P. 151 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 15001-15100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sgn1 15001	The signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
⊢ (sgn‘1) = 1
Theorem	sgnpnf 15002	The signum of +∞ is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
⊢ (sgn‘+∞) = 1
Theorem	sgnn 15003	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 15004	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 15005	Extend class notation to include complex conjugate function.
class ∗
Syntax	cre 15006	Extend class notation to include real part of a complex number.
class ℜ
Syntax	cim 15007	Extend class notation to include imaginary part of a complex number.
class ℑ
Definition	df-cj 15008*	Define the complex conjugate function. See cjcli 15078 for its closure and cjval 15011 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 15009	Define a function whose value is the real part of a complex number. See reval 15015 for its value, recli 15076 for its closure, and replim 15025 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
Definition	df-im 15010	Define a function whose value is the imaginary part of a complex number. See imval 15016 for its value, imcli 15077 for its closure, and replim 15025 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
Theorem	cjval 15011*	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 15012	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 15013	Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
⊢ ∗:ℂ⟶ℂ
Theorem	cjcl 15014	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 15015	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 15016	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 15017	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 15018	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 15019	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 15020	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 15021	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 15022	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 15023	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 15024	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 15025	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 15026	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 15027	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 15028	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 15029	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 15030	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 15031	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 15032	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 15033	Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
⊢ (𝐴 ∈ ℂ → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴))
Theorem	reneg 15034	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 15035	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 15036	Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 − 𝐵)) = ((ℜ‘𝐴) − (ℜ‘𝐵)))
Theorem	remullem 15037	Lemma for remul 15038, immul 15045, and cjmul 15051. (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 15038	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 15039	Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (𝐴 · (ℜ‘𝐵)))
Theorem	rediv 15040	Real part of a division. Related to remul2 15039. (Contributed by David A. Wheeler, 10-Jun-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (ℜ‘(𝐴 / 𝐵)) = ((ℜ‘𝐴) / 𝐵))
Theorem	imcj 15041	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 15042	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 15043	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 15044	Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵)))
Theorem	immul 15045	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 15046	Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (𝐴 · (ℑ‘𝐵)))
Theorem	imdiv 15047	Imaginary part of a division. Related to immul2 15046. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (ℑ‘(𝐴 / 𝐵)) = ((ℑ‘𝐴) / 𝐵))
Theorem	cjre 15048	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 15049	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 15050	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 15051	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 15052	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 15053	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 15054	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 15055	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 15056	Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
⊢ (𝐴 ∈ ℂ → (∗‘-𝐴) = -(∗‘𝐴))
Theorem	addcj 15057	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 15058	Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 − 𝐵)) = ((∗‘𝐴) − (∗‘𝐵)))
Theorem	cjexp 15059	Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (∗‘(𝐴↑𝑁)) = ((∗‘𝐴)↑𝑁))
Theorem	imval2 15060	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 15061	The real part of zero. (Contributed by NM, 27-Jul-1999.)
⊢ (ℜ‘0) = 0
Theorem	im0 15062	The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
⊢ (ℑ‘0) = 0
Theorem	re1 15063	The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
⊢ (ℜ‘1) = 1
Theorem	im1 15064	The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
⊢ (ℑ‘1) = 0
Theorem	rei 15065	The real part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
⊢ (ℜ‘i) = 0
Theorem	imi 15066	The imaginary part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
⊢ (ℑ‘i) = 1
Theorem	cj0 15067	The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
⊢ (∗‘0) = 0
Theorem	cji 15068	The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
⊢ (∗‘i) = -i
Theorem	cjreim 15069	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 15070	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)))
Theorem	cj11 15071	Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) = (∗‘𝐵) ↔ 𝐴 = 𝐵))
Theorem	cjne0 15072	A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.)
⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ (∗‘𝐴) ≠ 0))
Theorem	cjdiv 15073	Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵)))
Theorem	cnrecnv 15074*	The inverse to the canonical bijection from (ℝ × ℝ) to ℂ from cnref1o 12885. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    ⇒   ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)
Theorem	sqeqd 15075	A deduction for showing two numbers whose squares are equal are themselves equal. (Contributed by Mario Carneiro, 3-Apr-2015.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴↑2) = (𝐵↑2))    &   ⊢ (𝜑 → 0 ≤ (ℜ‘𝐴))    &   ⊢ (𝜑 → 0 ≤ (ℜ‘𝐵))    &   ⊢ ((𝜑 ∧ (ℜ‘𝐴) = 0 ∧ (ℜ‘𝐵) = 0) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	recli 15076	The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℜ‘𝐴) ∈ ℝ
Theorem	imcli 15077	The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℑ‘𝐴) ∈ ℝ
Theorem	cjcli 15078	Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (∗‘𝐴) ∈ ℂ
Theorem	replimi 15079	Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))
Theorem	cjcji 15080	The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by NM, 11-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (∗‘(∗‘𝐴)) = 𝐴
Theorem	reim0bi 15081	A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)
Theorem	rerebi 15082	A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 ∈ ℝ ↔ (ℜ‘𝐴) = 𝐴)
Theorem	cjrebi 15083	A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 ∈ ℝ ↔ (∗‘𝐴) = 𝐴)
Theorem	recji 15084	Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)
Theorem	imcji 15085	Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴)
Theorem	cjmulrcli 15086	A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · (∗‘𝐴)) ∈ ℝ
Theorem	cjmulvali 15087	A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · (∗‘𝐴)) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))
Theorem	cjmulge0i 15088	A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ 0 ≤ (𝐴 · (∗‘𝐴))
Theorem	renegi 15089	Real part of negative. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℜ‘-𝐴) = -(ℜ‘𝐴)
Theorem	imnegi 15090	Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (ℑ‘-𝐴) = -(ℑ‘𝐴)
Theorem	cjnegi 15091	Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (∗‘-𝐴) = -(∗‘𝐴)
Theorem	addcji 15092	A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 + (∗‘𝐴)) = (2 · (ℜ‘𝐴))
Theorem	readdi 15093	Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (ℜ‘(𝐴 + 𝐵)) = ((ℜ‘𝐴) + (ℜ‘𝐵))
Theorem	imaddi 15094	Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (ℑ‘(𝐴 + 𝐵)) = ((ℑ‘𝐴) + (ℑ‘𝐵))
Theorem	remuli 15095	Real part of a product. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵)))
Theorem	immuli 15096	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (ℑ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵)))
Theorem	cjaddi 15097	Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (∗‘(𝐴 + 𝐵)) = ((∗‘𝐴) + (∗‘𝐵))
Theorem	cjmuli 15098	Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (∗‘(𝐴 · 𝐵)) = ((∗‘𝐴) · (∗‘𝐵))
Theorem	ipcni 15099	Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (ℜ‘(𝐴 · (∗‘𝐵))) = (((ℜ‘𝐴) · (ℜ‘𝐵)) + ((ℑ‘𝐴) · (ℑ‘𝐵)))
Theorem	cjdivi 15100	Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵)))