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Theorem List for Metamath Proof Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremamgm2 12101 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
 ) )
 
Theoremsqrthi 12102 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 )
 
Theoremsqrcli 12103 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 )
 
Theoremsqrgt0i 12104 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 ) )
 
Theoremsqrmsqi 12105 Square root of square. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( sqr `  ( A  x.  A ) )  =  A )
 
Theoremsqrsqi 12106 Square root of square. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( sqr `  ( A ^ 2 ) )  =  A )
 
Theoremsqsqri 12107 Square of square root. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( ( sqr `  A ) ^ 2
 )  =  A )
 
Theoremsqrge0i 12108 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 12109 A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( abs `  A )  =  A )
 
Theoremabsnidi 12110 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 12111 A real number is less than or equal to its absolute value. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  A  <_  ( abs `  A )
 
Theoremabsori 12112 The absolute value of a real number is either that number or its negative. (Contributed by NM, 30-Sep-1999.)
 |-  A  e.  RR   =>    |-  ( ( abs `  A )  =  A  \/  ( abs `  A )  =  -u A )
 
Theoremabsrei 12113 Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( abs `  A )  =  ( sqr `  ( A ^ 2
 ) )
 
Theoremsqrpclii 12114 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
 
Theoremsqrgt0ii 12115 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 )
 
Theoremsqr11i 12116 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 ) )
 
Theoremsqrmuli 12117 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 ) ) )
 
Theoremsqrmulii 12118 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 ) )
 
Theoremsqrmsq2i 12119 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 ) ) )
 
Theoremsqrlei 12120 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 ) ) )
 
Theoremsqrlti 12121 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 12122 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 12123 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 12124 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( ( abs `  A ) ^ 2
 )  =  ( A  x.  ( * `  A ) )
 
Theoremabsvalsq2i 12125 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 12126 Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( abs `  A )  e.  RR
 
Theoremabsge0i 12127 Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  0  <_  ( abs `  A )
 
Theoremabsval2i 12128 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 12129 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 )
 
Theoremabsgt0i 12130 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 12131 Absolute value of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( abs `  -u A )  =  ( abs `  A )
 
Theoremabscji 12132 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 12133 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 12134 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 12135 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 12136 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 12137 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 ) ) ) ) )
 
Theoremabsdivzi 12138 Absolute value distributes over division. (Contributed by NM, 26-Mar-2005.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B  =/=  0  ->  ( abs `  ( A  /  B ) )  =  ( ( abs `  A )  /  ( abs `  B ) ) )
 
Theoremabstrii 12139 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( abs `  ( A  +  B )
 )  <_  ( ( abs `  A )  +  ( abs `  B )
 )
 
Theoremabs3difi 12140 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 12141 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 )
 
Theoremrpsqrcld 12142 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+ )
 
Theoremsqrgt0d 12143 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 12144 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 12145 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 ) )
 
Theoremabsord 12146 The absolute value of a real number is either that number or its negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  (
 ( abs `  A )  =  A  \/  ( abs `  A )  =  -u A ) )
 
Theoremabsred 12147 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( abs `  A )  =  ( sqr `  ( A ^ 2 ) ) )
 
Theoremresqrcld 12148 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 )
 
Theoremsqrmsqd 12149 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( sqr `  ( A  x.  A ) )  =  A )
 
Theoremsqrsqd 12150 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( sqr `  ( A ^
 2 ) )  =  A )
 
Theoremsqrge0d 12151 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 ) )
 
Theoremsqrnegd 12152 The square root of a negative number. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( sqr `  -u A )  =  ( _i  x.  ( sqr `  A ) ) )
 
Theoremabsidd 12153 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 )
 
Theoremsqrdivd 12154 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 ) ) )
 
Theoremsqrmuld 12155 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 ) ) )
 
Theoremsqrsq2d 12156 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 ) ) )
 
Theoremsqrled 12157 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 ) ) )
 
Theoremsqrltd 12158 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 12159 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 )
 
Theoremabsltd 12160 Absolute value and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( ( abs `  A )  <  B  <->  ( -u B  <  A  /\  A  <  B ) ) )
 
Theoremabsled 12161 Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( ( abs `  A )  <_  B  <->  ( -u B  <_  A  /\  A  <_  B ) ) )
 
Theoremabssubge0d 12162 Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( abs `  ( B  -  A ) )  =  ( B  -  A ) )
 
Theoremabssuble0d 12163 Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  =  ( B  -  A ) )
 
Theoremabsdifltd 12164 The absolute value of a difference and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( abs `  ( A  -  B ) )  <  C 
 <->  ( ( B  -  C )  <  A  /\  A  <  ( B  +  C ) ) ) )
 
Theoremabsdifled 12165 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( abs `  ( A  -  B ) )  <_  C 
 <->  ( ( B  -  C )  <_  A  /\  A  <_  ( B  +  C ) ) ) )
 
Theoremabscld 12166 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  A )  e. 
 RR )
 
Theoremsqrcld 12167 Closure of the square root function over the complexes. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( sqr `  A )  e. 
 CC )
 
Theoremsqrrege0d 12168 The real part of the square root function is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  0  <_  ( Re `  ( sqr `  A ) ) )
 
Theoremsqsqrd 12169 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( sqr `  A ) ^ 2 )  =  A )
 
Theoremmsqsqrd 12170 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( sqr `  A )  x.  ( sqr `  A ) )  =  A )
 
Theoremsqr00d 12171 A square root is zero iff its argument is 0. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( sqr `  A )  =  0 )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremabsvalsqd 12172 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( abs `  A ) ^ 2 )  =  ( A  x.  ( * `  A ) ) )
 
Theoremabsvalsq2d 12173 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( abs `  A ) ^ 2 )  =  ( ( ( Re
 `  A ) ^
 2 )  +  (
 ( Im `  A ) ^ 2 ) ) )
 
Theoremabsge0d 12174 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  0  <_  ( abs `  A ) )
 
Theoremabsval2d 12175 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  A )  =  ( sqr `  (
 ( ( Re `  A ) ^ 2
 )  +  ( ( Im `  A ) ^ 2 ) ) ) )
 
Theoremabs00d 12176 The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  =  0 )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremabsne0d 12177 The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( abs `  A )  =/=  0 )
 
Theoremabsrpcld 12178 The absolute value of a nonzero number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( abs `  A )  e.  RR+ )
 
Theoremabsnegd 12179 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  -u A )  =  ( abs `  A ) )
 
Theoremabscjd 12180 The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( * `  A ) )  =  ( abs `  A ) )
 
Theoremreleabsd 12181 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 Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  A )  <_  ( abs `  A )
 )
 
Theoremabsexpd 12182 Absolute value of natural number exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N ) )
 
Theoremabssubd 12183 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  =  ( abs `  ( B  -  A ) ) )
 
Theoremabsmuld 12184 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  x.  B ) )  =  ( ( abs `  A )  x.  ( abs `  B ) ) )
 
Theoremabsdivd 12185 Absolute value distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  ( abs `  ( A  /  B ) )  =  ( ( abs `  A )  /  ( abs `  B ) ) )
 
Theoremabstrid 12186 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  +  B ) )  <_  ( ( abs `  A )  +  ( abs `  B ) ) )
 
Theoremabs2difd 12187 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( abs `  A )  -  ( abs `  B ) )  <_  ( abs `  ( A  -  B ) ) )
 
Theoremabs2dif2d 12188 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  <_  ( ( abs `  A )  +  ( abs `  B ) ) )
 
Theoremabs2difabsd 12189 Absolute value of difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( ( abs `  A )  -  ( abs `  B )
 ) )  <_  ( abs `  ( A  -  B ) ) )
 
Theoremabs3difd 12190 Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  <_  ( ( abs `  ( A  -  C ) )  +  ( abs `  ( C  -  B ) ) ) )
 
Theoremabs3lemd 12191 Lemma involving absolute value of differences. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  ( abs `  ( A  -  C ) )  < 
 ( D  /  2
 ) )   &    |-  ( ph  ->  ( abs `  ( C  -  B ) )  < 
 ( D  /  2
 ) )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  <  D )
 
5.8  Elementary limits and convergence
 
5.8.1  Superior limit (lim sup)
 
Syntaxclsp 12192 Extend class notation to include the limsup function.
 class  limsup
 
Definitiondf-limsup 12193* Define the superior limit of an infinite sequence of extended real numbers. Definition 12-4.1 of [Gleason] p. 175. See limsupval 12196 for its value. (Contributed by NM, 26-Oct-2005.)
 |-  limsup  =  ( x  e. 
 _V  |->  sup ( ran  (
 k  e.  RR  |->  sup ( ( ( x
 " ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 ) ,  RR* ,  `'  <  ) )
 
Theoremlimsupgord 12194 Ordering property of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  sup ( ( ( F " ( B [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) 
 <_  sup ( ( ( F " ( A [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
 
Theoremlimsupcl 12195 Closure of the superior limit. (Contributed by NM, 26-Oct-2005.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  ( F  e.  V  ->  ( limsup `  F )  e.  RR* )
 
Theoremlimsupval 12196* The superior limit of an infinite sequence  F of extended real numbers, which is the infimum (indicated by  `'  <) of the set of suprema of all upper infinite subsequences of  F. Definition 12-4.1 of [Gleason] p. 175. (Contributed by NM, 26-Oct-2005.) (Revised by Mario Carneiro, 5-Sep-2014.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  ( F  e.  V  ->  ( limsup `  F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
 
Theoremlimsupgf 12197* Closure of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  G : RR --> RR*
 
Theoremlimsupgval 12198* Value of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  ( M  e.  RR  ->  ( G `  M )  =  sup ( ( ( F " ( M [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
 
Theoremlimsupgle 12199* The defining property of the superior limit function. (Contributed by Mario Carneiro, 5-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  ( ( ( B 
 C_  RR  /\  F : B
 --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( G `  C )  <_  A 
 <-> 
 A. j  e.  B  ( C  <_  j  ->  ( F `  j ) 
 <_  A ) ) )
 
Theoremlimsuple 12200* The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  ->  ( A  <_  ( limsup `
  F )  <->  A. j  e.  RR  A  <_  ( G `  j ) ) )
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