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	dfrtrcl2 15101	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 15102*	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 15103*	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 15104*	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 15105	Extend class notation with function shifter.
class shift
Definition	df-shft 15106*	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 15113 for its value. (Contributed by NM, 20-Jul-2005.)
⊢ shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ℂ ∧ (𝑦 − 𝑥)𝑓𝑧)})
Theorem	shftlem 15107*	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 15108*	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 15109*	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 15110*	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 15111	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 15112*	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 15113	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 15114	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 15115	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 15116	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 15117	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 15118*	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 15119	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 15120	Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
⊢ 𝐹 ∈ V    ⇒   ⊢ (𝐹 shift 0) = (𝐹 ↾ ℂ)
Theorem	shftidt 15121	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 15122	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 15123	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 15124	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 15125	Extend class notation to include the Signum function.
class sgn
Definition	df-sgn 15126	Signum function. We do not call it "sign", which is homophonic with "sine" (df-sin 16124). 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 16124. We define this over ℝ* (df-xr 11248) instead of ℝ so that it can accept +∞ and -∞. Note that df-psgn 19562 defines the sign of a permutation, which is different. Value shown in sgnval 15127. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))
Theorem	sgnval 15127	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 15128	The signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (sgn‘0) = 0
Theorem	sgnp 15129	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 15130	The signum of a positive real is 1. (Contributed by David A. Wheeler, 18-May-2015.)
⊢ (𝐴 ∈ ℝ+ → (sgn‘𝐴) = 1)
Theorem	sgn1 15131	The signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
⊢ (sgn‘1) = 1
Theorem	sgnpnf 15132	The signum of +∞ is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
⊢ (sgn‘+∞) = 1
Theorem	sgnn 15133	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 15134	The signum of -∞ is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
⊢ (sgn‘-∞) = -1
Theorem	sgndm 15135	The domain of the signum function. (Contributed by AV, 16-Jun-2026.)
⊢ dom sgn = ℝ*
Theorem	sgncl 15136	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 15137	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 15138	The signum function as onto function. (Contributed by AV, 16-Jun-2026.)
⊢ sgn:ℝ*–onto→{-1, 0, 1}
Theorem	sgnneg 15139	Negation of the signum. (Contributed by Thierry Arnoux, 1-Oct-2018.)
⊢ (𝐴 ∈ ℝ → (sgn‘-𝐴) = -(sgn‘𝐴))
Theorem	sgn3da 15140	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 15141	Closure of the signum for real numbers. (Contributed by Thierry Arnoux, 28-Sep-2018.)
⊢ (𝐴 ∈ ℝ → (sgn‘𝐴) ∈ ℝ)
Theorem	sgn0bi 15142	Zero signum. (Contributed by Thierry Arnoux, 10-Oct-2018.)
⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = 0 ↔ 𝐴 = 0))
Theorem	sgnnbi 15143	Negative signum. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = -1 ↔ 𝐴 < 0))
Theorem	sgnpbi 15144	Positive signum. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ (𝐴 ∈ ℝ* → ((sgn‘𝐴) = 1 ↔ 0 < 𝐴))
Theorem	sgnsub 15145	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 15146	Signum of a product. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (sgn‘(𝐴 · 𝐵)) = ((sgn‘𝐴) · (sgn‘𝐵)))
Theorem	sgnmulrp2 15147	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 15148	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 15149	Extend class notation to include complex conjugate function.
class ∗
Syntax	cre 15150	Extend class notation to include real part of a complex number.
class ℜ
Syntax	cim 15151	Extend class notation to include imaginary part of a complex number.
class ℑ
Definition	df-cj 15152*	Define the complex conjugate function. See cjcli 15222 for its closure and cjval 15155 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 15153	Define a function whose value is the real part of a complex number. See reval 15159 for its value, recli 15220 for its closure, and replim 15169 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
Definition	df-im 15154	Define a function whose value is the imaginary part of a complex number. See imval 15160 for its value, imcli 15221 for its closure, and replim 15169 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
Theorem	cjval 15155*	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 15156	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 15157	Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
⊢ ∗:ℂ⟶ℂ
Theorem	cjcl 15158	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 15159	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 15160	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 15161	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 15162	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 15163	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 15164	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 15165	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 15166	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 15167	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 15168	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 15169	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 15170	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 15171	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 15172	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 15173	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 15174	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 15175	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 15176	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 15177	Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
⊢ (𝐴 ∈ ℂ → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴))
Theorem	reneg 15178	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 15179	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 15180	Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 − 𝐵)) = ((ℜ‘𝐴) − (ℜ‘𝐵)))
Theorem	remullem 15181	Lemma for remul 15182, immul 15189, and cjmul 15195. (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 15182	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 15183	Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (𝐴 · (ℜ‘𝐵)))
Theorem	rediv 15184	Real part of a division. Related to remul2 15183. (Contributed by David A. Wheeler, 10-Jun-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (ℜ‘(𝐴 / 𝐵)) = ((ℜ‘𝐴) / 𝐵))
Theorem	imcj 15185	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 15186	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 15187	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 15188	Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵)))
Theorem	immul 15189	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 15190	Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (𝐴 · (ℑ‘𝐵)))
Theorem	imdiv 15191	Imaginary part of a division. Related to immul2 15190. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (ℑ‘(𝐴 / 𝐵)) = ((ℑ‘𝐴) / 𝐵))
Theorem	cjre 15192	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)
Theorem	cjcj 15193	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)
Theorem	cjadd 15194	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)))
Theorem	cjmul 15195	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)))
Theorem	ipcnval 15196	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))))
Theorem	cjmulrcl 15197	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)
Theorem	cjmulval 15198	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)))
Theorem	cjmulge0 15199	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)))
Theorem	cjneg 15200	Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
⊢ (𝐴 ∈ ℂ → (∗‘-𝐴) = -(∗‘𝐴))