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Theorem List for Metamath Proof Explorer - 14401-14500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrelexpuzrel 14401 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 14402 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 14403 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 14404 Extend class notation with recursively defined reflexive, transitive closure.
class t*rec

Definitiondf-rtrclrec 14405* 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 14406* 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 14407 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 14408 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 14409 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 14410* 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 14411 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 14412* 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)    &   (𝜂𝑅 ∈ V)    &   (𝜂𝑆 ∈ V)    &   (𝑖 = 𝑆 → (𝜑𝜒))    &   (𝑖 = 𝑥 → (𝜑𝜓))    &   (𝑖 = 𝑗 → (𝜑𝜃))    &   (𝜂𝜒)    &   (𝜂 → (𝑗𝑅𝑥 → (𝜃𝜓)))       (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜓)))

Theoremrelexpind 14413* 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 14414* 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 14415 Extend class notation with function shifter.
class shift

Definitiondf-shft 14416* 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 14423 for its value. (Contributed by NM, 20-Jul-2005.)
shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ℂ ∧ (𝑦𝑥)𝑓𝑧)})

Theoremshftlem 14417* 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 14418* 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 14419* 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 14420* 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 14421 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 14422* 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 14423 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 14424 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 14425 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 14426 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 14427 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 14428* 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 14429 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 14430 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       (𝐹 shift 0) = (𝐹 ↾ ℂ)

Theoremshftidt 14431 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 14432 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 14433 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 14434 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 14435 Extend class notation to include the Signum function.
class sgn

Definitiondf-sgn 14436 Signum function. We do not call it "sign", which is homophonic with "sine" (df-sin 15413). 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 15413. We define this over * (df-xr 10668) instead of so that it can accept +∞ and -∞. Note that df-psgn 18539 defines the sign of a permutation, which is different. Value shown in sgnval 14437. (Contributed by David A. Wheeler, 15-May-2015.)
sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))

Theoremsgnval 14437 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)))

Theoremsgn0 14438 The signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
(sgn‘0) = 0

Theoremsgnp 14439 The signum of a positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) = 1)

Theoremsgnrrp 14440 The signum of a positive real is 1. (Contributed by David A. Wheeler, 18-May-2015.)
(𝐴 ∈ ℝ+ → (sgn‘𝐴) = 1)

Theoremsgn1 14441 The signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
(sgn‘1) = 1

Theoremsgnpnf 14442 The signum of +∞ is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
(sgn‘+∞) = 1

Theoremsgnn 14443 The signum of a negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℝ*𝐴 < 0) → (sgn‘𝐴) = -1)

Theoremsgnmnf 14444 The signum of -∞ is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
(sgn‘-∞) = -1

5.9.3  Real and imaginary parts; conjugate

Syntaxccj 14445 Extend class notation to include complex conjugate function.
class

Syntaxcre 14446 Extend class notation to include real part of a complex number.
class

Syntaxcim 14447 Extend class notation to include imaginary part of a complex number.
class

Definitiondf-cj 14448* Define the complex conjugate function. See cjcli 14518 for its closure and cjval 14451 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 14449 Define a function whose value is the real part of a complex number. See reval 14455 for its value, recli 14516 for its closure, and replim 14465 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))

Definitiondf-im 14450 Define a function whose value is the imaginary part of a complex number. See imval 14456 for its value, imcli 14517 for its closure, and replim 14465 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))

Theoremcjval 14451* 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 14452 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ))

Theoremcjf 14453 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
∗:ℂ⟶ℂ

Theoremcjcl 14454 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 14455 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 14456 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 14457 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 14458 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 14459 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 14460 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 14461 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

Theoremimf 14462 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

Theoremcrre 14463 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)

Theoremcrim 14464 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)

Theoremreplim 14465 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))))

Theoremremim 14466 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))))

Theoremreim0 14467 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)

Theoremreim0b 14468 A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)
(𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0))

Theoremrereb 14469 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))

Theoremmulre 14470 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))

Theoremrere 14471 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)

Theoremcjreb 14472 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))

Theoremrecj 14473 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴))

Theoremreneg 14474 Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (ℜ‘-𝐴) = -(ℜ‘𝐴))

Theoremreadd 14475 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)))

Theoremresub 14476 Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴𝐵)) = ((ℜ‘𝐴) − (ℜ‘𝐵)))

Theoremremullem 14477 Lemma for remul 14478, immul 14485, and cjmul 14491. (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))))

Theoremremul 14478 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))))

Theoremremul2 14479 Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (𝐴 · (ℜ‘𝐵)))

Theoremrediv 14480 Real part of a division. Related to remul2 14479. (Contributed by David A. Wheeler, 10-Jun-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (ℜ‘(𝐴 / 𝐵)) = ((ℜ‘𝐴) / 𝐵))

Theoremimcj 14481 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))

Theoremimneg 14482 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))

Theoremimadd 14483 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)))

Theoremimsub 14484 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵)))

Theoremimmul 14485 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))))

Theoremimmul2 14486 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (𝐴 · (ℑ‘𝐵)))

Theoremimdiv 14487 Imaginary part of a division. Related to immul2 14486. (Contributed by Mario Carneiro, 20-Jun-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (ℑ‘(𝐴 / 𝐵)) = ((ℑ‘𝐴) / 𝐵))

Theoremcjre 14488 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)

Theoremcjcj 14489 The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (∗‘(∗‘𝐴)) = 𝐴)

Theoremcjadd 14490 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 + 𝐵)) = ((∗‘𝐴) + (∗‘𝐵)))

Theoremcjmul 14491 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 · 𝐵)) = ((∗‘𝐴) · (∗‘𝐵)))

Theoremipcnval 14492 Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · (∗‘𝐵))) = (((ℜ‘𝐴) · (ℜ‘𝐵)) + ((ℑ‘𝐴) · (ℑ‘𝐵))))

Theoremcjmulrcl 14493 A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (𝐴 · (∗‘𝐴)) ∈ ℝ)

Theoremcjmulval 14494 A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (𝐴 · (∗‘𝐴)) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2)))

Theoremcjmulge0 14495 A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → 0 ≤ (𝐴 · (∗‘𝐴)))

Theoremcjneg 14496 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (∗‘-𝐴) = -(∗‘𝐴))

Theoremaddcj 14497 A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (𝐴 + (∗‘𝐴)) = (2 · (ℜ‘𝐴)))

Theoremcjsub 14498 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴𝐵)) = ((∗‘𝐴) − (∗‘𝐵)))

Theoremcjexp 14499 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (∗‘(𝐴𝑁)) = ((∗‘𝐴)↑𝑁))

Theoremimval2 14500 The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ ℂ → (ℑ‘𝐴) = ((𝐴 − (∗‘𝐴)) / (2 · i)))

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