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Theorem List for Metamath Proof Explorer - 13601-13700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcotrtrclfv 13601 The transitive closure of a transitive relation. (Contributed by RP, 28-Apr-2020.)
((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (t+‘𝑅) = 𝑅)

Theoremtrclidm 13602 The transitive closure of a relation is idempotent. (Contributed by RP, 29-Apr-2020.)
(𝑅𝑉 → (t+‘(t+‘𝑅)) = (t+‘𝑅))

Theoremtrclun 13603 Transitive closure of a union of relations. (Contributed by RP, 5-May-2020.)
((𝑅𝑉𝑆𝑊) → (t+‘(𝑅𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆))))

Theoremtrclfvg 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) = ∅)

Theoremtrclfvcotrg 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)

Theoremreltrclfv 13606 The transitive closure of a relation is a relation. (Contributed by RP, 9-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅))

Theoremdmtrclfv 13607 The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 9-May-2020.)
(𝑅𝑉 → dom (t+‘𝑅) = dom 𝑅)

5.8.4  Exponentiation of relations

Syntaxcrelexp 13608 Extend class notation to include relation exponentiation.
class 𝑟

Definitiondf-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 ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))

Theoremrelexp0g 13610 A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
(𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))

Theoremrelexp0 13611 A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅) → (𝑅𝑟0) = ( I ↾ 𝑅))

Theoremrelexp0d 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)    &   (𝜑𝑅 ∈ V)       (𝜑 → (𝑅𝑟0) = ( I ↾ 𝑅))

Theoremrelexpsucnnr 13613 A reduction for relation exponentiation to the right. (Contributed by RP, 22-May-2020.)
((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅))

Theoremrelexp1g 13614 A relation composed once is itself. (Contributed by RP, 22-May-2020.)
(𝑅𝑉 → (𝑅𝑟1) = 𝑅)

Theoremdfid5 13615 Identity relation is equal to relational exponentiation to the first power. (Contributed by RP, 9-Jun-2020.)
I = (𝑥 ∈ V ↦ (𝑥𝑟1))

Theoremdfid6 13616* Identity relation expressed as indexed union of relational powers. (Contributed by RP, 9-Jun-2020.)
I = (𝑥 ∈ V ↦ 𝑛 ∈ {1} (𝑥𝑟𝑛))

Theoremrelexpsucr 13617 A reduction for relation exponentiation to the right. (Contributed by RP, 23-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅𝑁 ∈ ℕ0) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅))

Theoremrelexpsucrd 13618 A reduction for relation exponentiation to the right. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → (𝑁 ∈ ℕ0 → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅)))

Theoremrelexp1d 13619 A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝑅𝑟1) = 𝑅)

Theoremrelexpsucnnl 13620 A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))

Theoremrelexpsucl 13621 A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅𝑁 ∈ ℕ0) → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))

Theoremrelexpsucld 13622 A reduction for relation exponentiation to the left. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → (𝑁 ∈ ℕ0 → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁))))

Theoremrelexpcnv 13623 Commutation of converse and relation exponentiation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))

Theoremrelexpcnvd 13624 Commutation of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝑁 ∈ ℕ0(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))

Theoremrelexp0rel 13625 The exponentiation of a class to zero is a relation. (Contributed by RP, 23-May-2020.)
(𝑅𝑉 → Rel (𝑅𝑟0))

Theoremrelexprelg 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 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))

Theoremrelexprel 13627 The exponentiation of a relation is a relation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉 ∧ Rel 𝑅) → Rel (𝑅𝑟𝑁))

Theoremrelexpreld 13628 The exponentiation of a relation is a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → (𝑁 ∈ ℕ0 → Rel (𝑅𝑟𝑁)))

Theoremrelexpnndm 13629 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)

Theoremrelexpdmg 13630 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))

Theoremrelexpdm 13631 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ 𝑅)

Theoremrelexpdmd 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.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝑁 ∈ ℕ0 → dom (𝑅𝑟𝑁) ⊆ 𝑅))

Theoremrelexpnnrn 13633 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ ∧ 𝑅𝑉) → ran (𝑅𝑟𝑁) ⊆ ran 𝑅)

Theoremrelexprng 13634 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉) → ran (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))

Theoremrelexprn 13635 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉) → ran (𝑅𝑟𝑁) ⊆ 𝑅)

Theoremrelexprnd 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.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝑁 ∈ ℕ0 → ran (𝑅𝑟𝑁) ⊆ 𝑅))

Theoremrelexpfld 13637 The field of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅)

Theoremrelexpfldd 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.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝑁 ∈ ℕ0 (𝑅𝑟𝑁) ⊆ 𝑅))

Theoremrelexpaddnn 13639 Relation composition becomes addition under exponentiation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))

Theoremrelexpuzrel 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) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))

Theoremrelexpaddg 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 ∈ ℕ0𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))

Theoremrelexpaddd 13642 Relation composition becomes addition under exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → ((𝑁 ∈ ℕ0𝑀 ∈ ℕ0) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))

5.8.5  Reflexive-transitive closure as an indexed union

Syntaxcrtrcl 13643 Extend class notation with recursively defined reflexive, transitive closure.
class t*rec

Definitiondf-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 (𝑟𝑟𝑛))

Theoremdfrtrclrec2 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)    &   (𝜑𝑅 ∈ V)       (𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))

Theoremrtrclreclem1 13646 The reflexive, transitive closure is indeed reflexive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅))

Theoremrtrclreclem2 13647 The reflexive, transitive closure is indeed a closure. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (t*rec‘𝑅))

Theoremrtrclreclem3 13648 The reflexive, transitive closure is indeed transitive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))

Theoremrtrclreclem4 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)    &   (𝜑𝑅 ∈ V)       (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))

Theoremdfrtrcl2 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)    &   (𝜑𝑅 ∈ V)       (𝜑 → (t*‘𝑅) = (t*rec‘𝑅))

5.8.6  Principle of transitive induction.

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.

Theoremrelexpindlem 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)    &   (𝜂𝑅 ∈ V)    &   (𝜂𝑆 ∈ V)    &   (𝑖 = 𝑆 → (𝜑𝜒))    &   (𝑖 = 𝑥 → (𝜑𝜓))    &   (𝑖 = 𝑗 → (𝜑𝜃))    &   (𝜂𝜒)    &   (𝜂 → (𝑗𝑅𝑥 → (𝜃𝜓)))       (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜓)))

Theoremrelexpind 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)    &   (𝜂𝑅 ∈ V)    &   (𝜂𝑆 ∈ V)    &   (𝜂𝑋 ∈ V)    &   (𝑖 = 𝑆 → (𝜑𝜒))    &   (𝑖 = 𝑥 → (𝜑𝜓))    &   (𝑖 = 𝑗 → (𝜑𝜃))    &   (𝑥 = 𝑋 → (𝜓𝜏))    &   (𝜂𝜒)    &   (𝜂 → (𝑗𝑅𝑥 → (𝜃𝜓)))       (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑋𝜏)))

Theoremrtrclind 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)    &   (𝜂𝑅 ∈ V)    &   (𝜂𝑆 ∈ V)    &   (𝜂𝑋 ∈ V)    &   (𝑖 = 𝑆 → (𝜑𝜒))    &   (𝑖 = 𝑥 → (𝜑𝜓))    &   (𝑖 = 𝑗 → (𝜑𝜃))    &   (𝑥 = 𝑋 → (𝜓𝜏))    &   (𝜂𝜒)    &   (𝜂 → (𝑗𝑅𝑥 → (𝜃𝜓)))       (𝜂 → (𝑆(t*‘𝑅)𝑋𝜏))

5.9  Elementary real and complex functions

5.9.1  The "shift" operation

Syntaxcshi 13654 Extend class notation with function shifter.
class shift

Definitiondf-shft 13655* Define a function shifter. This operation offsets the value argument of a function (ordinarily on a subset of ) and produces a new function on . See shftval 13662 for its value. (Contributed by NM, 20-Jul-2005.)
shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ℂ ∧ (𝑦𝑥)𝑓𝑧)})

Theoremshftlem 13656* Two ways to write a shifted set (𝐵 + 𝐴). (Contributed by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ⊆ ℂ) → {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵} = {𝑥 ∣ ∃𝑦𝐵 𝑥 = (𝑦 + 𝐴)})

Theoremshftuz 13657* A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ (ℤ𝐵)} = (ℤ‘(𝐵 + 𝐴)))

Theoremshftfval 13658* The value of the sequence shifter operation is a function on . 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 ∧ (𝑥𝐴)𝐹𝑦)})

Theoremshftdm 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 ∣ (𝑥𝐴) ∈ dom 𝐹})

Theoremshftfib 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 “ {(𝐵𝐴)}))

Theoremshftfn 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 𝐵𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵})

Theoremshftval 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‘(𝐵𝐴)))

Theoremshftval2 13663 Value of a sequence shifted by 𝐴𝐵. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐹 shift (𝐴𝐵))‘(𝐴 + 𝐶)) = (𝐹‘(𝐵 + 𝐶)))

Theoremshftval3 13664 Value of a sequence shifted by 𝐴𝐵. (Contributed by NM, 20-Jul-2005.)
𝐹 ∈ V       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift (𝐴𝐵))‘𝐴) = (𝐹𝐵))

Theoremshftval4 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)))

Theoremshftval5 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)) = (𝐹𝐵))

Theoremshftf 13667* Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       ((𝐹:𝐵𝐶𝐴 ∈ ℂ) → (𝐹 shift 𝐴):{𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵}⟶𝐶)

Theorem2shfti 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)))

Theoremshftidt2 13669 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       (𝐹 shift 0) = (𝐹 ↾ ℂ)

Theoremshftidt 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) = (𝐹𝐴))

Theoremshftcan1 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) = (𝐹𝐵))

Theoremshftcan2 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) = (𝐹𝐵))

Theoremseqshft 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(𝑀𝑁)( + , 𝐹) shift 𝑁))

5.9.2  Signum (sgn or sign) function

Syntaxcsgn 13674 Extend class notation to include the Signum function.
class sgn

Definitiondf-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 * (df-xr 9957) instead of 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)))

Theoremsgnval 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)))

Theoremsgn0 13677 Proof that signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
(sgn‘0) = 0

Theoremsgnp 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)

Theoremsgnrrp 13679 Proof that signum of positive reals is 1. (Contributed by David A. Wheeler, 18-May-2015.)
(𝐴 ∈ ℝ+ → (sgn‘𝐴) = 1)

Theoremsgn1 13680 Proof that the signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
(sgn‘1) = 1

Theoremsgnpnf 13681 Proof that the signum of +∞ is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
(sgn‘+∞) = 1

Theoremsgnn 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)

Theoremsgnmnf 13683 Proof that the signum of -∞ is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
(sgn‘-∞) = -1

5.9.3  Real and imaginary parts; conjugate

Syntaxccj 13684 Extend class notation to include complex conjugate function.
class

Syntaxcre 13685 Extend class notation to include real part of a complex number.
class

Syntaxcim 13686 Extend class notation to include imaginary part of a complex number.
class

Definitiondf-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 · (𝑥𝑦)) ∈ ℝ)))

Definitiondf-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))

Definitiondf-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)))

Theoremcjval 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 · (𝐴𝑥)) ∈ ℝ)))

Theoremcjth 13691 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ))

Theoremcjf 13692 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
∗:ℂ⟶ℂ

Theoremcjcl 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)

Theoremreval 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))

Theoremimval 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)))

Theoremimre 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)))

Theoremreim 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)))

Theoremrecl 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)

Theoremimcl 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)

Theoremref 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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