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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | relexpreld 14401 | 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 14402 | 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 14403 | 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 14404 | 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 14405 | 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 14406 | 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 14407 | 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 14408 | 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 14409 | 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 14410 | 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 14411 | 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 14412 | 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 14413 | 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 14414 | 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 14415 | 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 14416 | Extend class notation with recursively defined reflexive, transitive closure. |
class t*rec | ||
Definition | df-rtrclrec 14417* | 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 14418* | 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 14419 | 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 14420 | 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 14421 | 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 14422* | 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 14423 | 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 14424* | 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.) |
⊢ (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 14425* | 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 14426* | 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 14427 | Extend class notation with function shifter. |
class shift | ||
Definition | df-shft 14428* | 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 14435 for its value. (Contributed by NM, 20-Jul-2005.) |
⊢ shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ℂ ∧ (𝑦 − 𝑥)𝑓𝑧)}) | ||
Theorem | shftlem 14429* | 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 14430* | 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 14431* | 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 14432* | 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 14433 | 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 14434* | 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 14435 | 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 14436 | 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 14437 | 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 14438 | 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 14439 | 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 14440* | 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 14441 | 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 14442 | Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐹 shift 0) = (𝐹 ↾ ℂ) | ||
Theorem | shftidt 14443 | 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 14444 | 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 14445 | 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 14446 | 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 14447 | Extend class notation to include the Signum function. |
class sgn | ||
Definition | df-sgn 14448 | Signum function. We do not call it "sign", which is homophonic with "sine" (df-sin 15425). 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 15425. We define this over ℝ* (df-xr 10681) instead of ℝ so that it can accept +∞ and -∞. Note that df-psgn 18621 defines the sign of a permutation, which is different. Value shown in sgnval 14449. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1))) | ||
Theorem | sgnval 14449 | 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 14450 | The signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ (sgn‘0) = 0 | ||
Theorem | sgnp 14451 | 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 14452 | The signum of a positive real is 1. (Contributed by David A. Wheeler, 18-May-2015.) |
⊢ (𝐴 ∈ ℝ+ → (sgn‘𝐴) = 1) | ||
Theorem | sgn1 14453 | The signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.) |
⊢ (sgn‘1) = 1 | ||
Theorem | sgnpnf 14454 | The signum of +∞ is 1. (Contributed by David A. Wheeler, 26-Jun-2016.) |
⊢ (sgn‘+∞) = 1 | ||
Theorem | sgnn 14455 | 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 14456 | The signum of -∞ is -1. (Contributed by David A. Wheeler, 26-Jun-2016.) |
⊢ (sgn‘-∞) = -1 | ||
Syntax | ccj 14457 | Extend class notation to include complex conjugate function. |
class ∗ | ||
Syntax | cre 14458 | Extend class notation to include real part of a complex number. |
class ℜ | ||
Syntax | cim 14459 | Extend class notation to include imaginary part of a complex number. |
class ℑ | ||
Definition | df-cj 14460* | Define the complex conjugate function. See cjcli 14530 for its closure and cjval 14463 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 14461 | Define a function whose value is the real part of a complex number. See reval 14467 for its value, recli 14528 for its closure, and replim 14477 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.) |
⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2)) | ||
Definition | df-im 14462 | Define a function whose value is the imaginary part of a complex number. See imval 14468 for its value, imcli 14529 for its closure, and replim 14477 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.) |
⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i))) | ||
Theorem | cjval 14463* | 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 14464 | 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 14465 | Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.) |
⊢ ∗:ℂ⟶ℂ | ||
Theorem | cjcl 14466 | 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 14467 | 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 14468 | 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 14469 | 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 14470 | 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 14471 | 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 14472 | 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 14473 | 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 | ||
Theorem | imf 14474 | Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.) |
⊢ ℑ:ℂ⟶ℝ | ||
Theorem | crre 14475 | The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℜ‘(𝐴 + (i · 𝐵))) = 𝐴) | ||
Theorem | crim 14476 | The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℑ‘(𝐴 + (i · 𝐵))) = 𝐵) | ||
Theorem | replim 14477 | Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.) |
⊢ (𝐴 ∈ ℂ → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) | ||
Theorem | remim 14478 | Value of the conjugate of a complex number. The value is the real part minus i times the imaginary part. Definition 10-3.2 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.) |
⊢ (𝐴 ∈ ℂ → (∗‘𝐴) = ((ℜ‘𝐴) − (i · (ℑ‘𝐴)))) | ||
Theorem | reim0 14479 | The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) |
⊢ (𝐴 ∈ ℝ → (ℑ‘𝐴) = 0) | ||
Theorem | reim0b 14480 | A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.) |
⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) | ||
Theorem | rereb 14481 | A number is real iff it equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 20-Aug-2008.) |
⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℜ‘𝐴) = 𝐴)) | ||
Theorem | mulre 14482 | A product with a nonzero real multiplier is real iff the multiplicand is real. (Contributed by NM, 21-Aug-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 ∈ ℝ ↔ (𝐵 · 𝐴) ∈ ℝ)) | ||
Theorem | rere 14483 | A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Paul Chapman, 7-Sep-2007.) |
⊢ (𝐴 ∈ ℝ → (ℜ‘𝐴) = 𝐴) | ||
Theorem | cjreb 14484 | A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (∗‘𝐴) = 𝐴)) | ||
Theorem | recj 14485 | Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.) |
⊢ (𝐴 ∈ ℂ → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)) | ||
Theorem | reneg 14486 | Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) = -(ℜ‘𝐴)) | ||
Theorem | readd 14487 | Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 + 𝐵)) = ((ℜ‘𝐴) + (ℜ‘𝐵))) | ||
Theorem | resub 14488 | Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 − 𝐵)) = ((ℜ‘𝐴) − (ℜ‘𝐵))) | ||
Theorem | remullem 14489 | Lemma for remul 14490, immul 14497, and cjmul 14503. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵))) ∧ (ℑ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵))) ∧ (∗‘(𝐴 · 𝐵)) = ((∗‘𝐴) · (∗‘𝐵)))) | ||
Theorem | remul 14490 | Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵)))) | ||
Theorem | remul2 14491 | Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (𝐴 · (ℜ‘𝐵))) | ||
Theorem | rediv 14492 | Real part of a division. Related to remul2 14491. (Contributed by David A. Wheeler, 10-Jun-2015.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (ℜ‘(𝐴 / 𝐵)) = ((ℜ‘𝐴) / 𝐵)) | ||
Theorem | imcj 14493 | Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
⊢ (𝐴 ∈ ℂ → (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴)) | ||
Theorem | imneg 14494 | The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
⊢ (𝐴 ∈ ℂ → (ℑ‘-𝐴) = -(ℑ‘𝐴)) | ||
Theorem | imadd 14495 | Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 + 𝐵)) = ((ℑ‘𝐴) + (ℑ‘𝐵))) | ||
Theorem | imsub 14496 | Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵))) | ||
Theorem | immul 14497 | Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵)))) | ||
Theorem | immul2 14498 | Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (𝐴 · (ℑ‘𝐵))) | ||
Theorem | imdiv 14499 | Imaginary part of a division. Related to immul2 14498. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (ℑ‘(𝐴 / 𝐵)) = ((ℑ‘𝐴) / 𝐵)) | ||
Theorem | cjre 14500 | A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.) |
⊢ (𝐴 ∈ ℝ → (∗‘𝐴) = 𝐴) |
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