HomeHome Intuitionistic Logic Explorer
Theorem List (p. 91 of 94)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexpnegapd 9001 Value of a complex number raised to a negative power. (Contributed by Jim Kingdon, 12-Jun-2020.)
 CC   &    #  0   &     N  ZZ   =>     ^ -u N  1  ^ N
 
Theoremexprecapd 9002 Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
 CC   &    #  0   &     N  ZZ   =>     1 
 ^ N  1  ^ N
 
Theoremexpp1zapd 9003 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
 CC   &    #  0   &     N  ZZ   =>     ^ N  +  1  ^ N  x.
 
Theoremexpm1apd 9004 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
 CC   &    #  0   &     N  ZZ   =>     ^ N  -  1  ^ N
 
Theoremexpsubapd 9005 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
 CC   &    #  0   &     N  ZZ   &     M  ZZ   =>     ^ M  -  N  ^ M  ^ N
 
Theoremsqmuld 9006 Distribution of square over multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 CC   &     CC   =>     x.  ^ 2  ^
 2  x.  ^ 2
 
Theoremsqdivapd 9007 Distribution of square over division. (Contributed by Jim Kingdon, 13-Jun-2020.)
 CC   &     CC   &    #  0   =>    
 ^ 2  ^
 2  ^ 2
 
Theoremexpdivapd 9008 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 13-Jun-2020.)
 CC   &     CC   &    #  0   &     N  NN0   =>    
 ^ N  ^ N 
 ^ N
 
Theoremmulexpd 9009 Positive integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016.)
 CC   &     CC   &     N  NN0   =>     x.  ^ N  ^ N  x.  ^ N
 
Theorem0expd 9010 Value of zero raised to a positive integer power. (Contributed by Mario Carneiro, 28-May-2016.)
 N  NN   =>    
 0 ^ N  0
 
Theoremreexpcld 9011 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     N  NN0   =>     ^ N 
 RR
 
Theoremexpge0d 9012 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     N  NN0   &     0  <_    =>     0  <_  ^ N
 
Theoremexpge1d 9013 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     N  NN0   &     1  <_    =>     1  <_  ^ N
 
Theoremnnsqcld 9014 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)
 NN   =>     ^ 2 
 NN
 
Theoremnnexpcld 9015 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
 NN   &     N  NN0   =>     ^ N 
 NN
 
Theoremnn0expcld 9016 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
 NN0   &     N  NN0   =>     ^ N 
 NN0
 
Theoremrpexpcld 9017 Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     N  ZZ   =>     ^ N  RR+
 
Theoremreexpclzapd 9018 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.)
 RR   &    #  0   &     N  ZZ   =>     ^ N 
 RR
 
Theoremresqcld 9019 Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     ^ 2 
 RR
 
Theoremsqge0d 9020 A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     0  <_  ^ 2
 
Theoremsqgt0apd 9021 The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.)
 RR   &    #  0   =>     0  <  ^ 2
 
Theoremleexp2ad 9022 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     1  <_    &     N  ZZ>= `  M   =>     ^ M  <_  ^ N
 
Theoremleexp2rd 9023 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     M  NN0   &     N  ZZ>= `  M   &     0  <_    &     <_  1   =>     ^ N  <_  ^ M
 
Theoremlt2sqd 9024 The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     0  <_    &     0  <_    =>     < 
 ^ 2  <  ^ 2
 
Theoremle2sqd 9025 The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     0  <_    &     0  <_    =>     <_ 
 ^ 2  <_  ^ 2
 
Theoremsq11d 9026 The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     0  <_    &     0  <_    &     ^ 2  ^ 2   =>   
 
3.6  Elementary real and complex functions
 
3.6.1  Real and imaginary parts; conjugate
 
Syntaxccj 9027 Extend class notation to include complex conjugate function.

 *
 
Syntaxcre 9028 Extend class notation to include real part of a complex number.

 Re
 
Syntaxcim 9029 Extend class notation to include imaginary part of a complex number.

 Im
 
Definitiondf-cj 9030* Define the complex conjugate function. See cjcli 9101 for its closure and cjval 9033 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 *  CC  |->  iota_  CC  +  RR  _i  x.  -  RR
 
Definitiondf-re 9031 Define a function whose value is the real part of a complex number. See reval 9037 for its value, recli 9099 for its closure, and replim 9047 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
 Re  CC  |->  +  * `
  2
 
Definitiondf-im 9032 Define a function whose value is the imaginary part of a complex number. See imval 9038 for its value, imcli 9100 for its closure, and replim 9047 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
 Im  CC  |->  Re
 `  _i
 
Theoremcjval 9033* The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
 CC  * `  iota_  CC  +  RR  _i  x.  -  RR
 
Theoremcjth 9034 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
 CC  +  * `  RR  _i  x.  -  * ` 
 RR
 
Theoremcjf 9035 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
 * : CC --> CC
 
Theoremcjcl 9036 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.)
 CC  * `  CC
 
Theoremreval 9037 The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 CC  Re `  +  * `  2
 
Theoremimval 9038 The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 CC  Im `  Re `  _i
 
Theoremimre 9039 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.)
 CC  Im `  Re `  -u _i  x.
 
Theoremreim 9040 The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 CC  Re `  Im `  _i  x.
 
Theoremrecl 9041 The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 CC  Re `  RR
 
Theoremimcl 9042 The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 CC  Im `  RR
 
Theoremref 9043 Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
 Re : CC --> RR
 
Theoremimf 9044 Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
 Im : CC --> RR
 
Theoremcrre 9045 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.)
 RR  RR  Re `  +  _i  x.
 
Theoremcrim 9046 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.)
 RR  RR  Im `  +  _i  x.
 
Theoremreplim 9047 Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
 CC  Re `  +  _i  x.  Im `
 
Theoremremim 9048 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.)
 CC  * `  Re `  -  _i  x.  Im `
 
Theoremreim0 9049 The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 RR  Im `  0
 
Theoremreim0b 9050 A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)
 CC  RR  Im `  0
 
Theoremrereb 9051 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.)
 CC  RR  Re `
 
Theoremmulreap 9052 A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.)
 CC  RR #  0  RR  x. 
 RR
 
Theoremrere 9053 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.)
 RR  Re `
 
Theoremcjreb 9054 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.)
 CC  RR  * `
 
Theoremrecj 9055 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
 CC  Re `  * `  Re `
 
Theoremreneg 9056 Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  Re `  -u  -u Re
 `
 
Theoremreadd 9057 Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  CC  Re `  +  Re `  +  Re `
 
Theoremresub 9058 Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
 CC  CC  Re `  -  Re ` 
 -  Re `
 
Theoremremullem 9059 Lemma for remul 9060, immul 9067, and cjmul 9073. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  CC  Re
 `  x.  Re `  x.  Re ` 
 -  Im
 `  x.  Im `  Im `  x.  Re `  x.  Im `  +  Im
 `  x.  Re `  * `  x.  * `  x.  * `
 
Theoremremul 9060 Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  CC  Re `  x.  Re `  x.  Re `  -  Im `  x.  Im `
 
Theoremremul2 9061 Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
 RR  CC  Re `  x.  x.  Re `
 
Theoremredivap 9062 Real part of a division. Related to remul2 9061. (Contributed by Jim Kingdon, 14-Jun-2020.)
 CC  RR #  0  Re `  Re `
 
Theoremimcj 9063 Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  Im `  * `  -u Im `
 
Theoremimneg 9064 The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  Im `  -u  -u Im
 `
 
Theoremimadd 9065 Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  CC  Im `  +  Im `  +  Im `
 
Theoremimsub 9066 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
 CC  CC  Im `  -  Im ` 
 -  Im `
 
Theoremimmul 9067 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  CC  Im `  x.  Re `  x.  Im `  +  Im `  x.  Re `
 
Theoremimmul2 9068 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
 RR  CC  Im `  x.  x.  Im `
 
Theoremimdivap 9069 Imaginary part of a division. Related to immul2 9068. (Contributed by Jim Kingdon, 14-Jun-2020.)
 CC  RR #  0  Im `  Im `
 
Theoremcjre 9070 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)
 RR  * `
 
Theoremcjcj 9071 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.)
 CC  * `  * `
 
Theoremcjadd 9072 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.)
 CC  CC  * `  +  * `  +  * `
 
Theoremcjmul 9073 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.)
 CC  CC  * `  x.  * `  x.  * `
 
Theoremipcnval 9074 Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  CC  Re `  x.  * `  Re `  x.  Re `  +  Im `  x.  Im `
 
Theoremcjmulrcl 9075 A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  x.  * `  RR
 
Theoremcjmulval 9076 A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  x.  * `  Re ` 
 ^ 2  +  Im `  ^ 2
 
Theoremcjmulge0 9077 A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  0  <_  x.  * `
 
Theoremcjneg 9078 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  * `  -u  -u * `
 
Theoremaddcj 9079 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.)
 CC  +  * `  2  x.  Re `
 
Theoremcjsub 9080 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
 CC  CC  * `  -  * ` 
 -  * `
 
Theoremcjexp 9081 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
 CC  N  NN0  * `  ^ N  * ` 
 ^ N
 
Theoremimval2 9082 The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)
 CC  Im `  -  * ` 
 2  x.  _i
 
Theoremre0 9083 The real part of zero. (Contributed by NM, 27-Jul-1999.)
 Re `  0  0
 
Theoremim0 9084 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
 Im `  0  0
 
Theoremre1 9085 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
 Re `  1  1
 
Theoremim1 9086 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
 Im `  1  0
 
Theoremrei 9087 The real part of  _i. (Contributed by Scott Fenton, 9-Jun-2006.)
 Re `  _i  0
 
Theoremimi 9088 The imaginary part of  _i. (Contributed by Scott Fenton, 9-Jun-2006.)
 Im `  _i  1
 
Theoremcj0 9089 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
 * `  0  0
 
Theoremcji 9090 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
 * `  _i  -u _i
 
Theoremcjreim 9091 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
 RR  RR  * `  +  _i  x. 
 -  _i  x.
 
Theoremcjreim2 9092 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.)
 RR  RR  * `  -  _i  x.  +  _i  x.
 
Theoremcj11 9093 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
 CC  CC  * `
  * `
 
Theoremcjap 9094 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
 CC  CC  * `
 #  * ` #
 
Theoremcjap0 9095 A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
 CC #  0  * `
 #  0
 
Theoremcjne0 9096 A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.)
 CC  =/=  0  * `  =/=  0
 
Theoremcjdivap 9097 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
 CC  CC #  0  * `  * `  * `
 
Theoremcnrecnv 9098* The inverse to the canonical bijection from  RR  X.  RR to  CC from cnref1o 8317. (Contributed by Mario Carneiro, 25-Aug-2014.)
 F  RR ,  RR  |->  +  _i  x.    =>     `' F  CC  |->  <. Re
 `  ,  Im ` 
 >.
 
Theoremrecli 9099 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
 CC   =>     Re `  RR
 
Theoremimcli 9100 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
 CC   =>     Im `  RR
    < Previous  Next >

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 9301-9381
  Copyright terms: Public domain < Previous  Next >