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Theorem List for Metamath Proof Explorer - 14901-15000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremshftidt 14901 Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V    β‡’   (𝐴 ∈ β„‚ β†’ ((𝐹 shift 0)β€˜π΄) = (πΉβ€˜π΄))
 
Theoremshftcan1 14902 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (((𝐹 shift 𝐴) shift -𝐴)β€˜π΅) = (πΉβ€˜π΅))
 
Theoremshftcan2 14903 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (((𝐹 shift -𝐴) shift 𝐴)β€˜π΅) = (πΉβ€˜π΅))
 
Theoremseqshft 14904 Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-Feb-2014.)
𝐹 ∈ V    β‡’   ((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ seq𝑀( + , (𝐹 shift 𝑁)) = (seq(𝑀 βˆ’ 𝑁)( + , 𝐹) shift 𝑁))
 
5.9.2  Signum (sgn or sign) function
 
Syntaxcsgn 14905 Extend class notation to include the Signum function.
class sgn
 
Definitiondf-sgn 14906 Signum function. We do not call it "sign", which is homophonic with "sine" (df-sin 15887). 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 15887. We define this over ℝ* (df-xr 11127) instead of ℝ so that it can accept +∞ and -∞. Note that df-psgn 19205 defines the sign of a permutation, which is different. Value shown in sgnval 14907. (Contributed by David A. Wheeler, 15-May-2015.)
sgn = (π‘₯ ∈ ℝ* ↦ if(π‘₯ = 0, 0, if(π‘₯ < 0, -1, 1)))
 
Theoremsgnval 14907 Value of the signum function. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴 ∈ ℝ* β†’ (sgnβ€˜π΄) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)))
 
Theoremsgn0 14908 The signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
(sgnβ€˜0) = 0
 
Theoremsgnp 14909 The signum of a positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) β†’ (sgnβ€˜π΄) = 1)
 
Theoremsgnrrp 14910 The signum of a positive real is 1. (Contributed by David A. Wheeler, 18-May-2015.)
(𝐴 ∈ ℝ+ β†’ (sgnβ€˜π΄) = 1)
 
Theoremsgn1 14911 The signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
(sgnβ€˜1) = 1
 
Theoremsgnpnf 14912 The signum of +∞ is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
(sgnβ€˜+∞) = 1
 
Theoremsgnn 14913 The signum of a negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.)
((𝐴 ∈ ℝ* ∧ 𝐴 < 0) β†’ (sgnβ€˜π΄) = -1)
 
Theoremsgnmnf 14914 The signum of -∞ is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
(sgnβ€˜-∞) = -1
 
5.9.3  Real and imaginary parts; conjugate
 
Syntaxccj 14915 Extend class notation to include complex conjugate function.
class βˆ—
 
Syntaxcre 14916 Extend class notation to include real part of a complex number.
class β„œ
 
Syntaxcim 14917 Extend class notation to include imaginary part of a complex number.
class β„‘
 
Definitiondf-cj 14918* Define the complex conjugate function. See cjcli 14988 for its closure and cjval 14921 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
βˆ— = (π‘₯ ∈ β„‚ ↦ (℩𝑦 ∈ β„‚ ((π‘₯ + 𝑦) ∈ ℝ ∧ (i Β· (π‘₯ βˆ’ 𝑦)) ∈ ℝ)))
 
Definitiondf-re 14919 Define a function whose value is the real part of a complex number. See reval 14925 for its value, recli 14986 for its closure, and replim 14935 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
β„œ = (π‘₯ ∈ β„‚ ↦ ((π‘₯ + (βˆ—β€˜π‘₯)) / 2))
 
Definitiondf-im 14920 Define a function whose value is the imaginary part of a complex number. See imval 14926 for its value, imcli 14987 for its closure, and replim 14935 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
β„‘ = (π‘₯ ∈ β„‚ ↦ (β„œβ€˜(π‘₯ / i)))
 
Theoremcjval 14921* The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ β„‚ β†’ (βˆ—β€˜π΄) = (β„©π‘₯ ∈ β„‚ ((𝐴 + π‘₯) ∈ ℝ ∧ (i Β· (𝐴 βˆ’ π‘₯)) ∈ ℝ)))
 
Theoremcjth 14922 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ β„‚ β†’ ((𝐴 + (βˆ—β€˜π΄)) ∈ ℝ ∧ (i Β· (𝐴 βˆ’ (βˆ—β€˜π΄))) ∈ ℝ))
 
Theoremcjf 14923 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
βˆ—:β„‚βŸΆβ„‚
 
Theoremcjcl 14924 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 ∈ β„‚ β†’ (βˆ—β€˜π΄) ∈ β„‚)
 
Theoremreval 14925 The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ β„‚ β†’ (β„œβ€˜π΄) = ((𝐴 + (βˆ—β€˜π΄)) / 2))
 
Theoremimval 14926 The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ β„‚ β†’ (β„‘β€˜π΄) = (β„œβ€˜(𝐴 / i)))
 
Theoremimre 14927 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 ∈ β„‚ β†’ (β„‘β€˜π΄) = (β„œβ€˜(-i Β· 𝐴)))
 
Theoremreim 14928 The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)
(𝐴 ∈ β„‚ β†’ (β„œβ€˜π΄) = (β„‘β€˜(i Β· 𝐴)))
 
Theoremrecl 14929 The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ β„‚ β†’ (β„œβ€˜π΄) ∈ ℝ)
 
Theoremimcl 14930 The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
(𝐴 ∈ β„‚ β†’ (β„‘β€˜π΄) ∈ ℝ)
 
Theoremref 14931 Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
β„œ:β„‚βŸΆβ„
 
Theoremimf 14932 Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
β„‘:β„‚βŸΆβ„
 
Theoremcrre 14933 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 ∧ 𝐡 ∈ ℝ) β†’ (β„œβ€˜(𝐴 + (i Β· 𝐡))) = 𝐴)
 
Theoremcrim 14934 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 ∧ 𝐡 ∈ ℝ) β†’ (β„‘β€˜(𝐴 + (i Β· 𝐡))) = 𝐡)
 
Theoremreplim 14935 Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
(𝐴 ∈ β„‚ β†’ 𝐴 = ((β„œβ€˜π΄) + (i Β· (β„‘β€˜π΄))))
 
Theoremremim 14936 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 ∈ β„‚ β†’ (βˆ—β€˜π΄) = ((β„œβ€˜π΄) βˆ’ (i Β· (β„‘β€˜π΄))))
 
Theoremreim0 14937 The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
(𝐴 ∈ ℝ β†’ (β„‘β€˜π΄) = 0)
 
Theoremreim0b 14938 A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)
(𝐴 ∈ β„‚ β†’ (𝐴 ∈ ℝ ↔ (β„‘β€˜π΄) = 0))
 
Theoremrereb 14939 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 ∈ β„‚ β†’ (𝐴 ∈ ℝ ↔ (β„œβ€˜π΄) = 𝐴))
 
Theoremmulre 14940 A product with a nonzero real multiplier is real iff the multiplicand is real. (Contributed by NM, 21-Aug-2008.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ ℝ ∧ 𝐡 β‰  0) β†’ (𝐴 ∈ ℝ ↔ (𝐡 Β· 𝐴) ∈ ℝ))
 
Theoremrere 14941 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 β†’ (β„œβ€˜π΄) = 𝐴)
 
Theoremcjreb 14942 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 ∈ β„‚ β†’ (𝐴 ∈ ℝ ↔ (βˆ—β€˜π΄) = 𝐴))
 
Theoremrecj 14943 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ β„‚ β†’ (β„œβ€˜(βˆ—β€˜π΄)) = (β„œβ€˜π΄))
 
Theoremreneg 14944 Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ β„‚ β†’ (β„œβ€˜-𝐴) = -(β„œβ€˜π΄))
 
Theoremreadd 14945 Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (β„œβ€˜(𝐴 + 𝐡)) = ((β„œβ€˜π΄) + (β„œβ€˜π΅)))
 
Theoremresub 14946 Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (β„œβ€˜(𝐴 βˆ’ 𝐡)) = ((β„œβ€˜π΄) βˆ’ (β„œβ€˜π΅)))
 
Theoremremullem 14947 Lemma for remul 14948, immul 14955, and cjmul 14961. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((β„œβ€˜(𝐴 Β· 𝐡)) = (((β„œβ€˜π΄) Β· (β„œβ€˜π΅)) βˆ’ ((β„‘β€˜π΄) Β· (β„‘β€˜π΅))) ∧ (β„‘β€˜(𝐴 Β· 𝐡)) = (((β„œβ€˜π΄) Β· (β„‘β€˜π΅)) + ((β„‘β€˜π΄) Β· (β„œβ€˜π΅))) ∧ (βˆ—β€˜(𝐴 Β· 𝐡)) = ((βˆ—β€˜π΄) Β· (βˆ—β€˜π΅))))
 
Theoremremul 14948 Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (β„œβ€˜(𝐴 Β· 𝐡)) = (((β„œβ€˜π΄) Β· (β„œβ€˜π΅)) βˆ’ ((β„‘β€˜π΄) Β· (β„‘β€˜π΅))))
 
Theoremremul2 14949 Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ β„‚) β†’ (β„œβ€˜(𝐴 Β· 𝐡)) = (𝐴 Β· (β„œβ€˜π΅)))
 
Theoremrediv 14950 Real part of a division. Related to remul2 14949. (Contributed by David A. Wheeler, 10-Jun-2015.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ ℝ ∧ 𝐡 β‰  0) β†’ (β„œβ€˜(𝐴 / 𝐡)) = ((β„œβ€˜π΄) / 𝐡))
 
Theoremimcj 14951 Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ β„‚ β†’ (β„‘β€˜(βˆ—β€˜π΄)) = -(β„‘β€˜π΄))
 
Theoremimneg 14952 The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ β„‚ β†’ (β„‘β€˜-𝐴) = -(β„‘β€˜π΄))
 
Theoremimadd 14953 Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (β„‘β€˜(𝐴 + 𝐡)) = ((β„‘β€˜π΄) + (β„‘β€˜π΅)))
 
Theoremimsub 14954 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (β„‘β€˜(𝐴 βˆ’ 𝐡)) = ((β„‘β€˜π΄) βˆ’ (β„‘β€˜π΅)))
 
Theoremimmul 14955 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (β„‘β€˜(𝐴 Β· 𝐡)) = (((β„œβ€˜π΄) Β· (β„‘β€˜π΅)) + ((β„‘β€˜π΄) Β· (β„œβ€˜π΅))))
 
Theoremimmul2 14956 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ β„‚) β†’ (β„‘β€˜(𝐴 Β· 𝐡)) = (𝐴 Β· (β„‘β€˜π΅)))
 
Theoremimdiv 14957 Imaginary part of a division. Related to immul2 14956. (Contributed by Mario Carneiro, 20-Jun-2015.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ ℝ ∧ 𝐡 β‰  0) β†’ (β„‘β€˜(𝐴 / 𝐡)) = ((β„‘β€˜π΄) / 𝐡))
 
Theoremcjre 14958 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)
(𝐴 ∈ ℝ β†’ (βˆ—β€˜π΄) = 𝐴)
 
Theoremcjcj 14959 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 ∈ β„‚ β†’ (βˆ—β€˜(βˆ—β€˜π΄)) = 𝐴)
 
Theoremcjadd 14960 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 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (βˆ—β€˜(𝐴 + 𝐡)) = ((βˆ—β€˜π΄) + (βˆ—β€˜π΅)))
 
Theoremcjmul 14961 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 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (βˆ—β€˜(𝐴 Β· 𝐡)) = ((βˆ—β€˜π΄) Β· (βˆ—β€˜π΅)))
 
Theoremipcnval 14962 Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (β„œβ€˜(𝐴 Β· (βˆ—β€˜π΅))) = (((β„œβ€˜π΄) Β· (β„œβ€˜π΅)) + ((β„‘β€˜π΄) Β· (β„‘β€˜π΅))))
 
Theoremcjmulrcl 14963 A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ β„‚ β†’ (𝐴 Β· (βˆ—β€˜π΄)) ∈ ℝ)
 
Theoremcjmulval 14964 A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ β„‚ β†’ (𝐴 Β· (βˆ—β€˜π΄)) = (((β„œβ€˜π΄)↑2) + ((β„‘β€˜π΄)↑2)))
 
Theoremcjmulge0 14965 A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ β„‚ β†’ 0 ≀ (𝐴 Β· (βˆ—β€˜π΄)))
 
Theoremcjneg 14966 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ β„‚ β†’ (βˆ—β€˜-𝐴) = -(βˆ—β€˜π΄))
 
Theoremaddcj 14967 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 ∈ β„‚ β†’ (𝐴 + (βˆ—β€˜π΄)) = (2 Β· (β„œβ€˜π΄)))
 
Theoremcjsub 14968 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (βˆ—β€˜(𝐴 βˆ’ 𝐡)) = ((βˆ—β€˜π΄) βˆ’ (βˆ—β€˜π΅)))
 
Theoremcjexp 14969 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„•0) β†’ (βˆ—β€˜(𝐴↑𝑁)) = ((βˆ—β€˜π΄)↑𝑁))
 
Theoremimval2 14970 The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)
(𝐴 ∈ β„‚ β†’ (β„‘β€˜π΄) = ((𝐴 βˆ’ (βˆ—β€˜π΄)) / (2 Β· i)))
 
Theoremre0 14971 The real part of zero. (Contributed by NM, 27-Jul-1999.)
(β„œβ€˜0) = 0
 
Theoremim0 14972 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
(β„‘β€˜0) = 0
 
Theoremre1 14973 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(β„œβ€˜1) = 1
 
Theoremim1 14974 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(β„‘β€˜1) = 0
 
Theoremrei 14975 The real part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(β„œβ€˜i) = 0
 
Theoremimi 14976 The imaginary part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(β„‘β€˜i) = 1
 
Theoremcj0 14977 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
(βˆ—β€˜0) = 0
 
Theoremcji 14978 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
(βˆ—β€˜i) = -i
 
Theoremcjreim 14979 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (βˆ—β€˜(𝐴 + (i Β· 𝐡))) = (𝐴 βˆ’ (i Β· 𝐡)))
 
Theoremcjreim2 14980 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 ∧ 𝐡 ∈ ℝ) β†’ (βˆ—β€˜(𝐴 βˆ’ (i Β· 𝐡))) = (𝐴 + (i Β· 𝐡)))
 
Theoremcj11 14981 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((βˆ—β€˜π΄) = (βˆ—β€˜π΅) ↔ 𝐴 = 𝐡))
 
Theoremcjne0 14982 A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.)
(𝐴 ∈ β„‚ β†’ (𝐴 β‰  0 ↔ (βˆ—β€˜π΄) β‰  0))
 
Theoremcjdiv 14983 Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐡 β‰  0) β†’ (βˆ—β€˜(𝐴 / 𝐡)) = ((βˆ—β€˜π΄) / (βˆ—β€˜π΅)))
 
Theoremcnrecnv 14984* The inverse to the canonical bijection from (ℝ Γ— ℝ) to β„‚ from cnref1o 12839. (Contributed by Mario Carneiro, 25-Aug-2014.)
𝐹 = (π‘₯ ∈ ℝ, 𝑦 ∈ ℝ ↦ (π‘₯ + (i Β· 𝑦)))    β‡’   β—‘𝐹 = (𝑧 ∈ β„‚ ↦ ⟨(β„œβ€˜π‘§), (β„‘β€˜π‘§)⟩)
 
Theoremsqeqd 14985 A deduction for showing two numbers whose squares are equal are themselves equal. (Contributed by Mario Carneiro, 3-Apr-2015.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ (𝐴↑2) = (𝐡↑2))    &   (πœ‘ β†’ 0 ≀ (β„œβ€˜π΄))    &   (πœ‘ β†’ 0 ≀ (β„œβ€˜π΅))    &   ((πœ‘ ∧ (β„œβ€˜π΄) = 0 ∧ (β„œβ€˜π΅) = 0) β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ 𝐴 = 𝐡)
 
Theoremrecli 14986 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
𝐴 ∈ β„‚    β‡’   (β„œβ€˜π΄) ∈ ℝ
 
Theoremimcli 14987 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
𝐴 ∈ β„‚    β‡’   (β„‘β€˜π΄) ∈ ℝ
 
Theoremcjcli 14988 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
𝐴 ∈ β„‚    β‡’   (βˆ—β€˜π΄) ∈ β„‚
 
Theoremreplimi 14989 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)
𝐴 ∈ β„‚    β‡’   π΄ = ((β„œβ€˜π΄) + (i Β· (β„‘β€˜π΄)))
 
Theoremcjcji 14990 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 ∈ β„‚    β‡’   (βˆ—β€˜(βˆ—β€˜π΄)) = 𝐴
 
Theoremreim0bi 14991 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)
𝐴 ∈ β„‚    β‡’   (𝐴 ∈ ℝ ↔ (β„‘β€˜π΄) = 0)
 
Theoremrerebi 14992 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)
𝐴 ∈ β„‚    β‡’   (𝐴 ∈ ℝ ↔ (β„œβ€˜π΄) = 𝐴)
 
Theoremcjrebi 14993 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 ∈ β„‚    β‡’   (𝐴 ∈ ℝ ↔ (βˆ—β€˜π΄) = 𝐴)
 
Theoremrecji 14994 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ β„‚    β‡’   (β„œβ€˜(βˆ—β€˜π΄)) = (β„œβ€˜π΄)
 
Theoremimcji 14995 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ β„‚    β‡’   (β„‘β€˜(βˆ—β€˜π΄)) = -(β„‘β€˜π΄)
 
Theoremcjmulrcli 14996 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
𝐴 ∈ β„‚    β‡’   (𝐴 Β· (βˆ—β€˜π΄)) ∈ ℝ
 
Theoremcjmulvali 14997 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ β„‚    β‡’   (𝐴 Β· (βˆ—β€˜π΄)) = (((β„œβ€˜π΄)↑2) + ((β„‘β€˜π΄)↑2))
 
Theoremcjmulge0i 14998 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
𝐴 ∈ β„‚    β‡’   0 ≀ (𝐴 Β· (βˆ—β€˜π΄))
 
Theoremrenegi 14999 Real part of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ β„‚    β‡’   (β„œβ€˜-𝐴) = -(β„œβ€˜π΄)
 
Theoremimnegi 15000 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ β„‚    β‡’   (β„‘β€˜-𝐴) = -(β„‘β€˜π΄)
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