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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cotrtrclfv 13601 | The transitive closure of a transitive relation. (Contributed by RP, 28-Apr-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → (t+‘𝑅) = 𝑅) | ||
Theorem | trclidm 13602 | The transitive closure of a relation is idempotent. (Contributed by RP, 29-Apr-2020.) |
⊢ (𝑅 ∈ 𝑉 → (t+‘(t+‘𝑅)) = (t+‘𝑅)) | ||
Theorem | trclun 13603 | Transitive closure of a union of relations. (Contributed by RP, 5-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘(𝑅 ∪ 𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆)))) | ||
Theorem | trclfvg 13604 | The value of the transitive closure of a relation is a superset or (for proper classes) the empty set. (Contributed by RP, 8-May-2020.) |
⊢ (𝑅 ⊆ (t+‘𝑅) ∨ (t+‘𝑅) = ∅) | ||
Theorem | trclfvcotrg 13605 | The value of the transitive closure of a relation is always a transitive relation. (Contributed by RP, 8-May-2020.) |
⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) | ||
Theorem | reltrclfv 13606 | The transitive closure of a relation is a relation. (Contributed by RP, 9-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅)) | ||
Theorem | dmtrclfv 13607 | The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 9-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) = dom 𝑅) | ||
Syntax | crelexp 13608 | Extend class notation to include relation exponentiation. |
class ↑𝑟 | ||
Definition | df-relexp 13609* | Definition of repeated composition of a relation with itself, aka relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 22-May-2020.) |
⊢ ↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) | ||
Theorem | relexp0g 13610 | A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) | ||
Theorem | relexp0 13611 | A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) | ||
Theorem | relexp0d 13612 | A relation composed zero times is the (restricted) identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) | ||
Theorem | relexpsucnnr 13613 | A reduction for relation exponentiation to the right. (Contributed by RP, 22-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) | ||
Theorem | relexp1g 13614 | A relation composed once is itself. (Contributed by RP, 22-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅) | ||
Theorem | dfid5 13615 | Identity relation is equal to relational exponentiation to the first power. (Contributed by RP, 9-Jun-2020.) |
⊢ I = (𝑥 ∈ V ↦ (𝑥↑𝑟1)) | ||
Theorem | dfid6 13616* | Identity relation expressed as indexed union of relational powers. (Contributed by RP, 9-Jun-2020.) |
⊢ I = (𝑥 ∈ V ↦ ∪ 𝑛 ∈ {1} (𝑥↑𝑟𝑛)) | ||
Theorem | relexpsucr 13617 | A reduction for relation exponentiation to the right. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅 ∧ 𝑁 ∈ ℕ0) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) | ||
Theorem | relexpsucrd 13618 | A reduction for relation exponentiation to the right. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑁 ∈ ℕ0 → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅))) | ||
Theorem | relexp1d 13619 | A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑅↑𝑟1) = 𝑅) | ||
Theorem | relexpsucnnl 13620 | A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) | ||
Theorem | relexpsucl 13621 | A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅 ∧ 𝑁 ∈ ℕ0) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) | ||
Theorem | relexpsucld 13622 | A reduction for relation exponentiation to the left. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑁 ∈ ℕ0 → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))) | ||
Theorem | relexpcnv 13623 | Commutation of converse and relation exponentiation. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)) | ||
Theorem | relexpcnvd 13624 | Commutation of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑁 ∈ ℕ0 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))) | ||
Theorem | relexp0rel 13625 | The exponentiation of a class to zero is a relation. (Contributed by RP, 23-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → Rel (𝑅↑𝑟0)) | ||
Theorem | relexprelg 13626 | The exponentiation of a class is a relation except when the exponent is one and the class is not a relation. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)) | ||
Theorem | relexprel 13627 | The exponentiation of a relation is a relation. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel (𝑅↑𝑟𝑁)) | ||
Theorem | relexpreld 13628 | The exponentiation of a relation is a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑁 ∈ ℕ0 → Rel (𝑅↑𝑟𝑁))) | ||
Theorem | relexpnndm 13629 | The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅) | ||
Theorem | relexpdmg 13630 | The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) | ||
Theorem | relexpdm 13631 | The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅) | ||
Theorem | relexpdmd 13632 | The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑁 ∈ ℕ0 → dom (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)) | ||
Theorem | relexpnnrn 13633 | The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ ran 𝑅) | ||
Theorem | relexprng 13634 | The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) | ||
Theorem | relexprn 13635 | The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅) | ||
Theorem | relexprnd 13636 | The range of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑁 ∈ ℕ0 → ran (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)) | ||
Theorem | relexpfld 13637 | The field of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅) | ||
Theorem | relexpfldd 13638 | The field of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑁 ∈ ℕ0 → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)) | ||
Theorem | relexpaddnn 13639 | Relation composition becomes addition under exponentiation. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))) | ||
Theorem | relexpuzrel 13640 | The exponentiation of a class to an integer not smaller than 2 is a relation. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁)) | ||
Theorem | relexpaddg 13641 | Relation composition becomes addition under exponentiation except when the exponents total to one and the class isn't a relation. (Contributed by RP, 30-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))) | ||
Theorem | relexpaddd 13642 | Relation composition becomes addition under exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))) | ||
Syntax | crtrcl 13643 | Extend class notation with recursively defined reflexive, transitive closure. |
class t*rec | ||
Definition | df-rtrclrec 13644* | The reflexive, transitive closure of a relation constructed as the union of all finite exponentiations. (Contributed by Drahflow, 12-Nov-2015.) |
⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) | ||
Theorem | dfrtrclrec2 13645* | If two elements are connected by a reflexive, transitive closure, then they are connected via 𝑛 instances the relation, for some 𝑛. (Contributed by Drahflow, 12-Nov-2015.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)) | ||
Theorem | rtrclreclem1 13646 | The reflexive, transitive closure is indeed reflexive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)) | ||
Theorem | rtrclreclem2 13647 | The reflexive, transitive closure is indeed a closure. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → 𝑅 ⊆ (t*rec‘𝑅)) | ||
Theorem | rtrclreclem3 13648 | The reflexive, transitive closure is indeed transitive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)) | ||
Theorem | rtrclreclem4 13649* | The reflexive, transitive closure of 𝑅 is the smallest reflexive, transitive relation which contains 𝑅 and the identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠)) | ||
Theorem | dfrtrcl2 13650 | The two definitions t* and t*rec of the reflexive, transitive closure coincide if 𝑅 is indeed a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (t*‘𝑅) = (t*rec‘𝑅)) | ||
If we have a statement that holds for some element, and a relation between elements that implies if it holds for the first element then it must hold for the second element, the principle of transitive induction shows the statement holds for any element related to the first by the (reflexive-)transitive closure of the relation. | ||
Theorem | relexpindlem 13651* | Principle of transitive induction, finite and non-class version. The first three hypotheses give various existences, the next three give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜂 → Rel 𝑅) & ⊢ (𝜂 → 𝑅 ∈ V) & ⊢ (𝜂 → 𝑆 ∈ V) & ⊢ (𝑖 = 𝑆 → (𝜑 ↔ 𝜒)) & ⊢ (𝑖 = 𝑥 → (𝜑 ↔ 𝜓)) & ⊢ (𝑖 = 𝑗 → (𝜑 ↔ 𝜃)) & ⊢ (𝜂 → 𝜒) & ⊢ (𝜂 → (𝑗𝑅𝑥 → (𝜃 → 𝜓))) ⇒ ⊢ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓))) | ||
Theorem | relexpind 13652* | Principle of transitive induction, finite version. The first three hypotheses give various existences, the next three give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜂 → Rel 𝑅) & ⊢ (𝜂 → 𝑅 ∈ V) & ⊢ (𝜂 → 𝑆 ∈ V) & ⊢ (𝜂 → 𝑋 ∈ V) & ⊢ (𝑖 = 𝑆 → (𝜑 ↔ 𝜒)) & ⊢ (𝑖 = 𝑥 → (𝜑 ↔ 𝜓)) & ⊢ (𝑖 = 𝑗 → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜏)) & ⊢ (𝜂 → 𝜒) & ⊢ (𝜂 → (𝑗𝑅𝑥 → (𝜃 → 𝜓))) ⇒ ⊢ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏))) | ||
Theorem | rtrclind 13653* | Principle of transitive induction. The first four hypotheses give various existences, the next four give necessary substitutions and the last two are the basis and the induction step. (Contributed by Drahflow, 12-Nov-2015.) |
⊢ (𝜂 → Rel 𝑅) & ⊢ (𝜂 → 𝑅 ∈ V) & ⊢ (𝜂 → 𝑆 ∈ V) & ⊢ (𝜂 → 𝑋 ∈ V) & ⊢ (𝑖 = 𝑆 → (𝜑 ↔ 𝜒)) & ⊢ (𝑖 = 𝑥 → (𝜑 ↔ 𝜓)) & ⊢ (𝑖 = 𝑗 → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜏)) & ⊢ (𝜂 → 𝜒) & ⊢ (𝜂 → (𝑗𝑅𝑥 → (𝜃 → 𝜓))) ⇒ ⊢ (𝜂 → (𝑆(t*‘𝑅)𝑋 → 𝜏)) | ||
Syntax | cshi 13654 | Extend class notation with function shifter. |
class shift | ||
Definition | df-shft 13655* | Define a function shifter. This operation offsets the value argument of a function (ordinarily on a subset of ℂ) and produces a new function on ℂ. See shftval 13662 for its value. (Contributed by NM, 20-Jul-2005.) |
⊢ shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ℂ ∧ (𝑦 − 𝑥)𝑓𝑧)}) | ||
Theorem | shftlem 13656* | Two ways to write a shifted set (𝐵 + 𝐴). (Contributed by Mario Carneiro, 3-Nov-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ⊆ ℂ) → {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵} = {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = (𝑦 + 𝐴)}) | ||
Theorem | shftuz 13657* | A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵)} = (ℤ≥‘(𝐵 + 𝐴))) | ||
Theorem | shftfval 13658* | The value of the sequence shifter operation is a function on ℂ. 𝐴 is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴)𝐹𝑦)}) | ||
Theorem | shftdm 13659* | Domain of a relation shifted by 𝐴. The set on the right is more commonly notated as (dom 𝐹 + 𝐴) (meaning add 𝐴 to every element of dom 𝐹). (Contributed by Mario Carneiro, 3-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐴 ∈ ℂ → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ dom 𝐹}) | ||
Theorem | shftfib 13660 | Value of a fiber of the relation 𝐹. (Contributed by Mario Carneiro, 4-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴) “ {𝐵}) = (𝐹 “ {(𝐵 − 𝐴)})) | ||
Theorem | shftfn 13661* | Functionality and domain of a sequence shifted by 𝐴. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}) | ||
Theorem | shftval 13662 | Value of a sequence shifted by 𝐴. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴))) | ||
Theorem | shftval2 13663 | Value of a sequence shifted by 𝐴 − 𝐵. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐹 shift (𝐴 − 𝐵))‘(𝐴 + 𝐶)) = (𝐹‘(𝐵 + 𝐶))) | ||
Theorem | shftval3 13664 | Value of a sequence shifted by 𝐴 − 𝐵. (Contributed by NM, 20-Jul-2005.) |
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift (𝐴 − 𝐵))‘𝐴) = (𝐹‘𝐵)) | ||
Theorem | shftval4 13665 | Value of a sequence shifted by -𝐴. (Contributed by NM, 18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift -𝐴)‘𝐵) = (𝐹‘(𝐴 + 𝐵))) | ||
Theorem | shftval5 13666 | Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘(𝐵 + 𝐴)) = (𝐹‘𝐵)) | ||
Theorem | shftf 13667* | Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴):{𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}⟶𝐶) | ||
Theorem | 2shfti 13668 | Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴) shift 𝐵) = (𝐹 shift (𝐴 + 𝐵))) | ||
Theorem | shftidt2 13669 | Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐹 shift 0) = (𝐹 ↾ ℂ) | ||
Theorem | shftidt 13670 | Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐴 ∈ ℂ → ((𝐹 shift 0)‘𝐴) = (𝐹‘𝐴)) | ||
Theorem | shftcan1 13671 | Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐹 shift 𝐴) shift -𝐴)‘𝐵) = (𝐹‘𝐵)) | ||
Theorem | shftcan2 13672 | Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐹 shift -𝐴) shift 𝐴)‘𝐵) = (𝐹‘𝐵)) | ||
Theorem | seqshft 13673 | Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-Feb-2014.) |
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → seq𝑀( + , (𝐹 shift 𝑁)) = (seq(𝑀 − 𝑁)( + , 𝐹) shift 𝑁)) | ||
Syntax | csgn 13674 | Extend class notation to include the Signum function. |
class sgn | ||
Definition | df-sgn 13675 | Signum function. Pronounced "signum" , otherwise it might be confused with "sine". Defined as "sgn" in ISO 80000-2:2009(E) operation 2-9.13. It is named "sign" (with the same definition) in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. We define this over ℝ* (df-xr 9957) instead of ℝ so that it can accept +∞ and -∞. Note that df-psgn 17734 defines the sign of a permutation, which is different. Value shown in sgnval 13676. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1))) | ||
Theorem | sgnval 13676 | Value of Signum function. Pronounced "signum" . See df-sgn 13675. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) | ||
Theorem | sgn0 13677 | Proof that signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ (sgn‘0) = 0 | ||
Theorem | sgnp 13678 | Proof that signum of positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) = 1) | ||
Theorem | sgnrrp 13679 | Proof that signum of positive reals is 1. (Contributed by David A. Wheeler, 18-May-2015.) |
⊢ (𝐴 ∈ ℝ+ → (sgn‘𝐴) = 1) | ||
Theorem | sgn1 13680 | Proof that the signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.) |
⊢ (sgn‘1) = 1 | ||
Theorem | sgnpnf 13681 | Proof that the signum of +∞ is 1. (Contributed by David A. Wheeler, 26-Jun-2016.) |
⊢ (sgn‘+∞) = 1 | ||
Theorem | sgnn 13682 | Proof that signum of negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1) | ||
Theorem | sgnmnf 13683 | Proof that the signum of -∞ is -1. (Contributed by David A. Wheeler, 26-Jun-2016.) |
⊢ (sgn‘-∞) = -1 | ||
Syntax | ccj 13684 | Extend class notation to include complex conjugate function. |
class ∗ | ||
Syntax | cre 13685 | Extend class notation to include real part of a complex number. |
class ℜ | ||
Syntax | cim 13686 | Extend class notation to include imaginary part of a complex number. |
class ℑ | ||
Definition | df-cj 13687* | Define the complex conjugate function. See cjcli 13757 for its closure and cjval 13690 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
⊢ ∗ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥 − 𝑦)) ∈ ℝ))) | ||
Definition | df-re 13688 | Define a function whose value is the real part of a complex number. See reval 13694 for its value, recli 13755 for its closure, and replim 13704 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.) |
⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2)) | ||
Definition | df-im 13689 | Define a function whose value is the imaginary part of a complex number. See imval 13695 for its value, imcli 13756 for its closure, and replim 13704 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.) |
⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i))) | ||
Theorem | cjval 13690* | The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.) |
⊢ (𝐴 ∈ ℂ → (∗‘𝐴) = (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ))) | ||
Theorem | cjth 13691 | The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.) |
⊢ (𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ)) | ||
Theorem | cjf 13692 | Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.) |
⊢ ∗:ℂ⟶ℂ | ||
Theorem | cjcl 13693 | The conjugate of a complex number is a complex number (closure law). (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | ||
Theorem | reval 13694 | The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2)) | ||
Theorem | imval 13695 | The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(𝐴 / i))) | ||
Theorem | imre 13696 | The imaginary part of a complex number in terms of the real part function. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 6-Nov-2013.) |
⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(-i · 𝐴))) | ||
Theorem | reim 13697 | The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = (ℑ‘(i · 𝐴))) | ||
Theorem | recl 13698 | The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | ||
Theorem | imcl 13699 | The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | ||
Theorem | ref 13700 | Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.) |
⊢ ℜ:ℂ⟶ℝ |
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