Home Metamath Proof ExplorerTheorem List (p. 139 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27745) Hilbert Space Explorer (27746-29270) Users' Mathboxes (29271-42316)

Theorem List for Metamath Proof Explorer - 13801-13900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem2shfti 13801 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 13802 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V       (𝐹 shift 0) = (𝐹 ↾ ℂ)

Theoremshftidt 13803 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 13804 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 13805 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 13806 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 13807 Extend class notation to include the Signum function.
class sgn

Definitiondf-sgn 13808 Signum function. Pronounced "signum" , otherwise it might be confused with "sine". 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. We define this over * (df-xr 10063) instead of so that it can accept +∞ and -∞. Note that df-psgn 17892 defines the sign of a permutation, which is different. Value shown in sgnval 13809. (Contributed by David A. Wheeler, 15-May-2015.)
sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))

Theoremsgnval 13809 Value of Signum function. Pronounced "signum" . See df-sgn 13808. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴 ∈ ℝ* → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)))

Theoremsgn0 13810 Proof that signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
(sgn‘0) = 0

Theoremsgnp 13811 Proof that signum of positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) = 1)

Theoremsgnrrp 13812 Proof that signum of positive reals is 1. (Contributed by David A. Wheeler, 18-May-2015.)
(𝐴 ∈ ℝ+ → (sgn‘𝐴) = 1)

Theoremsgn1 13813 Proof that the signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
(sgn‘1) = 1

Theoremsgnpnf 13814 Proof that the signum of +∞ is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
(sgn‘+∞) = 1

Theoremsgnn 13815 Proof that signum of negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℝ*𝐴 < 0) → (sgn‘𝐴) = -1)

Theoremsgnmnf 13816 Proof that the signum of -∞ is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
(sgn‘-∞) = -1

5.9.3  Real and imaginary parts; conjugate

Syntaxccj 13817 Extend class notation to include complex conjugate function.
class

Syntaxcre 13818 Extend class notation to include real part of a complex number.
class

Syntaxcim 13819 Extend class notation to include imaginary part of a complex number.
class

Definitiondf-cj 13820* Define the complex conjugate function. See cjcli 13890 for its closure and cjval 13823 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 13821 Define a function whose value is the real part of a complex number. See reval 13827 for its value, recli 13888 for its closure, and replim 13837 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))

Definitiondf-im 13822 Define a function whose value is the imaginary part of a complex number. See imval 13828 for its value, imcli 13889 for its closure, and replim 13837 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))

Theoremcjval 13823* 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 13824 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ))

Theoremcjf 13825 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
∗:ℂ⟶ℂ

Theoremcjcl 13826 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 13827 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 13828 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 13829 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 13830 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 13831 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 13832 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 13833 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 13834 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 13835 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 13836 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 13837 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 13838 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 13839 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 13840 A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)
(𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0))

Theoremrereb 13841 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 13842 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 13843 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 13844 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 13845 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴))

Theoremreneg 13846 Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (ℜ‘-𝐴) = -(ℜ‘𝐴))

Theoremreadd 13847 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 13848 Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴𝐵)) = ((ℜ‘𝐴) − (ℜ‘𝐵)))

Theoremremullem 13849 Lemma for remul 13850, immul 13857, and cjmul 13863. (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 13850 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 13851 Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (𝐴 · (ℜ‘𝐵)))

Theoremrediv 13852 Real part of a division. Related to remul2 13851. (Contributed by David A. Wheeler, 10-Jun-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (ℜ‘(𝐴 / 𝐵)) = ((ℜ‘𝐴) / 𝐵))

Theoremimcj 13853 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 13854 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 13855 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 13856 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵)))

Theoremimmul 13857 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 13858 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (𝐴 · (ℑ‘𝐵)))

Theoremimdiv 13859 Imaginary part of a division. Related to immul2 13858. (Contributed by Mario Carneiro, 20-Jun-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (ℑ‘(𝐴 / 𝐵)) = ((ℑ‘𝐴) / 𝐵))

Theoremcjre 13860 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 13861 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 13862 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 13863 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 13864 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 13865 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 13866 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 13867 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 13868 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (∗‘-𝐴) = -(∗‘𝐴))

Theoremaddcj 13869 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 13870 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴𝐵)) = ((∗‘𝐴) − (∗‘𝐵)))

Theoremcjexp 13871 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (∗‘(𝐴𝑁)) = ((∗‘𝐴)↑𝑁))

Theoremimval2 13872 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 13873 The real part of zero. (Contributed by NM, 27-Jul-1999.)
(ℜ‘0) = 0

Theoremim0 13874 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
(ℑ‘0) = 0

Theoremre1 13875 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℜ‘1) = 1

Theoremim1 13876 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℑ‘1) = 0

Theoremrei 13877 The real part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℜ‘i) = 0

Theoremimi 13878 The imaginary part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(ℑ‘i) = 1

Theoremcj0 13879 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
(∗‘0) = 0

Theoremcji 13880 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
(∗‘i) = -i

Theoremcjreim 13881 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 13882 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 13883 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 13884 A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.)
(𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ (∗‘𝐴) ≠ 0))

Theoremcjdiv 13885 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 13886* The inverse to the canonical bijection from (ℝ × ℝ) to from cnref1o 11812. (Contributed by Mario Carneiro, 25-Aug-2014.)
𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))       𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)

Theoremsqeqd 13887 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 13888 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (ℜ‘𝐴) ∈ ℝ

Theoremimcli 13889 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (ℑ‘𝐴) ∈ ℝ

Theoremcjcli 13890 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (∗‘𝐴) ∈ ℂ

Theoremreplimi 13891 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 13892 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 13893 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)
𝐴 ∈ ℂ       (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)

Theoremrerebi 13894 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 13895 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 13896 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)

Theoremimcji 13897 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴)

Theoremcjmulrcli 13898 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ       (𝐴 · (∗‘𝐴)) ∈ ℝ

Theoremcjmulvali 13899 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       (𝐴 · (∗‘𝐴)) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))

Theoremcjmulge0i 13900 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
𝐴 ∈ ℂ       0 ≤ (𝐴 · (∗‘𝐴))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
 Copyright terms: Public domain < Previous  Next >