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Theorem List for Metamath Proof Explorer - 14801-14900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremshftidt2 14801 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       (𝐹 shift 0) = (𝐹 ↾ ℂ)
 
Theoremshftidt 14802 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 14803 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 14804 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 14805 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 14806 Extend class notation to include the Signum function.
class sgn
 
Definitiondf-sgn 14807 Signum function. We do not call it "sign", which is homophonic with "sine" (df-sin 15788). 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 15788. We define this over * (df-xr 11022) instead of so that it can accept +∞ and -∞. Note that df-psgn 19108 defines the sign of a permutation, which is different. Value shown in sgnval 14808. (Contributed by David A. Wheeler, 15-May-2015.)
sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))
 
Theoremsgnval 14808 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 14809 The signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
(sgn‘0) = 0
 
Theoremsgnp 14810 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 14811 The signum of a positive real is 1. (Contributed by David A. Wheeler, 18-May-2015.)
(𝐴 ∈ ℝ+ → (sgn‘𝐴) = 1)
 
Theoremsgn1 14812 The signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
(sgn‘1) = 1
 
Theoremsgnpnf 14813 The signum of +∞ is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
(sgn‘+∞) = 1
 
Theoremsgnn 14814 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 14815 The signum of -∞ is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
(sgn‘-∞) = -1
 
5.9.3  Real and imaginary parts; conjugate
 
Syntaxccj 14816 Extend class notation to include complex conjugate function.
class
 
Syntaxcre 14817 Extend class notation to include real part of a complex number.
class
 
Syntaxcim 14818 Extend class notation to include imaginary part of a complex number.
class
 
Definitiondf-cj 14819* Define the complex conjugate function. See cjcli 14889 for its closure and cjval 14822 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 14820 Define a function whose value is the real part of a complex number. See reval 14826 for its value, recli 14887 for its closure, and replim 14836 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
 
Definitiondf-im 14821 Define a function whose value is the imaginary part of a complex number. See imval 14827 for its value, imcli 14888 for its closure, and replim 14836 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
 
Theoremcjval 14822* 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 14823 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ))
 
Theoremcjf 14824 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
∗:ℂ⟶ℂ
 
Theoremcjcl 14825 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 14826 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 14827 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 14828 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 14829 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 14830 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 14831 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 14832 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 14833 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 14834 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 14835 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 14836 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 14837 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 14838 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 14839 A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)
(𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0))
 
Theoremrereb 14840 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 14841 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 14842 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 14843 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 14844 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴))
 
Theoremreneg 14845 Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (ℜ‘-𝐴) = -(ℜ‘𝐴))
 
Theoremreadd 14846 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 14847 Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴𝐵)) = ((ℜ‘𝐴) − (ℜ‘𝐵)))
 
Theoremremullem 14848 Lemma for remul 14849, immul 14856, and cjmul 14862. (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 14849 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 14850 Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (𝐴 · (ℜ‘𝐵)))
 
Theoremrediv 14851 Real part of a division. Related to remul2 14850. (Contributed by David A. Wheeler, 10-Jun-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (ℜ‘(𝐴 / 𝐵)) = ((ℜ‘𝐴) / 𝐵))
 
Theoremimcj 14852 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 14853 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 14854 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 14855 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵)))
 
Theoremimmul 14856 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 14857 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (𝐴 · (ℑ‘𝐵)))
 
Theoremimdiv 14858 Imaginary part of a division. Related to immul2 14857. (Contributed by Mario Carneiro, 20-Jun-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (ℑ‘(𝐴 / 𝐵)) = ((ℑ‘𝐴) / 𝐵))
 
Theoremcjre 14859 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 14860 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 14861 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 14862 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 14863 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 14864 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 14865 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 14866 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 14867 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (∗‘-𝐴) = -(∗‘𝐴))
 
Theoremaddcj 14868 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 14869 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴𝐵)) = ((∗‘𝐴) − (∗‘𝐵)))
 
Theoremcjexp 14870 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (∗‘(𝐴𝑁)) = ((∗‘𝐴)↑𝑁))
 
Theoremimval2 14871 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)))
 
Theoremre0 14872 The real part of zero. (Contributed by NM, 27-Jul-1999.)
(ℜ‘0) = 0
 
Theoremim0 14873 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
(ℑ‘0) = 0
 
Theoremre1 14874 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℜ‘1) = 1
 
Theoremim1 14875 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℑ‘1) = 0
 
Theoremrei 14876 The real part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℜ‘i) = 0
 
Theoremimi 14877 The imaginary part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℑ‘i) = 1
 
Theoremcj0 14878 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
(∗‘0) = 0
 
Theoremcji 14879 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
(∗‘i) = -i
 
Theoremcjreim 14880 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)))
 
Theoremcjreim2 14881 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)))
 
Theoremcj11 14882 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))
 
Theoremcjne0 14883 A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.)
(𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ (∗‘𝐴) ≠ 0))
 
Theoremcjdiv 14884 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)))
 
Theoremcnrecnv 14885* The inverse to the canonical bijection from (ℝ × ℝ) to from cnref1o 12734. (Contributed by Mario Carneiro, 25-Aug-2014.)
𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))       𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)
 
Theoremsqeqd 14886 A deduction for showing two numbers whose squares are equal are themselves equal. (Contributed by Mario Carneiro, 3-Apr-2015.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴↑2) = (𝐵↑2))    &   (𝜑 → 0 ≤ (ℜ‘𝐴))    &   (𝜑 → 0 ≤ (ℜ‘𝐵))    &   ((𝜑 ∧ (ℜ‘𝐴) = 0 ∧ (ℜ‘𝐵) = 0) → 𝐴 = 𝐵)       (𝜑𝐴 = 𝐵)
 
Theoremrecli 14887 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (ℜ‘𝐴) ∈ ℝ
 
Theoremimcli 14888 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (ℑ‘𝐴) ∈ ℝ
 
Theoremcjcli 14889 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (∗‘𝐴) ∈ ℂ
 
Theoremreplimi 14890 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)
𝐴 ∈ ℂ       𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))
 
Theoremcjcji 14891 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
 
Theoremreim0bi 14892 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)
𝐴 ∈ ℂ       (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)
 
Theoremrerebi 14893 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)
 
Theoremcjrebi 14894 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)
 
Theoremrecji 14895 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)
 
Theoremimcji 14896 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴)
 
Theoremcjmulrcli 14897 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (𝐴 · (∗‘𝐴)) ∈ ℝ
 
Theoremcjmulvali 14898 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       (𝐴 · (∗‘𝐴)) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))
 
Theoremcjmulge0i 14899 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
𝐴 ∈ ℂ       0 ≤ (𝐴 · (∗‘𝐴))
 
Theoremrenegi 14900 Real part of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (ℜ‘-𝐴) = -(ℜ‘𝐴)
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