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Theorem List for Metamath Proof Explorer - 11801-11900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremabsdiflt 11801 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 11802 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 11803 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 11804 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 11805 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 ) )
 
Theoremrecval 11806 Reciprocal expressed with a real denominator. (Contributed by Mario Carneiro, 1-Apr-2015.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( 1  /  A )  =  ( ( * `  A )  /  ( ( abs `  A ) ^ 2
 ) ) )
 
Theoremabsidm 11807 The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)
 |-  ( A  e.  CC  ->  ( abs `  ( abs `  A ) )  =  ( abs `  A ) )
 
Theoremabsgt0 11808 The absolute value of a nonzero number is positive. (Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( A  e.  CC  ->  ( A  =/=  0  <->  0  <  ( abs `  A ) ) )
 
Theoremnnabscl 11809 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 11810 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 11811 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 11812 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 ) )
 
Theoremabsmax 11813 The maximum of two numbers using absolute value. (Contributed by NM, 7-Aug-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  if ( A 
 <_  B ,  B ,  A )  =  (
 ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) ) 
 /  2 ) )
 
Theoremabstri 11814 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 11815 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 11816 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 11817 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 11818 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 ) ) )
 
Theoremabs1m 11819* For any complex number, there exists a unit-magnitude multiplier that produces its absolute value. Part of proof of Theorem 13-2.12 of [Gleason] p. 195. (Contributed by NM, 26-Mar-2005.)
 |-  ( A  e.  CC  ->  E. x  e.  CC  ( ( abs `  x )  =  1  /\  ( abs `  A )  =  ( x  x.  A ) ) )
 
Theoremrecan 11820* 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 11821 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 11822 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 ) )
 
Theoremabslem2 11823 Lemma involving absolute values. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( (
 ( * `  ( A  /  ( abs `  A ) ) )  x.  A )  +  (
 ( A  /  ( abs `  A ) )  x.  ( * `  A ) ) )  =  ( 2  x.  ( abs `  A ) ) )
 
Theoremrddif 11824 The difference between a real number and its nearest integer is less than or equal to one half. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Sep-2015.)
 |-  ( A  e.  RR  ->  ( abs `  (
 ( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A ) )  <_  ( 1  /  2 ) )
 
Theoremabsrdbnd 11825 Bound on the absolute value of a real number rounded to the nearest integer. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Sep-2015.)
 |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  ( 1  /  2
 ) ) ) ) 
 <_  ( ( |_ `  ( abs `  A ) )  +  1 ) )
 
Theoremfzomaxdiflem 11826 Lemma for fzomaxdif 11827. (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 11827 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 ) ) )
 
Theoremuzin2 11828 The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)
 |-  ( ( A  e.  ran  ZZ>= 
 /\  B  e.  ran  ZZ>= )  ->  ( A  i^i  B )  e.  ran  ZZ>= )
 
Theoremrexanuz 11829* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)
 |-  ( E. j  e. 
 ZZ  A. k  e.  ( ZZ>=
 `  j ) (
 ph  /\  ps )  <->  ( E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ph  /\  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ps ) )
 
Theoremrexanre 11830* Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.)
 |-  ( A  C_  RR  ->  ( E. j  e. 
 RR  A. k  e.  A  ( j  <_  k  ->  ( ph  /\  ps )
 ) 
 <->  ( E. j  e. 
 RR  A. k  e.  A  ( j  <_  k  ->  ph )  /\  E. j  e.  RR  A. k  e.  A  ( j  <_  k  ->  ps ) ) ) )
 
Theoremrexfiuz 11831* Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  ( A  e.  Fin  ->  ( E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) A. n  e.  A  ph  <->  A. n  e.  A  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ph )
 )
 
Theoremrexuz3 11832* Rextrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ph 
 <-> 
 E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ph )
 )
 
Theoremrexanuz2 11833* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( ph  /\  ps )  <->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ph  /\  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ps ) )
 
Theoremr19.29uz 11834* A version of 19.29 1583 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( A. k  e.  Z  ph  /\  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ps )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( ph  /\  ps )
 )
 
Theoremr19.2uz 11835* A version of r19.2z 3543 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ph  ->  E. k  e.  Z  ph )
 
Theoremrexuzre 11836* Convert an upper real quantifier to an upper integer quantifier. (Contributed by Mario Carneiro, 7-May-2016.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ph 
 <-> 
 E. j  e.  RR  A. k  e.  Z  ( j  <_  k  ->  ph ) ) )
 
Theoremrexico 11837* Rextrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  ( ( A  C_  RR  /\  B  e.  RR )  ->  ( E. j  e.  ( B [,)  +oo ) A. k  e.  A  ( j  <_  k  ->  ph )  <->  E. j  e.  RR  A. k  e.  A  ( j  <_  k  ->  ph ) ) )
 
Theoremcau3lem 11838* Lemma for cau3 11839. (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 11839* Convert between three-quantifier and four-quantifier versions of the Cauchy criterion. (In particular, the four-quantifier version has no occurence of  j in the assertion, so it can be used with rexanuz 11829 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 11840* 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 11841* 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
 )
 
Theoremcaubnd 11842* A Cauchy sequence of complex numbers is bounded. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 14-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( A. k  e.  Z  ( F `  k )  e.  CC  /\ 
 A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( F `  k )  -  ( F `  j ) ) )  <  x ) 
 ->  E. y  e.  RR  A. k  e.  Z  ( abs `  ( F `  k ) )  < 
 y )
 
Theoremsqreulem 11843 Lemma for sqreu 11844: write a general complex square root in terms of the square root function over nonnegative reals. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  B  =  ( ( sqr `  ( abs `  A ) )  x.  ( ( ( abs `  A )  +  A )  /  ( abs `  (
 ( abs `  A )  +  A ) ) ) )   =>    |-  ( ( A  e.  CC  /\  ( ( abs `  A )  +  A )  =/=  0 )  ->  ( ( B ^
 2 )  =  A  /\  0  <_  ( Re
 `  B )  /\  ( _i  x.  B )  e/  RR+ ) )
 
Theoremsqreu 11844* Existence and uniqueness for the square root function in general. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( A  e.  CC  ->  E! x  e.  CC  ( ( x ^
 2 )  =  A  /\  0  <_  ( Re
 `  x )  /\  ( _i  x.  x )  e/  RR+ ) )
 
Theoremsqrcl 11845 Closure of the square root function over the complexes. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  ( A  e.  CC  ->  ( sqr `  A )  e.  CC )
 
Theoremsqrthlem 11846 Lemma for sqrth 11848. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  ( A  e.  CC  ->  ( ( ( sqr `  A ) ^ 2
 )  =  A  /\  0  <_  ( Re `  ( sqr `  A )
 )  /\  ( _i  x.  ( sqr `  A ) )  e/  RR+ )
 )
 
Theoremsqrf 11847 Mapping domain and codomain of the square root function. (Contributed by Mario Carneiro, 13-Sep-2015.)
 |- 
 sqr : CC --> CC
 
Theoremsqrth 11848 Square root theorem over the complexes. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  ( A  e.  CC  ->  ( ( sqr `  A ) ^ 2 )  =  A )
 
Theoremsqrrege0 11849 The square root function must make a choice between the two roots, which differ by a sign change. In the general complex case, the choice of "positive" and "negative" is not so clear. The convention we use is to take the root with positive real part, unless  A is a non-positive real (in which case both roots have 0 real part); in this case we take the one in the positive imaginary direction. Another way to look at this is that we choose the root that is largest with respect to lexicographic order on the complexes (sorting by real part first, then by imaginary part as tie-breaker). (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  ( A  e.  CC  ->  0  <_  ( Re `  ( sqr `  A ) ) )
 
Theoremeqsqror 11850 Solve an equation containing a square. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^ 2 )  =  B  <->  ( A  =  ( sqr `  B )  \/  A  =  -u ( sqr `  B ) ) ) )
 
Theoremeqsqrd 11851 A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A ^
 2 )  =  B )   &    |-  ( ph  ->  0  <_  ( Re `  A ) )   &    |-  ( ph  ->  -.  ( _i  x.  A )  e.  RR+ )   =>    |-  ( ph  ->  A  =  ( sqr `  B ) )
 
Theoremeqsqr2d 11852 A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A ^
 2 )  =  B )   &    |-  ( ph  ->  0  <  ( Re `  A ) )   =>    |-  ( ph  ->  A  =  ( sqr `  B ) )
 
Theoremamgm2 11853 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 11854 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 11855 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 11856 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 11857 Square root of square. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( sqr `  ( A  x.  A ) )  =  A )
 
Theoremsqrsqi 11858 Square root of square. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( sqr `  ( A ^ 2 ) )  =  A )
 
Theoremsqsqri 11859 Square of square root. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( ( sqr `  A ) ^ 2
 )  =  A )
 
Theoremsqrge0i 11860 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 11861 A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <_  A  ->  ( abs `  A )  =  A )
 
Theoremabsnidi 11862 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 11863 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 11864 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 11865 Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)
 |-  A  e.  RR   =>    |-  ( abs `  A )  =  ( sqr `  ( A ^ 2
 ) )
 
Theoremsqrpclii 11866 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 11867 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 11868 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 11869 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 11870 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 11871 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 11872 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 11873 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 11874 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 11875 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 11876 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( ( abs `  A ) ^ 2
 )  =  ( A  x.  ( * `  A ) )
 
Theoremabsvalsq2i 11877 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 11878 Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( abs `  A )  e.  RR
 
Theoremabsge0i 11879 Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  0  <_  ( abs `  A )
 
Theoremabsval2i 11880 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 11881 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 11882 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 11883 Absolute value of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( abs `  -u A )  =  ( abs `  A )
 
Theoremabscji 11884 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 11885 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 11886 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 11887 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 11888 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 11889 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 11890 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 11891 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 11892 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 11893 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 11894 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 11895 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 11896 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 11897 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 11898 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 11899 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( abs `  A )  =  ( sqr `  ( A ^ 2 ) ) )
 
Theoremresqrcld 11900 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 )
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