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Theorem List for Intuitionistic Logic Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremabstri 10501 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 10502 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 10503 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 10504 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 10505 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 10506* 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 10507 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 10508 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 10509 Lemma for fzomaxdif 10510. (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 10510 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 10511* Lemma for cau3 10512. (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 10512* 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 10385 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 10513* 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 10514* 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 10515 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 10516 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 10517 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 10518 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 10519 Square root of square. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( sqr `  ( A  x.  A ) )  =  A )
 
Theoremsqrtsqi 10520 Square root of square. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( sqr `  ( A ^ 2 ) )  =  A )
 
Theoremsqsqrti 10521 Square of square root. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( ( sqr `  A ) ^ 2
 )  =  A )
 
Theoremsqrtge0i 10522 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 10523 A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( abs `  A )  =  A )
 
Theoremabsnidi 10524 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 10525 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 10526 Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( abs `  A )  =  ( sqr `  ( A ^ 2
 ) )
 
Theoremsqrtpclii 10527 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 10528 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 10529 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 10530 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 10531 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 10532 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 10533 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 10534 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 10535 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 10536 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 10537 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( ( abs `  A ) ^ 2
 )  =  ( A  x.  ( * `  A ) )
 
Theoremabsvalsq2i 10538 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 10539 Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( abs `  A )  e.  RR
 
Theoremabsge0i 10540 Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  0  <_  ( abs `  A )
 
Theoremabsval2i 10541 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 10542 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 10543 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 10544 Absolute value of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( abs `  -u A )  =  ( abs `  A )
 
Theoremabscji 10545 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 10546 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 10547 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 10548 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 10549 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 10550 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 10551 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 10552 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 10553 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 10554 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 10555 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 10556 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 10557 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 10558 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 10559 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( abs `  A )  =  ( sqr `  ( A ^ 2 ) ) )
 
Theoremresqrtcld 10560 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 10561 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 10562 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( sqr `  ( A ^
 2 ) )  =  A )
 
Theoremsqrtge0d 10563 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 10564 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 10565 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 10566 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 10567 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 10568 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 10569 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 10570 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 10571 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 10572 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 10573 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 10574 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 10575 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 10576 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 ) ) ) )
 
Theoremicodiamlt 10577 Two elements in a half-open interval have separation strictly less than the difference between the endpoints. (Contributed by Stefan O'Rear, 12-Sep-2014.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,) B )  /\  D  e.  ( A [,) B ) ) )  ->  ( abs `  ( C  -  D ) )  < 
 ( B  -  A ) )
 
Theoremabscld 10578 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  A )  e. 
 RR )
 
Theoremabsvalsqd 10579 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 10580 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 10581 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  0  <_  ( abs `  A ) )
 
Theoremabsval2d 10582 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 10583 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 10584 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 )
 
Theoremabsrpclapd 10585 The absolute value of a complex number apart from zero is a positive real. (Contributed by Jim Kingdon, 13-Aug-2021.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  ( abs `  A )  e.  RR+ )
 
Theoremabsnegd 10586 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  -u A )  =  ( abs `  A ) )
 
Theoremabscjd 10587 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 10588 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 10589 Absolute value of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N ) )
 
Theoremabssubd 10590 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 10591 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 ) ) )
 
Theoremabsdivapd 10592 Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( abs `  ( A  /  B ) )  =  ( ( abs `  A )  /  ( abs `  B ) ) )
 
Theoremabstrid 10593 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 10594 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 10595 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 10596 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 10597 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 10598 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 )
 
Theoremqdenre 10599* The rational numbers are dense in 
RR: any real number can be approximated with arbitrary precision by a rational number. For order theoretic density, see qbtwnre 9633. (Contributed by BJ, 15-Oct-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  E. x  e.  QQ  ( abs `  ( x  -  A ) )  <  B )
 
3.7.5  The maximum of two real numbers
 
Theoremmaxcom 10600 The maximum of two reals is commutative. Lemma 3.9 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)
 |- 
 sup ( { A ,  B } ,  RR ,  <  )  =  sup ( { B ,  A } ,  RR ,  <  )
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