributed by BJ, 22-Jun-2019.) |
⊢ Id = (𝑥 ∈ V ↦ ( I ↾ 𝑥)) | ||
Theorem | bj-diagval 36647 | Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the functionalized identity, whereas bj-diagval2 36648 views it as the diagonal function. See df-bj-diag 36646 for the terminology. (Contributed by BJ, 22-Jun-2019.) |
⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ↾ 𝐴)) | ||
Theorem | bj-diagval2 36648 | Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the diagonal function, whereas bj-diagval 36647 views it as the functionalized identity. See df-bj-diag 36646 for the terminology. (Contributed by BJ, 22-Jun-2019.) |
⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴))) | ||
Theorem | bj-eldiag 36649 | Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid6 36643. (Contributed by BJ, 22-Jun-2019.) |
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵)))) | ||
Theorem | bj-eldiag2 36650 | Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid7 36644. (Contributed by BJ, 22-Jun-2019.) |
⊢ (𝐴 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (Id‘𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))) | ||
Definitions of the functionalized direct image and inverse image. The functionalized direct (resp. inverse) image is the morphism component of the covariant (resp. contravariant) powerset endofunctor of the category of sets and relations (and, up to restriction, of its subcategory of sets and functions). Its object component is the powerset operation 𝒫 defined in df-pw 4600. | ||
Syntax | cimdir 36651 | Syntax for the functionalized direct image. |
class 𝒫* | ||
Definition | df-imdir 36652* | Definition of the functionalized direct image, which maps a binary relation between two given sets to its associated direct image relation. (Contributed by BJ, 16-Dec-2023.) |
⊢ 𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ (𝑟 “ 𝑥) = 𝑦)})) | ||
Theorem | bj-imdirvallem 36653* | Lemma for bj-imdirval 36654 and bj-iminvval 36666. (Contributed by BJ, 23-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)})) ⇒ ⊢ (𝜑 → (𝐴𝐶𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)})) | ||
Theorem | bj-imdirval 36654* | Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦)})) | ||
Theorem | bj-imdirval2lem 36655* | Lemma for bj-imdirval2 36656 and bj-iminvval2 36667. (Contributed by BJ, 23-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ∈ V) | ||
Theorem | bj-imdirval2 36656* | Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) ⇒ ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)}) | ||
Theorem | bj-imdirval3 36657 | Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) ⇒ ⊢ (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌))) | ||
Theorem | bj-imdiridlem 36658* | Lemma for bj-imdirid 36659 and bj-iminvid 36668. (Contributed by BJ, 26-May-2024.) |
⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → (𝜑 ↔ 𝑥 = 𝑦)) ⇒ ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝜑)} = ( I ↾ 𝒫 𝐴) | ||
Theorem | bj-imdirid 36659 | Functorial property of the direct image: the direct image by the identity on a set is the identity on the powerset. (Contributed by BJ, 24-Dec-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴)) | ||
Theorem | bj-opelopabid 36660* | Membership in an ordered-pair class abstraction. One can remove the DV condition on 𝑥, 𝑦 by using opabid 5521 in place of opabidw 5520. (Contributed by BJ, 22-May-2024.) |
⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 𝜑) | ||
Theorem | bj-opabco 36661* | Composition of ordered-pair class abstractions. (Contributed by BJ, 22-May-2024.) |
⊢ ({〈𝑦, 𝑧〉 ∣ 𝜓} ∘ {〈𝑥, 𝑦〉 ∣ 𝜑}) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝜑 ∧ 𝜓)} | ||
Theorem | bj-xpcossxp 36662 | The composition of two Cartesian products is included in the expected Cartesian product. There is equality if (𝐵 ∩ 𝐶) ≠ ∅, see xpcogend 14947. (Contributed by BJ, 22-May-2024.) |
⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐷) | ||
Theorem | bj-imdirco 36663 | Functorial property of the direct image: the direct image by a composition is the composition of the direct images. (Contributed by BJ, 23-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) & ⊢ (𝜑 → 𝑆 ⊆ (𝐵 × 𝐶)) ⇒ ⊢ (𝜑 → ((𝐴𝒫*𝐶)‘(𝑆 ∘ 𝑅)) = (((𝐵𝒫*𝐶)‘𝑆) ∘ ((𝐴𝒫*𝐵)‘𝑅))) | ||
Syntax | ciminv 36664 | Syntax for the functionalized inverse image. |
class 𝒫* | ||
Definition | df-iminv 36665* | Definition of the functionalized inverse image, which maps a binary relation between two given sets to its associated inverse image relation. (Contributed by BJ, 23-Dec-2023.) |
⊢ 𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝑥 = (◡𝑟 “ 𝑦))})) | ||
Theorem | bj-iminvval 36666* | Value of the functionalized inverse image. (Contributed by BJ, 23-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦))})) | ||
Theorem | bj-iminvval2 36667* | Value of the functionalized inverse image. (Contributed by BJ, 23-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) ⇒ ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))}) | ||
Theorem | bj-iminvid 36668 | Functorial property of the inverse image: the inverse image by the identity on a set is the identity on the powerset. (Contributed by BJ, 26-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴)) | ||
We parameterize the set of infinite extended complex numbers ℂ∞ (df-bj-ccinfty 36685) using the real numbers ℝ (df-r 11142) via the function +∞eiτ. Since at that point, we have only defined the set of real numbers but no operations on it, we define a temporary "fractional part" function, which is more convenient to define on the temporary reals R (df-nr 11073) since we can use operations on the latter. We also define the temporary real "one-half" in order to define minus infinity (df-bj-minfty 36697) and then we can define the sets of extended real numbers and of extended complex numbers, and the projective real and complex lines, as well as addition and negation on these, and also the order relation on the extended reals (which bypasses the intermediate definition of a temporary order on the real numbers and then a superseding one on the extended real numbers). | ||
Syntax | cfractemp 36669 | Syntax for the fractional part of a tempopary real. |
class {R | ||
Definition | df-bj-fractemp 36670* |
Temporary definition: fractional part of a temporary real.
To understand this definition, recall the canonical injection ω⟶R, 𝑛 ↦ [{𝑥 ∈ Q ∣ 𝑥 <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 = (𝑥 ∈ R ↦ (℩𝑦 ∈ R ((𝑦 = 0R ∨ (0R <R 𝑦 ∧ 𝑦 <R 1R)) ∧ ∃𝑛 ∈ ω ([〈{𝑧 ∈ Q ∣ 𝑧 <Q 〈suc 𝑛, 1o〉}, 1P〉] ~R +R 𝑦) = 𝑥))) | ||
Syntax | cinftyexpitau 36671 | Syntax for the function +∞eiτ parameterizing ℂ∞. |
class +∞eiτ | ||
Definition | df-bj-inftyexpitau 36672 | Definition of the auxiliary function +∞eiτ parameterizing the circle at infinity ℂ∞ in ℂ̅. We use coupling with {R} to simplify the proof of bj-inftyexpitaudisj 36678. (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) → ((+∞eiτ‘𝐴) = (+∞eiτ‘𝐵) ↔ (𝐴 − 𝐵) ∈ ℤ)). (New usage is discouraged.) |
⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ 〈({R‘(1st ‘𝑥)), {R}〉) | ||
Syntax | cccinftyN 36673 | Syntax for the circle at infinity ℂ∞N. |
class ℂ∞N | ||
Definition | df-bj-ccinftyN 36674 | Definition of the circle at infinity ℂ∞N. (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.) |
⊢ ℂ∞N = ran +∞eiτ | ||
Theorem | bj-inftyexpitaufo 36675 | The function +∞eiτ written as a surjection with domain and range. (Contributed by BJ, 4-Feb-2023.) |
⊢ +∞eiτ:ℝ–onto→ℂ∞N | ||
Syntax | chalf 36676 | Syntax for the temporary one-half. |
class 1/2 | ||
Definition | df-bj-onehalf 36677 |
Define the temporary real "one-half". Once the machinery is
developed,
the real number "one-half" is commonly denoted by (1 / 2).
(Contributed by BJ, 4-Feb-2023.) (New usage is discouraged.)
TODO: $p |- 1/2 e. R. $= ? $. (riotacl 7388) $p |- -. 0R = 1/2 $= ? $. (since -. ( 0R +R 0R ) = 1R ) $p |- 0R <R 1/2 $= ? $. $p |- 1/2 <R 1R $= ? $. $p |- ( {R ` 0R ) = 0R $= ? $. $p |- ( {R ` 1/2 ) = 1/2 $= ? $. df-minfty $a |- minfty = ( inftyexpitau ` <. 1/2 , 0R >. ) $. |
⊢ 1/2 = (℩𝑥 ∈ R (𝑥 +R 𝑥) = 1R) | ||
Theorem | bj-inftyexpitaudisj 36678 | An element of the circle at infinity is not a complex number. (Contributed by BJ, 4-Feb-2023.) |
⊢ ¬ (+∞eiτ‘𝐴) ∈ ℂ | ||
Syntax | cinftyexpi 36679 | Syntax for the function +∞ei parameterizing ℂ∞. |
class +∞ei | ||
Definition | df-bj-inftyexpi 36680 | Definition of the auxiliary function +∞ei parameterizing the circle at infinity ℂ∞ in ℂ̅. We use coupling with ℂ to simplify the proof of bj-ccinftydisj 36686. It could seem more natural to define +∞ei on all of ℝ, but we want to use only basic functions in the definition of ℂ̅. TODO: transition to df-bj-inftyexpitau 36672 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〉) | ||
Theorem | bj-inftyexpiinv 36681 | 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 ‘(+∞ei‘𝐴)) = 𝐴) | ||
Theorem | bj-inftyexpiinj 36682 | Injectivity of the parameterization +∞ei. Remark: a more conceptual proof would use bj-inftyexpiinv 36681 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 ↔ (+∞ei‘𝐴) = (+∞ei‘𝐵))) | ||
Theorem | bj-inftyexpidisj 36683 | 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.) |
⊢ ¬ (+∞ei‘𝐴) ∈ ℂ | ||
Syntax | cccinfty 36684 | Syntax for the circle at infinity ℂ∞. |
class ℂ∞ | ||
Definition | df-bj-ccinfty 36685 | Definition of the circle at infinity ℂ∞. (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.) |
⊢ ℂ∞ = ran +∞ei | ||
Theorem | bj-ccinftydisj 36686 | The circle at infinity is disjoint from the set of complex numbers. (Contributed by BJ, 22-Jun-2019.) |
⊢ (ℂ ∩ ℂ∞) = ∅ | ||
Theorem | bj-elccinfty 36687 | A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.) |
⊢ (𝐴 ∈ (-π(,]π) → (+∞ei‘𝐴) ∈ ℂ∞) | ||
Syntax | cccbar 36688 | Syntax for the set of extended complex numbers ℂ̅. |
class ℂ̅ | ||
Definition | df-bj-ccbar 36689 | Definition of the set of extended complex numbers ℂ̅. (Contributed by BJ, 22-Jun-2019.) |
⊢ ℂ̅ = (ℂ ∪ ℂ∞) | ||
Theorem | bj-ccssccbar 36690 | Complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.) |
⊢ ℂ ⊆ ℂ̅ | ||
Theorem | bj-ccinftyssccbar 36691 | Infinite extended complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.) |
⊢ ℂ∞ ⊆ ℂ̅ | ||
Syntax | cpinfty 36692 | Syntax for "plus infinity". |
class +∞ | ||
Definition | df-bj-pinfty 36693 | Definition of "plus infinity". (Contributed by BJ, 27-Jun-2019.) |
⊢ +∞ = (+∞ei‘0) | ||
Theorem | bj-pinftyccb 36694 | The class +∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.) |
⊢ +∞ ∈ ℂ̅ | ||
Theorem | bj-pinftynrr 36695 | The extended complex number +∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.) |
⊢ ¬ +∞ ∈ ℂ | ||
Syntax | cminfty 36696 | Syntax for "minus infinity". |
class -∞ | ||
Definition | df-bj-minfty 36697 | Definition of "minus infinity". (Contributed by BJ, 27-Jun-2019.) |
⊢ -∞ = (+∞ei‘π) | ||
Theorem | bj-minftyccb 36698 | The class -∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.) |
⊢ -∞ ∈ ℂ̅ | ||
Theorem | bj-minftynrr 36699 | The extended complex number -∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.) |
⊢ ¬ -∞ ∈ ℂ | ||
Theorem | bj-pinftynminfty 36700 | The extended complex numbers +∞ and -∞ are different. (Contributed by BJ, 27-Jun-2019.) |
⊢ +∞ ≠ -∞ |
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