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Theorem List for Metamath Proof Explorer - 13701-13800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem0trrel 13701 The empty class is a transitive relation. (Contributed by RP, 24-Dec-2019.)
(∅ ∘ ∅) ⊆ ∅

5.8.2  Basic properties of closures

Theoremcleq1lem 13702 Equality implies bijection. (Contributed by RP, 9-May-2020.)
(𝐴 = 𝐵 → ((𝐴𝐶𝜑) ↔ (𝐵𝐶𝜑)))

Theoremcleq1 13703* Equality of relations implies equality of closures. (Contributed by RP, 9-May-2020.)
(𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})

Theoremclsslem 13704* The closure of a subclass is a subclass of the closure. (Contributed by RP, 16-May-2020.)
(𝑅𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} ⊆ {𝑟 ∣ (𝑆𝑟𝜑)})

5.8.3  Definitions and basic properties of transitive closures

Syntaxctcl 13705 Extend class notation to include the transitive closure symbol.
class t+

Syntaxcrtcl 13706 Extend class notation with reflexive-transitive closure.
class t*

Definitiondf-trcl 13707* Transitive closure of a relation. This is the smallest superset which has the transitive property. (Contributed by FL, 27-Jun-2011.)
t+ = (𝑥 ∈ V ↦ {𝑧 ∣ (𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})

Definitiondf-rtrcl 13708* Reflexive-transitive closure of a relation. This is the smallest superset which is reflexive property over all elements of its domain and range and has the transitive property. (Contributed by FL, 27-Jun-2011.)
t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})

Theoremtrcleq1 13709* Equality of relations implies equality of transitive closures. (Contributed by RP, 9-May-2020.)
(𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} = {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})

Theoremtrclsslem 13710* The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.)
(𝑅𝑆 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})

Theoremtrcleq2lem 13711 Equality implies bijection. (Contributed by RP, 5-May-2020.)
(𝐴 = 𝐵 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑅𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))

Theoremcvbtrcl 13712* Change of bound variable in class of all transitive relations which are supersets of a relation. (Contributed by RP, 5-May-2020.)
{𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)}

Theoremtrcleq12lem 13713 Equality implies bijection. (Contributed by RP, 9-May-2020.)
((𝑅 = 𝑆𝐴 = 𝐵) → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))

Theoremtrclexlem 13714 Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 5-May-2020.)
(𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)

Theoremtrclublem 13715* If a relation exists then the class of transitive relations which are supersets of that relation is not empty. (Contributed by RP, 28-Apr-2020.)
(𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})

Theoremtrclubi 13716* The Cartesian product of the domain and range of a relation is an upper bound for its transitive closure. (Contributed by RP, 2-Jan-2020.) (Revised by RP, 28-Apr-2020.) (Revised by AV, 26-Mar-2021.)
Rel 𝑅    &   𝑅 ∈ V        {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ (dom 𝑅 × ran 𝑅)

TheoremtrclubiOLD 13717* Obsolete version of trclubi 13716 as of 26-Mar-2021. (Contributed by RP, 2-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Rel 𝑅    &   𝑅𝑉        {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ (dom 𝑅 × ran 𝑅)

Theoremtrclubgi 13718* The union with the Cartesian product of its domain and range is an upper bound for a set's transitive closure. (Contributed by RP, 3-Jan-2020.) (Revised by RP, 28-Apr-2020.) (Revised by AV, 26-Mar-2021.)
𝑅 ∈ V        {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))

TheoremtrclubgiOLD 13719* Obsolete version of trclubgi 13718 as of 26-Mar-2021. (Contributed by RP, 3-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑅𝑉        {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))

Theoremtrclub 13720* The Cartesian product of the domain and range of a relation is an upper bound for its transitive closure. (Contributed by RP, 17-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅) → {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ (dom 𝑅 × ran 𝑅))

Theoremtrclubg 13721* The union with the Cartesian product of its domain and range is an upper bound for a set's transitive closure (as a relation). (Contributed by RP, 17-May-2020.)
(𝑅𝑉 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

Theoremtrclfv 13722* The transitive closure of a relation. (Contributed by RP, 28-Apr-2020.)
(𝑅𝑉 → (t+‘𝑅) = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})

Theorembrintclab 13723* Two ways to express a binary relation which is the intersection of a class. (Contributed by RP, 4-Apr-2020.)
(𝐴 {𝑥𝜑}𝐵 ↔ ∀𝑥(𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑥))

Theorembrtrclfv 13724* Two ways of expressing the transitive closure of a binary relation. (Contributed by RP, 9-May-2020.)
(𝑅𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵)))

Theorembrcnvtrclfv 13725* Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 9-May-2020.)
((𝑅𝑈𝐴𝑉𝐵𝑊) → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐵𝑟𝐴)))

Theorembrtrclfvcnv 13726* Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 10-May-2020.)
(𝑅𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵)))

Theorembrcnvtrclfvcnv 13727* Two ways of expressing the transitive closure of the converse of the converse of a binary relation. (Contributed by RP, 10-May-2020.)
((𝑅𝑈𝐴𝑉𝐵𝑊) → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐵𝑟𝐴)))

Theoremtrclfvss 13728 The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.)
((𝑅𝑉𝑆𝑊𝑅𝑆) → (t+‘𝑅) ⊆ (t+‘𝑆))

Theoremtrclfvub 13729 The transitive closure of a relation has an upper bound. (Contributed by RP, 28-Apr-2020.)
(𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

Theoremtrclfvlb 13730 The transitive closure of a relation has a lower bound. (Contributed by RP, 28-Apr-2020.)
(𝑅𝑉𝑅 ⊆ (t+‘𝑅))

Theoremtrclfvcotr 13731 The transitive closure of a relation is a transitive relation. (Contributed by RP, 29-Apr-2020.)
(𝑅𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))

Theoremtrclfvlb2 13732 The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)
(𝑅𝑉 → (𝑅𝑅) ⊆ (t+‘𝑅))

Theoremtrclfvlb3 13733 The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)
(𝑅𝑉 → (𝑅 ∪ (𝑅𝑅)) ⊆ (t+‘𝑅))

Theoremcotrtrclfv 13734 The transitive closure of a transitive relation. (Contributed by RP, 28-Apr-2020.)
((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (t+‘𝑅) = 𝑅)

Theoremtrclidm 13735 The transitive closure of a relation is idempotent. (Contributed by RP, 29-Apr-2020.)
(𝑅𝑉 → (t+‘(t+‘𝑅)) = (t+‘𝑅))

Theoremtrclun 13736 Transitive closure of a union of relations. (Contributed by RP, 5-May-2020.)
((𝑅𝑉𝑆𝑊) → (t+‘(𝑅𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆))))

Theoremtrclfvg 13737 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 13738 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 13739 The transitive closure of a relation is a relation. (Contributed by RP, 9-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅))

Theoremdmtrclfv 13740 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 13741 Extend class notation to include relation exponentiation.
class 𝑟

Definitiondf-relexp 13742* 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 13743 A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
(𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))

Theoremrelexp0 13744 A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅) → (𝑅𝑟0) = ( I ↾ 𝑅))

Theoremrelexp0d 13745 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 13746 A reduction for relation exponentiation to the right. (Contributed by RP, 22-May-2020.)
((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅))

Theoremrelexp1g 13747 A relation composed once is itself. (Contributed by RP, 22-May-2020.)
(𝑅𝑉 → (𝑅𝑟1) = 𝑅)

Theoremdfid5 13748 Identity relation is equal to relational exponentiation to the first power. (Contributed by RP, 9-Jun-2020.)
I = (𝑥 ∈ V ↦ (𝑥𝑟1))

Theoremdfid6 13749* Identity relation expressed as indexed union of relational powers. (Contributed by RP, 9-Jun-2020.)
I = (𝑥 ∈ V ↦ 𝑛 ∈ {1} (𝑥𝑟𝑛))

Theoremrelexpsucr 13750 A reduction for relation exponentiation to the right. (Contributed by RP, 23-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅𝑁 ∈ ℕ0) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅))

Theoremrelexpsucrd 13751 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 13752 A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝑅𝑟1) = 𝑅)

Theoremrelexpsucnnl 13753 A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))

Theoremrelexpsucl 13754 A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅𝑁 ∈ ℕ0) → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))

Theoremrelexpsucld 13755 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 13756 Commutation of converse and relation exponentiation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))

Theoremrelexpcnvd 13757 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 13758 The exponentiation of a class to zero is a relation. (Contributed by RP, 23-May-2020.)
(𝑅𝑉 → Rel (𝑅𝑟0))

Theoremrelexprelg 13759 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 13760 The exponentiation of a relation is a relation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉 ∧ Rel 𝑅) → Rel (𝑅𝑟𝑁))

Theoremrelexpreld 13761 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 13762 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 13763 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 13764 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 13765 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 13766 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 13767 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 13768 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 13769 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 13770 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 13771 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 13772 Relation composition becomes addition under exponentiation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))

Theoremrelexpuzrel 13773 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 13774 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 13775 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 13776 Extend class notation with recursively defined reflexive, transitive closure.
class t*rec

Definitiondf-rtrclrec 13777* 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 13778* 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 13779 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 13780 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 13781 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 13782* 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 13783 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 13784* 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 13785* 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 13786* 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 13787 Extend class notation with function shifter.
class shift

Definitiondf-shft 13788* 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 13795 for its value. (Contributed by NM, 20-Jul-2005.)
shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ℂ ∧ (𝑦𝑥)𝑓𝑧)})

Theoremshftlem 13789* 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 13790* 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 13791* 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 13792* 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 13793 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 13794* 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 13795 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 13796 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 13797 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 13798 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 13799 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 13800* 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)

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