P. 152 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 15101-15200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	relexp0d 15101	A relation composed zero times is the (restricted) identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → Rel 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))
Theorem	relexpsucnnr 15102	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 15103	A relation composed once is itself. (Contributed by RP, 22-May-2020.)
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
Theorem	dfid5 15104	Identity relation is equal to relational exponentiation to the first power. (Contributed by RP, 9-Jun-2020.)
⊢ I = (𝑥 ∈ V ↦ (𝑥↑𝑟1))
Theorem	dfid6 15105*	Identity relation expressed as indexed union of relational powers. (Contributed by RP, 9-Jun-2020.)
⊢ I = (𝑥 ∈ V ↦ ∪ 𝑛 ∈ {1} (𝑥↑𝑟𝑛))
Theorem	relexp1d 15106	A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → 𝑅 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑅↑𝑟1) = 𝑅)
Theorem	relexpsucnnl 15107	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 15108	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	relexpsucr 15109	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 15110	A reduction for relation exponentiation to the right. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → Rel 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅))
Theorem	relexpsucld 15111	A reduction for relation exponentiation to the left. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → Rel 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))
Theorem	relexpcnv 15112	Commutation of converse and relation exponentiation. (Contributed by RP, 23-May-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))
Theorem	relexpcnvd 15113	Commutation of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))
Theorem	relexp0rel 15114	The exponentiation of a class to zero is a relation. (Contributed by RP, 23-May-2020.)
⊢ (𝑅 ∈ 𝑉 → Rel (𝑅↑𝑟0))
Theorem	relexprelg 15115	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 15116	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 15117	The exponentiation of a relation is a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → Rel 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → Rel (𝑅↑𝑟𝑁))
Theorem	relexpnndm 15118	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 15119	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 15120	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 15121	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.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → dom (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)
Theorem	relexpnnrn 15122	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 15123	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 15124	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 15125	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.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ran (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)
Theorem	relexpfld 15126	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 15127	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.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)
Theorem	relexpaddnn 15128	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 15129	The exponentiation of a class to an integer greater than 1 is a relation. (Contributed by RP, 23-May-2020.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁))
Theorem	relexpaddg 15130	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 15131	Relation composition becomes addition under exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → Rel 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
5.8.5  Reflexive-transitive closure as an indexed union
Syntax	crtrcl 15132	Extend class notation with recursively defined reflexive, transitive closure.
class t*rec
Definition	df-rtrclrec 15133*	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	rtrclreclem1 15134	The reflexive, transitive closure is indeed a closure. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → 𝑅 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝑅 ⊆ (t*rec‘𝑅))
Theorem	dfrtrclrec2 15135*	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.) (Revised by AV, 13-Jul-2024.)
⊢ (𝜑 → Rel 𝑅)    ⇒   ⊢ (𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))
Theorem	rtrclreclem2 15136	The reflexive, transitive closure is indeed reflexive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 13-Jul-2024.)
⊢ (𝜑 → Rel 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅))
Theorem	rtrclreclem3 15137	The reflexive, transitive closure is indeed transitive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 13-Jul-2024.)
⊢ (𝜑 → Rel 𝑅)    ⇒   ⊢ (𝜑 → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))
Theorem	rtrclreclem4 15138*	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.) (Revised by AV, 13-Jul-2024.)
⊢ (𝜑 → Rel 𝑅)    ⇒   ⊢ (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
Theorem	dfrtrcl2 15139	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.) (Revised by AV, 13-Jul-2024.)
⊢ (𝜑 → Rel 𝑅)    ⇒   ⊢ (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.
Theorem	relexpindlem 15140*	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.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) (Revised by AV, 13-Jul-2024.)
⊢ (𝜂 → Rel 𝑅)    &   ⊢ (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 15141*	Principle of transitive induction, finite version. The first three hypotheses give various existences, the next four 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.) (Revised by AV, 13-Jul-2024.)
⊢ (𝜂 → Rel 𝑅)    &   ⊢ (𝜂 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜂 → 𝑋 ∈ 𝑊)    &   ⊢ (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 15142*	Principle of transitive induction. The first three 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.) (Revised by AV, 13-Jul-2024.)
⊢ (𝜂 → Rel 𝑅)    &   ⊢ (𝜂 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜂 → 𝑋 ∈ 𝑊)    &   ⊢ (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))
5.9  Elementary real and complex functions
5.9.1  The "shift" operation
Syntax	cshi 15143	Extend class notation with function shifter.
class shift
Definition	df-shft 15144*	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 15151 for its value. (Contributed by NM, 20-Jul-2005.)
⊢ shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ℂ ∧ (𝑦 − 𝑥)𝑓𝑧)})
Theorem	shftlem 15145*	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 15146*	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 15147*	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 15148*	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 15149	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 15150*	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 15151	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 15152	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 15153	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 15154	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 15155	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 15156*	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 15157	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 15158	Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
⊢ 𝐹 ∈ V    ⇒   ⊢ (𝐹 shift 0) = (𝐹 ↾ ℂ)
Theorem	shftidt 15159	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 15160	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 15161	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 15162	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))
5.9.2  Signum (sgn or sign) function
Syntax	csgn 15163	Extend class notation to include the Signum function.
class sgn
Definition	df-sgn 15164	Signum function. We do not call it "sign", which is homophonic with "sine" (df-sin 16161). 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 16161. We define this over ℝ* (df-xr 11275) instead of ℝ so that it can accept +∞ and -∞. Note that df-psgn 19624 defines the sign of a permutation, which is different. Value shown in sgnval 15165. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))
Theorem	sgnval 15165	Value of the signum function. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)))
Theorem	sgn0 15166	The signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (sgn‘0) = 0
Theorem	sgnp 15167	The signum of a positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) = 1)
Theorem	sgnrrp 15168	The signum of a positive real is 1. (Contributed by David A. Wheeler, 18-May-2015.)
⊢ (𝐴 ∈ ℝ+ → (sgn‘𝐴) = 1)
Theorem	sgn1 15169	The signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
⊢ (sgn‘1) = 1
Theorem	sgnpnf 15170	The signum of +∞ is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
⊢ (sgn‘+∞) = 1
Theorem	sgnn 15171	The signum of a negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1)
Theorem	sgnmnf 15172	The signum of -∞ is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
⊢ (sgn‘-∞) = -1
Theorem	sgndm 15173	The domain of the signum function. (Contributed by AV, 16-Jun-2026.)
⊢ dom sgn = ℝ*
Theorem	sgncl 15174	Closure of the signum: The signum of an extended real is either -1 or 0 or 1. (Contributed by Thierry Arnoux, 28-Sep-2018.)
⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) ∈ {-1, 0, 1})
Theorem	sgnrn 15175	The range of the signum function. (Contributed by AV, 16-Jun-2026.) (Proof shortened by TA, 21-Jun-2026.)
⊢ ran sgn = {-1, 0, 1}
Theorem	sgnfo 15176	The signum function as onto function. (Contributed by AV, 16-Jun-2026.)
⊢ sgn:ℝ*–onto→{-1, 0, 1}
Theorem	sgnneg 15177	Negation of the signum. (Contributed by Thierry Arnoux, 1-Oct-2018.)
⊢ (𝐴 ∈ ℝ → (sgn‘-𝐴) = -(sgn‘𝐴))
Theorem	sgn3da 15178	A conditional containing a signum is true if it is true in all three possible cases. (Contributed by Thierry Arnoux, 1-Oct-2018.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ ((sgn‘𝐴) = 0 → (𝜓 ↔ 𝜒))    &   ⊢ ((sgn‘𝐴) = 1 → (𝜓 ↔ 𝜃))    &   ⊢ ((sgn‘𝐴) = -1 → (𝜓 ↔ 𝜏))    &   ⊢ ((𝜑 ∧ 𝐴 = 0) → 𝜒)    &   ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝜏)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	sgnclre 15179	Closure of the signum for real numbers. (Contributed by Thierry Arnoux, 28-Sep-2018.)
⊢ (𝐴 ∈ ℝ → (sgn‘𝐴) ∈ ℝ)
Theorem	sgn0bi 15180	Zero signum. (Contributed by Thierry Arnoux, 10-Oct-2018.)
⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = 0 ↔ 𝐴 = 0))
Theorem	sgnnbi 15181	Negative signum. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = -1 ↔ 𝐴 < 0))
Theorem	sgnpbi 15182	Positive signum. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = 1 ↔ 0 < 𝐴))
Theorem	sgnsub 15183	Signum of a difference with a number of opposite sign. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) → (sgn‘(𝐴 − 𝐵)) = (sgn‘𝐴))
Theorem	sgnmul 15184	Signum of a product. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (sgn‘(𝐴 · 𝐵)) = ((sgn‘𝐴) · (sgn‘𝐵)))
Theorem	sgnmulrp2 15185	Multiplication by a positive number does not affect signum. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (sgn‘(𝐴 · 𝐵)) = (sgn‘𝐴))
Theorem	sgnmulsgn 15186	If two real numbers are of different signs, so are their signums. (Contributed by Thierry Arnoux, 12-Oct-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 𝐵) < 0 ↔ ((sgn‘𝐴) · (sgn‘𝐵)) < 0))
5.9.3  Real and imaginary parts; conjugate
Syntax	ccj 15187	Extend class notation to include complex conjugate function.
class ∗
Syntax	cre 15188	Extend class notation to include real part of a complex number.
class ℜ
Syntax	cim 15189	Extend class notation to include imaginary part of a complex number.
class ℑ
Definition	df-cj 15190*	Define the complex conjugate function. See cjcli 15260 for its closure and cjval 15193 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 15191	Define a function whose value is the real part of a complex number. See reval 15197 for its value, recli 15258 for its closure, and replim 15207 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
Definition	df-im 15192	Define a function whose value is the imaginary part of a complex number. See imval 15198 for its value, imcli 15259 for its closure, and replim 15207 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
Theorem	cjval 15193*	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 15194	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 15195	Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
⊢ ∗:ℂ⟶ℂ
Theorem	cjcl 15196	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 15197	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 15198	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 15199	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 15200	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)))