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Theorem List for Intuitionistic Logic Explorer - 9901-10000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremleabs 9901 A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005.)
 |-  ( A  e.  RR  ->  A  <_  ( abs `  A ) )
 
Theoremqabsor 9902 The absolute value of a rational number is either that number or its negative. (Contributed by Jim Kingdon, 8-Nov-2021.)
 |-  ( A  e.  QQ  ->  ( ( abs `  A )  =  A  \/  ( abs `  A )  =  -u A ) )
 
Theoremqabsord 9903 The absolute value of a rational number is either that number or its negative. (Contributed by Jim Kingdon, 8-Nov-2021.)
 |-  ( ph  ->  A  e.  QQ )   =>    |-  ( ph  ->  (
 ( abs `  A )  =  A  \/  ( abs `  A )  =  -u A ) )
 
Theoremabsre 9904 Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)
 |-  ( A  e.  RR  ->  ( abs `  A )  =  ( sqr `  ( A ^ 2
 ) ) )
 
Theoremabsresq 9905 Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)
 |-  ( A  e.  RR  ->  ( ( abs `  A ) ^ 2 )  =  ( A ^ 2
 ) )
 
Theoremabsexp 9906 Absolute value of positive integer exponentiation. (Contributed by NM, 5-Jan-2006.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N ) )
 
Theoremabsexpzap 9907 Absolute value of integer exponentiation. (Contributed by Jim Kingdon, 11-Aug-2021.)
 |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N ) )
 
Theoremabssq 9908 Square can be moved in and out of absolute value. (Contributed by Scott Fenton, 18-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( A  e.  CC  ->  ( ( abs `  A ) ^ 2 )  =  ( abs `  ( A ^ 2 ) ) )
 
Theoremsqabs 9909 The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A ^ 2 )  =  ( B ^ 2
 ) 
 <->  ( abs `  A )  =  ( abs `  B ) ) )
 
Theoremabsrele 9910 The absolute value of a complex number is greater than or equal to the absolute value of its real part. (Contributed by NM, 1-Apr-2005.)
 |-  ( A  e.  CC  ->  ( abs `  ( Re `  A ) ) 
 <_  ( abs `  A ) )
 
Theoremabsimle 9911 The absolute value of a complex number is greater than or equal to the absolute value of its imaginary part. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( A  e.  CC  ->  ( abs `  ( Im `  A ) ) 
 <_  ( abs `  A ) )
 
Theoremnn0abscl 9912 The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.)
 |-  ( A  e.  ZZ  ->  ( abs `  A )  e.  NN0 )
 
Theoremzabscl 9913 The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)
 |-  ( A  e.  ZZ  ->  ( abs `  A )  e.  ZZ )
 
Theoremltabs 9914 A number which is less than its absolute value is negative. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  ( ( A  e.  RR  /\  A  <  ( abs `  A ) ) 
 ->  A  <  0 )
 
Theoremabslt 9915 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A )  <  B  <->  (
 -u B  <  A  /\  A  <  B ) ) )
 
Theoremabsle 9916 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A )  <_  B  <->  (
 -u B  <_  A  /\  A  <_  B )
 ) )
 
Theoremabssubap0 9917 If the absolute value of a complex number is less than a real, its difference from the real is apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  ( ( A  e.  CC  /\  B  e.  RR  /\  ( abs `  A )  <  B )  ->  ( B  -  A ) #  0 )
 
Theoremabssubne0 9918 If the absolute value of a complex number is less than a real, its difference from the real is nonzero. See also abssubap0 9917 which is the same with not equal changed to apart. (Contributed by NM, 2-Nov-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  RR  /\  ( abs `  A )  <  B )  ->  ( B  -  A )  =/=  0 )
 
Theoremabsdiflt 9919 The absolute value of a difference and 'less than' relation. (Contributed by Paul Chapman, 18-Sep-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( abs `  ( A  -  B ) )  <  C  <->  ( ( B  -  C )  <  A  /\  A  <  ( B  +  C )
 ) ) )
 
Theoremabsdifle 9920 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Paul Chapman, 18-Sep-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( abs `  ( A  -  B ) ) 
 <_  C  <->  ( ( B  -  C )  <_  A  /\  A  <_  ( B  +  C )
 ) ) )
 
Theoremelicc4abs 9921 Membership in a symmetric closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  (
 ( A  -  B ) [,] ( A  +  B ) )  <->  ( abs `  ( C  -  A ) ) 
 <_  B ) )
 
Theoremlenegsq 9922 Comparison to a nonnegative number based on comparison to squares. (Contributed by NM, 16-Jan-2006.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <_  B )  ->  ( ( A  <_  B 
 /\  -u A  <_  B ) 
 <->  ( A ^ 2
 )  <_  ( B ^ 2 ) ) )
 
Theoremreleabs 9923 The real part of a number is less than or equal to its absolute value. Proposition 10-3.7(d) of [Gleason] p. 133. (Contributed by NM, 1-Apr-2005.)
 |-  ( A  e.  CC  ->  ( Re `  A )  <_  ( abs `  A ) )
 
Theoremrecvalap 9924 Reciprocal expressed with a real denominator. (Contributed by Jim Kingdon, 13-Aug-2021.)
 |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  ( 1  /  A )  =  ( ( * `  A )  /  ( ( abs `  A ) ^ 2 ) ) )
 
Theoremabsidm 9925 The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)
 |-  ( A  e.  CC  ->  ( abs `  ( abs `  A ) )  =  ( abs `  A ) )
 
Theoremabsgt0ap 9926 The absolute value of a number apart from zero is positive. (Contributed by Jim Kingdon, 13-Aug-2021.)
 |-  ( A  e.  CC  ->  ( A #  0  <->  0  <  ( abs `  A ) ) )
 
Theoremnnabscl 9927 The absolute value of a nonzero integer is a positive integer. (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ( N  e.  ZZ  /\  N  =/=  0
 )  ->  ( abs `  N )  e.  NN )
 
Theoremabssub 9928 Swapping order of subtraction doesn't change the absolute value. (Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  -  B ) )  =  ( abs `  ( B  -  A ) ) )
 
Theoremabssubge0 9929 Absolute value of a nonnegative difference. (Contributed by NM, 14-Feb-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( abs `  ( B  -  A ) )  =  ( B  -  A ) )
 
Theoremabssuble0 9930 Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( abs `  ( A  -  B ) )  =  ( B  -  A ) )
 
Theoremabstri 9931 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by NM, 7-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  +  B )
 )  <_  ( ( abs `  A )  +  ( abs `  B )
 ) )
 
Theoremabs3dif 9932 Absolute value of differences around common element. (Contributed by FL, 9-Oct-2006.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( abs `  ( A  -  B ) ) 
 <_  ( ( abs `  ( A  -  C ) )  +  ( abs `  ( C  -  B ) ) ) )
 
Theoremabs2dif 9933 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  A )  -  ( abs `  B ) ) 
 <_  ( abs `  ( A  -  B ) ) )
 
Theoremabs2dif2 9934 Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  -  B ) ) 
 <_  ( ( abs `  A )  +  ( abs `  B ) ) )
 
Theoremabs2difabs 9935 Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  (
 ( abs `  A )  -  ( abs `  B ) ) )  <_  ( abs `  ( A  -  B ) ) )
 
Theoremrecan 9936* Cancellation law involving the real part of a complex number. (Contributed by NM, 12-May-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A. x  e.  CC  ( Re `  ( x  x.  A ) )  =  ( Re `  ( x  x.  B ) )  <->  A  =  B ) )
 
Theoremabsf 9937 Mapping domain and codomain of the absolute value function. (Contributed by NM, 30-Aug-2007.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |- 
 abs : CC --> RR
 
Theoremabs3lem 9938 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  ->  ( ( ( abs `  ( A  -  C ) )  <  ( D 
 /  2 )  /\  ( abs `  ( C  -  B ) )  < 
 ( D  /  2
 ) )  ->  ( abs `  ( A  -  B ) )  <  D ) )
 
Theoremfzomaxdiflem 9939 Lemma for fzomaxdif 9940. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( ( ( A  e.  ( C..^ D )  /\  B  e.  ( C..^ D ) )  /\  A  <_  B )  ->  ( abs `  ( B  -  A ) )  e.  ( 0..^ ( D  -  C ) ) )
 
Theoremfzomaxdif 9940 A bound on the separation of two points in a half-open range. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( ( A  e.  ( C..^ D )  /\  B  e.  ( C..^ D ) )  ->  ( abs `  ( A  -  B ) )  e.  ( 0..^ ( D  -  C ) ) )
 
Theoremcau3lem 9941* Lemma for cau3 9942. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 1-May-2014.)
 |-  Z  C_  ZZ   &    |-  ( ta  ->  ps )   &    |-  ( ( F `
  k )  =  ( F `  j
 )  ->  ( ps  <->  ch ) )   &    |-  ( ( F `
  k )  =  ( F `  m )  ->  ( ps  <->  th ) )   &    |-  (
 ( ph  /\  ch  /\  ps )  ->  ( G `  ( ( F `  j ) D ( F `  k ) ) )  =  ( G `  ( ( F `  k ) D ( F `  j ) ) ) )   &    |-  ( ( ph  /\ 
 th  /\  ch )  ->  ( G `  (
 ( F `  m ) D ( F `  j ) ) )  =  ( G `  ( ( F `  j ) D ( F `  m ) ) ) )   &    |-  (
 ( ph  /\  ( ps 
 /\  th )  /\  ( ch  /\  x  e.  RR ) )  ->  ( ( ( G `  (
 ( F `  k
 ) D ( F `
  j ) ) )  <  ( x 
 /  2 )  /\  ( G `  ( ( F `  j ) D ( F `  m ) ) )  <  ( x  / 
 2 ) )  ->  ( G `  ( ( F `  k ) D ( F `  m ) ) )  <  x ) )   =>    |-  ( ph  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( ta  /\  ( G `  ( ( F `
  k ) D ( F `  j
 ) ) )  < 
 x )  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( ta 
 /\  A. m  e.  ( ZZ>=
 `  k ) ( G `  ( ( F `  k ) D ( F `  m ) ) )  <  x ) ) )
 
Theoremcau3 9942* Convert between three-quantifier and four-quantifier versions of the Cauchy criterion. (In particular, the four-quantifier version has no occurrence of  j in the assertion, so it can be used with rexanuz 9815 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( ( F `  k )  e.  CC  /\  ( abs `  ( ( F `
  k )  -  ( F `  j ) ) )  <  x ) 
 <-> 
 A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( ( F `  k )  e.  CC  /\  A. m  e.  ( ZZ>= `  k ) ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x ) )
 
Theoremcau4 9943* Change the base of a Cauchy criterion. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  N )   =>    |-  ( N  e.  Z  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( ( F `  k )  e.  CC  /\  ( abs `  (
 ( F `  k
 )  -  ( F `
  j ) ) )  <  x )  <->  A. x  e.  RR+  E. j  e.  W  A. k  e.  ( ZZ>= `  j )
 ( ( F `  k )  e.  CC  /\  ( abs `  (
 ( F `  k
 )  -  ( F `
  j ) ) )  <  x ) ) )
 
Theoremcaubnd2 9944* A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( ( F `  k )  e.  CC  /\  ( abs `  ( ( F `
  k )  -  ( F `  j ) ) )  <  x )  ->  E. y  e.  RR  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( F `  k
 ) )  <  y
 )
 
Theoremamgm2 9945 Arithmetic-geometric mean inequality for  n  =  2. (Contributed by Mario Carneiro, 2-Jul-2014.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( sqr `  ( A  x.  B ) )  <_  ( ( A  +  B )  /  2
 ) )
 
Theoremsqrtthi 9946 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( ( sqr `  A )  x.  ( sqr `  A ) )  =  A )
 
Theoremsqrtcli 9947 The square root of a nonnegative real is a real. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( sqr `  A )  e.  RR )
 
Theoremsqrtgt0i 9948 The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)
 |-  A  e.  RR   =>    |-  ( 0  <  A  ->  0  <  ( sqr `  A ) )
 
Theoremsqrtmsqi 9949 Square root of square. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( sqr `  ( A  x.  A ) )  =  A )
 
Theoremsqrtsqi 9950 Square root of square. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( sqr `  ( A ^ 2 ) )  =  A )
 
Theoremsqsqrti 9951 Square of square root. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( ( sqr `  A ) ^ 2
 )  =  A )
 
Theoremsqrtge0i 9952 The square root of a nonnegative real is nonnegative. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  0  <_  ( sqr `  A ) )
 
Theoremabsidi 9953 A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( abs `  A )  =  A )
 
Theoremabsnidi 9954 A negative number is the negative of its own absolute value. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( A  <_  0 
 ->  ( abs `  A )  =  -u A )
 
Theoremleabsi 9955 A real number is less than or equal to its absolute value. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  A  <_  ( abs `  A )
 
Theoremabsrei 9956 Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( abs `  A )  =  ( sqr `  ( A ^ 2
 ) )
 
Theoremsqrtpclii 9957 The square root of a positive real is a real. (Contributed by Mario Carneiro, 6-Sep-2013.)
 |-  A  e.  RR   &    |-  0  <  A   =>    |-  ( sqr `  A )  e.  RR
 
Theoremsqrtgt0ii 9958 The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)
 |-  A  e.  RR   &    |-  0  <  A   =>    |-  0  <  ( sqr `  A )
 
Theoremsqrt11i 9959 The square root function is one-to-one. (Contributed by NM, 27-Jul-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( ( sqr `  A )  =  ( sqr `  B )  <->  A  =  B ) )
 
Theoremsqrtmuli 9960 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( sqr `  ( A  x.  B ) )  =  ( ( sqr `  A )  x.  ( sqr `  B ) ) )
 
Theoremsqrtmulii 9961 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  0  <_  A   &    |-  0  <_  B   =>    |-  ( sqr `  ( A  x.  B ) )  =  ( ( sqr `  A )  x.  ( sqr `  B ) )
 
Theoremsqrtmsq2i 9962 Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( ( sqr `  A )  =  B  <->  A  =  ( B  x.  B ) ) )
 
Theoremsqrtlei 9963 Square root is monotonic. (Contributed by NM, 3-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( A  <_  B  <-> 
 ( sqr `  A )  <_  ( sqr `  B ) ) )
 
Theoremsqrtlti 9964 Square root is strictly monotonic. (Contributed by Roy F. Longton, 8-Aug-2005.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( A  <  B  <-> 
 ( sqr `  A )  <  ( sqr `  B ) ) )
 
Theoremabslti 9965 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( abs `  A )  <  B  <->  ( -u B  <  A  /\  A  <  B ) )
 
Theoremabslei 9966 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( abs `  A )  <_  B  <->  ( -u B  <_  A  /\  A  <_  B ) )
 
Theoremabsvalsqi 9967 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( ( abs `  A ) ^ 2
 )  =  ( A  x.  ( * `  A ) )
 
Theoremabsvalsq2i 9968 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( ( abs `  A ) ^ 2
 )  =  ( ( ( Re `  A ) ^ 2 )  +  ( ( Im `  A ) ^ 2
 ) )
 
Theoremabscli 9969 Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( abs `  A )  e.  RR
 
Theoremabsge0i 9970 Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  0  <_  ( abs `  A )
 
Theoremabsval2i 9971 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( abs `  A )  =  ( sqr `  ( ( ( Re
 `  A ) ^
 2 )  +  (
 ( Im `  A ) ^ 2 ) ) )
 
Theoremabs00i 9972 The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   =>    |-  ( ( abs `  A )  =  0  <->  A  =  0 )
 
Theoremabsgt0api 9973 The absolute value of a nonzero number is positive. Remark in [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( A #  0  <->  0  <  ( abs `  A ) )
 
Theoremabsnegi 9974 Absolute value of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( abs `  -u A )  =  ( abs `  A )
 
Theoremabscji 9975 The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( abs `  ( * `  A ) )  =  ( abs `  A )
 
Theoremreleabsi 9976 The real part of a number is less than or equal to its absolute value. Proposition 10-3.7(d) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( Re `  A )  <_  ( abs `  A )
 
Theoremabssubi 9977 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( abs `  ( A  -  B ) )  =  ( abs `  ( B  -  A ) )
 
Theoremabsmuli 9978 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 1-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( abs `  ( A  x.  B ) )  =  ( ( abs `  A )  x.  ( abs `  B ) )
 
Theoremsqabsaddi 9979 Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( abs `  ( A  +  B )
 ) ^ 2 )  =  ( ( ( ( abs `  A ) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  +  ( 2  x.  ( Re `  ( A  x.  ( * `  B ) ) ) ) )
 
Theoremsqabssubi 9980 Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-Jan-2007.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( abs `  ( A  -  B ) ) ^ 2 )  =  ( ( ( ( abs `  A ) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  -  ( 2  x.  ( Re `  ( A  x.  ( * `  B ) ) ) ) )
 
Theoremabsdivapzi 9981 Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( abs `  ( A  /  B ) )  =  ( ( abs `  A )  /  ( abs `  B ) ) )
 
Theoremabstrii 9982 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. This is Metamath 100 proof #91. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( abs `  ( A  +  B )
 )  <_  ( ( abs `  A )  +  ( abs `  B )
 )
 
Theoremabs3difi 9983 Absolute value of differences around common element. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( abs `  ( A  -  B ) )  <_  ( ( abs `  ( A  -  C ) )  +  ( abs `  ( C  -  B ) ) )
 
Theoremabs3lemi 9984 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  RR   =>    |-  ( ( ( abs `  ( A  -  C ) )  <  ( D 
 /  2 )  /\  ( abs `  ( C  -  B ) )  < 
 ( D  /  2
 ) )  ->  ( abs `  ( A  -  B ) )  <  D )
 
Theoremrpsqrtcld 9985 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( sqr `  A )  e.  RR+ )
 
Theoremsqrtgt0d 9986 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  0  <  ( sqr `  A ) )
 
Theoremabsnidd 9987 A negative number is the negative of its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A 
 <_  0 )   =>    |-  ( ph  ->  ( abs `  A )  =  -u A )
 
Theoremleabsd 9988 A real number is less than or equal to its absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  <_  ( abs `  A ) )
 
Theoremabsred 9989 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( abs `  A )  =  ( sqr `  ( A ^ 2 ) ) )
 
Theoremresqrtcld 9990 The square root of a nonnegative real is a real. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( sqr `  A )  e. 
 RR )
 
Theoremsqrtmsqd 9991 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( sqr `  ( A  x.  A ) )  =  A )
 
Theoremsqrtsqd 9992 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( sqr `  ( A ^
 2 ) )  =  A )
 
Theoremsqrtge0d 9993 The square root of a nonnegative real is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  0  <_  ( sqr `  A ) )
 
Theoremabsidd 9994 A nonnegative number is its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( abs `  A )  =  A )
 
Theoremsqrtdivd 9995 Square root distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( sqr `  ( A  /  B ) )  =  ( ( sqr `  A )  /  ( sqr `  B ) ) )
 
Theoremsqrtmuld 9996 Square root distributes over multiplication. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  ( sqr `  ( A  x.  B ) )  =  ( ( sqr `  A )  x.  ( sqr `  B ) ) )
 
Theoremsqrtsq2d 9997 Relationship between square root and squares. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  (
 ( sqr `  A )  =  B  <->  A  =  ( B ^ 2 ) ) )
 
Theoremsqrtled 9998 Square root is monotonic. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( sqr `  A )  <_  ( sqr `  B ) ) )
 
Theoremsqrtltd 9999 Square root is strictly monotonic. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  ( A  <  B  <->  ( sqr `  A )  <  ( sqr `  B ) ) )
 
Theoremsqr11d 10000 The square root function is one-to-one. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   &    |-  ( ph  ->  ( sqr `  A )  =  ( sqr `  B ) )   =>    |-  ( ph  ->  A  =  B )
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