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Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsqrt0rlem 10401 Lemma for sqrt0 10402. (Contributed by Jim Kingdon, 26-Aug-2020.)
 |-  ( ( A  e.  RR  /\  ( ( A ^ 2 )  =  0  /\  0  <_  A ) )  <->  A  =  0
 )
 
Theoremsqrt0 10402 Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( sqr `  0
 )  =  0
 
Theoremresqrexlem1arp 10403 Lemma for resqrex 10424.  1  +  A is a positive real (expressed in a way that will help apply seqf 9845 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ( ph  /\  N  e.  NN )  ->  (
 ( NN  X.  {
 ( 1  +  A ) } ) `  N )  e.  RR+ )
 
Theoremresqrexlemp1rp 10404* Lemma for resqrex 10424. Applying the recursion rule yields a positive real (expressed in a way that will help apply seqf 9845 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ( ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  ( B ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) C )  e.  RR+ )
 
Theoremresqrexlemf 10405* Lemma for resqrex 10424. The sequence is a function. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   =>    |-  ( ph  ->  F : NN --> RR+ )
 
Theoremresqrexlemf1 10406* Lemma for resqrex 10424. Initial value. Although this sequence converges to the square root with any positive initial value, this choice makes various steps in the proof of convergence easier. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   =>    |-  ( ph  ->  ( F `  1 )  =  ( 1  +  A ) )
 
Theoremresqrexlemfp1 10407* Lemma for resqrex 10424. Recursion rule. This sequence is the ancient method for computing square roots, often known as the babylonian method, although known to many ancient cultures. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   =>    |-  ( ( ph  /\  N  e.  NN )  ->  ( F `  ( N  +  1 ) )  =  ( ( ( F `
  N )  +  ( A  /  ( F `  N ) ) )  /  2 ) )
 
Theoremresqrexlemover 10408* Lemma for resqrex 10424. Each element of the sequence is an overestimate. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   =>    |-  ( ( ph  /\  N  e.  NN )  ->  A  <  ( ( F `  N ) ^ 2
 ) )
 
Theoremresqrexlemdec 10409* Lemma for resqrex 10424. The sequence is decreasing. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   =>    |-  ( ( ph  /\  N  e.  NN )  ->  ( F `  ( N  +  1 ) )  < 
 ( F `  N ) )
 
Theoremresqrexlemdecn 10410* Lemma for resqrex 10424. The sequence is decreasing. (Contributed by Jim Kingdon, 31-Jul-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  N  <  M )   =>    |-  ( ph  ->  ( F `  M )  < 
 ( F `  N ) )
 
Theoremresqrexlemlo 10411* Lemma for resqrex 10424. A (variable) lower bound for each term of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   =>    |-  ( ( ph  /\  N  e.  NN )  ->  (
 1  /  ( 2 ^ N ) )  < 
 ( F `  N ) )
 
Theoremresqrexlemcalc1 10412* Lemma for resqrex 10424. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   =>    |-  ( ( ph  /\  N  e.  NN )  ->  (
 ( ( F `  ( N  +  1
 ) ) ^ 2
 )  -  A )  =  ( ( ( ( ( F `  N ) ^ 2
 )  -  A ) ^ 2 )  /  ( 4  x.  (
 ( F `  N ) ^ 2 ) ) ) )
 
Theoremresqrexlemcalc2 10413* Lemma for resqrex 10424. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   =>    |-  ( ( ph  /\  N  e.  NN )  ->  (
 ( ( F `  ( N  +  1
 ) ) ^ 2
 )  -  A ) 
 <_  ( ( ( ( F `  N ) ^ 2 )  -  A )  /  4
 ) )
 
Theoremresqrexlemcalc3 10414* Lemma for resqrex 10424. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   =>    |-  ( ( ph  /\  N  e.  NN )  ->  (
 ( ( F `  N ) ^ 2
 )  -  A ) 
 <_  ( ( ( F `
  1 ) ^
 2 )  /  (
 4 ^ ( N  -  1 ) ) ) )
 
Theoremresqrexlemnmsq 10415* Lemma for resqrex 10424. The difference between the squares of two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 30-Jul-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  N  <_  M )   =>    |-  ( ph  ->  (
 ( ( F `  N ) ^ 2
 )  -  ( ( F `  M ) ^ 2 ) )  <  ( ( ( F `  1 ) ^ 2 )  /  ( 4 ^ ( N  -  1 ) ) ) )
 
Theoremresqrexlemnm 10416* Lemma for resqrex 10424. The difference between two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 31-Jul-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  N  <_  M )   =>    |-  ( ph  ->  (
 ( F `  N )  -  ( F `  M ) )  < 
 ( ( ( ( F `  1 ) ^ 2 )  x.  2 )  /  (
 2 ^ ( N  -  1 ) ) ) )
 
Theoremresqrexlemcvg 10417* Lemma for resqrex 10424. The sequence has a limit. (Contributed by Jim Kingdon, 6-Aug-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   =>    |-  ( ph  ->  E. r  e.  RR  A. x  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j ) ( ( F `  i )  <  ( r  +  x )  /\  r  < 
 ( ( F `  i )  +  x ) ) )
 
Theoremresqrexlemgt0 10418* Lemma for resqrex 10424. A limit is nonnegative. (Contributed by Jim Kingdon, 7-Aug-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  L  e.  RR )   &    |-  ( ph  ->  A. e  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
 ( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `
  i )  +  e ) ) )   =>    |-  ( ph  ->  0  <_  L )
 
Theoremresqrexlemoverl 10419* Lemma for resqrex 10424. Every term in the sequence is an overestimate compared with the limit 
L. Although this theorem is stated in terms of a particular sequence the proof could be adapted for any decreasing convergent sequence. (Contributed by Jim Kingdon, 9-Aug-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  L  e.  RR )   &    |-  ( ph  ->  A. e  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
 ( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `
  i )  +  e ) ) )   &    |-  ( ph  ->  K  e.  NN )   =>    |-  ( ph  ->  L  <_  ( F `  K ) )
 
Theoremresqrexlemglsq 10420* Lemma for resqrex 10424. The sequence formed by squaring each term of  F converges to  ( L ^
2 ). (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  L  e.  RR )   &    |-  ( ph  ->  A. e  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
 ( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `
  i )  +  e ) ) )   &    |-  G  =  ( x  e.  NN  |->  ( ( F `
  x ) ^
 2 ) )   =>    |-  ( ph  ->  A. e  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
 ( ( G `  k )  <  ( ( L ^ 2 )  +  e )  /\  ( L ^ 2 )  <  ( ( G `
  k )  +  e ) ) )
 
Theoremresqrexlemga 10421* Lemma for resqrex 10424. The sequence formed by squaring each term of  F converges to  A. (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  L  e.  RR )   &    |-  ( ph  ->  A. e  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
 ( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `
  i )  +  e ) ) )   &    |-  G  =  ( x  e.  NN  |->  ( ( F `
  x ) ^
 2 ) )   =>    |-  ( ph  ->  A. e  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
 ( ( G `  k )  <  ( A  +  e )  /\  A  <  ( ( G `
  k )  +  e ) ) )
 
Theoremresqrexlemsqa 10422* Lemma for resqrex 10424. The square of a limit is  A. (Contributed by Jim Kingdon, 7-Aug-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  L  e.  RR )   &    |-  ( ph  ->  A. e  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
 ( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `
  i )  +  e ) ) )   =>    |-  ( ph  ->  ( L ^ 2 )  =  A )
 
Theoremresqrexlemex 10423* Lemma for resqrex 10424. Existence of square root given a sequence which converges to the square root. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
 |-  F  =  seq 1
 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
 ) )  /  2
 ) ) ,  ( NN  X.  { ( 1  +  A ) }
 ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   =>    |-  ( ph  ->  E. x  e.  RR  ( 0  <_  x  /\  ( x ^
 2 )  =  A ) )
 
Theoremresqrex 10424* Existence of a square root for positive reals. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  E. x  e.  RR  ( 0  <_  x  /\  ( x ^ 2
 )  =  A ) )
 
Theoremrsqrmo 10425* Uniqueness for the square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  E* x  e.  RR  ( ( x ^
 2 )  =  A  /\  0  <_  x ) )
 
Theoremrersqreu 10426* Existence and uniqueness for the real square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  E! x  e. 
 RR  ( ( x ^ 2 )  =  A  /\  0  <_  x ) )
 
Theoremresqrtcl 10427 Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( sqr `  A )  e.  RR )
 
Theoremrersqrtthlem 10428 Lemma for resqrtth 10429. (Contributed by Jim Kingdon, 10-Aug-2021.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( ( ( sqr `  A ) ^ 2 )  =  A  /\  0  <_  ( sqr `  A )
 ) )
 
Theoremresqrtth 10429 Square root theorem over the reals. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( ( sqr `  A ) ^ 2
 )  =  A )
 
Theoremremsqsqrt 10430 Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( ( sqr `  A )  x.  ( sqr `  A ) )  =  A )
 
Theoremsqrtge0 10431 The square root function is nonnegative for nonnegative input. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  0  <_  ( sqr `  A ) )
 
Theoremsqrtgt0 10432 The square root function is positive for positive input. (Contributed by Mario Carneiro, 10-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2013.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  0  <  ( sqr `  A ) )
 
Theoremsqrtmul 10433 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( sqr `  ( A  x.  B ) )  =  ( ( sqr `  A )  x.  ( sqr `  B ) ) )
 
Theoremsqrtle 10434 Square root is monotonic. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( A  <_  B  <->  ( sqr `  A )  <_  ( sqr `  B ) ) )
 
Theoremsqrtlt 10435 Square root is strictly monotonic. Closed form of sqrtlti 10535. (Contributed by Scott Fenton, 17-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( A  <  B  <->  ( sqr `  A )  <  ( sqr `  B ) ) )
 
Theoremsqrt11ap 10436 Analogue to sqrt11 10437 but for apartness. (Contributed by Jim Kingdon, 11-Aug-2021.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( sqr `  A ) #  ( sqr `  B ) 
 <->  A #  B ) )
 
Theoremsqrt11 10437 The square root function is one-to-one. Also see sqrt11ap 10436 which would follow easily from this given excluded middle, but which is proved another way without it. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( sqr `  A )  =  ( sqr `  B )  <->  A  =  B ) )
 
Theoremsqrt00 10438 A square root is zero iff its argument is 0. (Contributed by NM, 27-Jul-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( ( sqr `  A )  =  0  <->  A  =  0 )
 )
 
Theoremrpsqrtcl 10439 The square root of a positive real is a positive real. (Contributed by NM, 22-Feb-2008.)
 |-  ( A  e.  RR+  ->  ( sqr `  A )  e.  RR+ )
 
Theoremsqrtdiv 10440 Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  B  e.  RR+ )  ->  ( sqr `  ( A  /  B ) )  =  ( ( sqr `  A )  /  ( sqr `  B ) ) )
 
Theoremsqrtsq2 10441 Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( sqr `  A )  =  B  <->  A  =  ( B ^ 2 ) ) )
 
Theoremsqrtsq 10442 Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( sqr `  ( A ^ 2 ) )  =  A )
 
Theoremsqrtmsq 10443 Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( sqr `  ( A  x.  A ) )  =  A )
 
Theoremsqrt1 10444 The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)
 |-  ( sqr `  1
 )  =  1
 
Theoremsqrt4 10445 The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)
 |-  ( sqr `  4
 )  =  2
 
Theoremsqrt9 10446 The square root of 9 is 3. (Contributed by NM, 11-May-2004.)
 |-  ( sqr `  9
 )  =  3
 
Theoremsqrt2gt1lt2 10447 The square root of 2 is bounded by 1 and 2. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 6-Sep-2013.)
 |-  ( 1  <  ( sqr `  2 )  /\  ( sqr `  2 )  <  2 )
 
Theoremabsneg 10448 Absolute value of negative. (Contributed by NM, 27-Feb-2005.)
 |-  ( A  e.  CC  ->  ( abs `  -u A )  =  ( abs `  A ) )
 
Theoremabscl 10449 Real closure of absolute value. (Contributed by NM, 3-Oct-1999.)
 |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
 
Theoremabscj 10450 The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 28-Apr-2005.)
 |-  ( A  e.  CC  ->  ( abs `  ( * `  A ) )  =  ( abs `  A ) )
 
Theoremabsvalsq 10451 Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.)
 |-  ( A  e.  CC  ->  ( ( abs `  A ) ^ 2 )  =  ( A  x.  ( * `  A ) ) )
 
Theoremabsvalsq2 10452 Square of value of absolute value function. (Contributed by NM, 1-Feb-2007.)
 |-  ( A  e.  CC  ->  ( ( abs `  A ) ^ 2 )  =  ( ( ( Re
 `  A ) ^
 2 )  +  (
 ( Im `  A ) ^ 2 ) ) )
 
Theoremsqabsadd 10453 Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  +  B ) ) ^ 2
 )  =  ( ( ( ( abs `  A ) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  +  ( 2  x.  ( Re `  ( A  x.  ( * `  B ) ) ) ) ) )
 
Theoremsqabssub 10454 Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  -  B ) ) ^ 2
 )  =  ( ( ( ( abs `  A ) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  -  ( 2  x.  ( Re `  ( A  x.  ( * `  B ) ) ) ) ) )
 
Theoremabsval2 10455 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.)
 |-  ( A  e.  CC  ->  ( abs `  A )  =  ( sqr `  ( ( ( Re
 `  A ) ^
 2 )  +  (
 ( Im `  A ) ^ 2 ) ) ) )
 
Theoremabs0 10456 The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( abs `  0
 )  =  0
 
Theoremabsi 10457 The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
 |-  ( abs `  _i )  =  1
 
Theoremabsge0 10458 Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( A  e.  CC  ->  0  <_  ( abs `  A ) )
 
Theoremabsrpclap 10459 The absolute value of a number apart from zero is a positive real. (Contributed by Jim Kingdon, 11-Aug-2021.)
 |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  ( abs `  A )  e.  RR+ )
 
Theoremabs00ap 10460 The absolute value of a number is apart from zero iff the number is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
 |-  ( A  e.  CC  ->  ( ( abs `  A ) #  0  <->  A #  0 )
 )
 
Theoremabsext 10461 Strong extensionality for absolute value. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  A ) #  ( abs `  B )  ->  A #  B ) )
 
Theoremabs00 10462 The absolute value of a number is zero iff the number is zero. Also see abs00ap 10460 which is similar but for apartness. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by NM, 26-Sep-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( A  e.  CC  ->  ( ( abs `  A )  =  0  <->  A  =  0
 ) )
 
Theoremabs00ad 10463 A complex number is zero iff its absolute value is zero. Deduction form of abs00 10462. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( abs `  A )  =  0  <->  A  =  0
 ) )
 
Theoremabs00bd 10464 If a complex number is zero, its absolute value is zero. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  =  0 )   =>    |-  ( ph  ->  ( abs `  A )  =  0 )
 
Theoremabsreimsq 10465 Square of the absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 1-Feb-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  ( A  +  ( _i  x.  B ) ) ) ^ 2 )  =  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )
 
Theoremabsreim 10466 Absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 14-Jan-2006.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( A  +  ( _i  x.  B ) ) )  =  ( sqr `  (
 ( A ^ 2
 )  +  ( B ^ 2 ) ) ) )
 
Theoremabsmul 10467 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  x.  B ) )  =  ( ( abs `  A )  x.  ( abs `  B ) ) )
 
Theoremabsdivap 10468 Absolute value distributes over division. (Contributed by Jim Kingdon, 11-Aug-2021.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( abs `  ( A  /  B ) )  =  ( ( abs `  A )  /  ( abs `  B ) ) )
 
Theoremabsid 10469 A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( abs `  A )  =  A )
 
Theoremabs1 10470 The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
 |-  ( abs `  1
 )  =  1
 
Theoremabsnid 10471 A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005.)
 |-  ( ( A  e.  RR  /\  A  <_  0
 )  ->  ( abs `  A )  =  -u A )
 
Theoremleabs 10472 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 10473 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 10474 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 10475 Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)
 |-  ( A  e.  RR  ->  ( abs `  A )  =  ( sqr `  ( A ^ 2
 ) ) )
 
Theoremabsresq 10476 Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)
 |-  ( A  e.  RR  ->  ( ( abs `  A ) ^ 2 )  =  ( A ^ 2
 ) )
 
Theoremabsexp 10477 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 10478 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 10479 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 10480 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 10481 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 10482 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 10483 The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.)
 |-  ( A  e.  ZZ  ->  ( abs `  A )  e.  NN0 )
 
Theoremzabscl 10484 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 10485 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 10486 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 10487 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 10488 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 10489 If the absolute value of a complex number is less than a real, its difference from the real is nonzero. See also abssubap0 10488 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 10490 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 10491 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 10492 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 10493 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 10494 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 10495 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 10496 The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)
 |-  ( A  e.  CC  ->  ( abs `  ( abs `  A ) )  =  ( abs `  A ) )
 
Theoremabsgt0ap 10497 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 10498 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 10499 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 10500 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 ) )
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