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 16105). 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 16105. We define this over ℝ* (df-xr 11299) instead of ℝ so that it can accept +∞ and -∞. Note that df-psgn 19509 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
5.9.3  Real and imaginary parts; conjugate
Syntax	ccj 15135	Extend class notation to include complex conjugate function.
class ∗
Syntax	cre 15136	Extend class notation to include real part of a complex number.
class ℜ
Syntax	cim 15137	Extend class notation to include imaginary part of a complex number.
class ℑ
Definition	df-cj 15138*	Define the complex conjugate function. See cjcli 15208 for its closure and cjval 15141 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 15139	Define a function whose value is the real part of a complex number. See reval 15145 for its value, recli 15206 for its closure, and replim 15155 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
Definition	df-im 15140	Define a function whose value is the imaginary part of a complex number. See imval 15146 for its value, imcli 15207 for its closure, and replim 15155 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
Theorem	cjval 15141*	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 15142	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 15143	Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
⊢ ∗:ℂ⟶ℂ
Theorem	cjcl 15144	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 15145	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 15146	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 15147	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 15148	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 15149	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 15150	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 15151	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 15152	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 15153	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 15154	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 15155	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 15156	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 15157	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 15158	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 15159	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 15160	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 15161	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 15162	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 15163	Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
⊢ (𝐴 ∈ ℂ → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴))
Theorem	reneg 15164	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 15165	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 15166	Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 − 𝐵)) = ((ℜ‘𝐴) − (ℜ‘𝐵)))
Theorem	remullem 15167	Lemma for remul 15168, immul 15175, and cjmul 15181. (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 15168	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 15169	Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (𝐴 · (ℜ‘𝐵)))
Theorem	rediv 15170	Real part of a division. Related to remul2 15169. (Contributed by David A. Wheeler, 10-Jun-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (ℜ‘(𝐴 / 𝐵)) = ((ℜ‘𝐴) / 𝐵))
Theorem	imcj 15171	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 15172	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 15173	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 15174	Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵)))
Theorem	immul 15175	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 15176	Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (𝐴 · (ℑ‘𝐵)))
Theorem	imdiv 15177	Imaginary part of a division. Related to immul2 15176. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (ℑ‘(𝐴 / 𝐵)) = ((ℑ‘𝐴) / 𝐵))
Theorem	cjre 15178	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 15179	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 15180	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 15181	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 15182	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 15183	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 15184	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 15185	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 15186	Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
⊢ (𝐴 ∈ ℂ → (∗‘-𝐴) = -(∗‘𝐴))
Theorem	addcj 15187	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)))
Theorem	cjsub 15188	Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 − 𝐵)) = ((∗‘𝐴) − (∗‘𝐵)))
Theorem	cjexp 15189	Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (∗‘(𝐴↑𝑁)) = ((∗‘𝐴)↑𝑁))
Theorem	imval2 15190	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)))
Theorem	re0 15191	The real part of zero. (Contributed by NM, 27-Jul-1999.)
⊢ (ℜ‘0) = 0
Theorem	im0 15192	The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
⊢ (ℑ‘0) = 0
Theorem	re1 15193	The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
⊢ (ℜ‘1) = 1
Theorem	im1 15194	The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
⊢ (ℑ‘1) = 0
Theorem	rei 15195	The real part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
⊢ (ℜ‘i) = 0
Theorem	imi 15196	The imaginary part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
⊢ (ℑ‘i) = 1
Theorem	cj0 15197	The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
⊢ (∗‘0) = 0
Theorem	cji 15198	The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
⊢ (∗‘i) = -i
Theorem	cjreim 15199	The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (∗‘(𝐴 + (i · 𝐵))) = (𝐴 − (i · 𝐵)))
Theorem	cjreim2 15200	The conjugate of the representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (∗‘(𝐴 − (i · 𝐵))) = (𝐴 + (i · 𝐵)))