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Theorem List for Intuitionistic Logic Explorer - 10601-10700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremremimd 10601 Value of the conjugate of a complex number. The value is the real part minus  _i times the imaginary part. Definition 10-3.2 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  A )  =  ( ( Re `  A )  -  ( _i  x.  ( Im `  A ) ) ) )
 
Theoremcjcjd 10602 The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  ( * `  A ) )  =  A )
 
Theoremreim0bd 10603 A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( Im `  A )  =  0 )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremrerebd 10604 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( Re `  A )  =  A )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremcjrebd 10605 A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( * `  A )  =  A )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremcjne0d 10606 A number which is nonzero has a complex conjugate which is nonzero. Also see cjap0d 10607 which is similar but for apartness. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( * `  A )  =/=  0 )
 
Theoremcjap0d 10607 A number which is apart from zero has a complex conjugate which is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  ( * `  A ) #  0 )
 
Theoremrecjd 10608 Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( * `  A ) )  =  ( Re `  A ) )
 
Theoremimcjd 10609 Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( * `  A ) )  =  -u ( Im `  A ) )
 
Theoremcjmulrcld 10610 A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( * `  A ) )  e. 
 RR )
 
Theoremcjmulvald 10611 A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( * `  A ) )  =  ( ( ( Re
 `  A ) ^
 2 )  +  (
 ( Im `  A ) ^ 2 ) ) )
 
Theoremcjmulge0d 10612 A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  0  <_  ( A  x.  ( * `  A ) ) )
 
Theoremrenegd 10613 Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  -u A )  =  -u ( Re `  A ) )
 
Theoremimnegd 10614 Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Im `  -u A )  =  -u ( Im `  A ) )
 
Theoremcjnegd 10615 Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  -u A )  =  -u ( * `  A ) )
 
Theoremaddcjd 10616 A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  +  ( * `  A ) )  =  ( 2  x.  ( Re `  A ) ) )
 
Theoremcjexpd 10617 Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( * `  ( A ^ N ) )  =  ( ( * `  A ) ^ N ) )
 
Theoremreaddd 10618 Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( A  +  B ) )  =  ( ( Re
 `  A )  +  ( Re `  B ) ) )
 
Theoremimaddd 10619 Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( A  +  B ) )  =  ( ( Im
 `  A )  +  ( Im `  B ) ) )
 
Theoremresubd 10620 Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( A  -  B ) )  =  ( ( Re
 `  A )  -  ( Re `  B ) ) )
 
Theoremimsubd 10621 Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( A  -  B ) )  =  ( ( Im
 `  A )  -  ( Im `  B ) ) )
 
Theoremremuld 10622 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( A  x.  B ) )  =  ( ( ( Re `  A )  x.  ( Re `  B ) )  -  ( ( Im `  A )  x.  ( Im `  B ) ) ) )
 
Theoremimmuld 10623 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( A  x.  B ) )  =  ( ( ( Re `  A )  x.  ( Im `  B ) )  +  ( ( Im `  A )  x.  ( Re `  B ) ) ) )
 
Theoremcjaddd 10624 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( * `  ( A  +  B ) )  =  ( ( * `
  A )  +  ( * `  B ) ) )
 
Theoremcjmuld 10625 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( * `  ( A  x.  B ) )  =  ( ( * `
  A )  x.  ( * `  B ) ) )
 
Theoremipcnd 10626 Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( A  x.  ( * `  B ) ) )  =  ( ( ( Re `  A )  x.  ( Re `  B ) )  +  ( ( Im `  A )  x.  ( Im `  B ) ) ) )
 
Theoremcjdivapd 10627 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( * `  ( A  /  B ) )  =  ( ( * `  A ) 
 /  ( * `  B ) ) )
 
Theoremrered 10628 A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( Re `  A )  =  A )
 
Theoremreim0d 10629 The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( Im `  A )  =  0 )
 
Theoremcjred 10630 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( * `  A )  =  A )
 
Theoremremul2d 10631 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( A  x.  B ) )  =  ( A  x.  ( Re `  B ) ) )
 
Theoremimmul2d 10632 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( A  x.  B ) )  =  ( A  x.  ( Im `  B ) ) )
 
Theoremredivapd 10633 Real part of a division. Related to remul2 10532. (Contributed by Jim Kingdon, 15-Jun-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  ( Re `  ( B  /  A ) )  =  ( ( Re `  B ) 
 /  A ) )
 
Theoremimdivapd 10634 Imaginary part of a division. Related to remul2 10532. (Contributed by Jim Kingdon, 15-Jun-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  ( Im `  ( B  /  A ) )  =  ( ( Im `  B ) 
 /  A ) )
 
Theoremcrred 10635 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  =  A )
 
Theoremcrimd 10636 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  B )
 
3.7.3  Sequence convergence
 
Theoremcaucvgrelemrec 10637* Two ways to express a reciprocal. (Contributed by Jim Kingdon, 20-Jul-2021.)
 |-  ( ( A  e.  RR  /\  A #  0 ) 
 ->  ( iota_ r  e.  RR  ( A  x.  r
 )  =  1 )  =  ( 1  /  A ) )
 
Theoremcaucvgrelemcau 10638* Lemma for caucvgre 10639. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n ) ( ( F `  n )  <  ( ( F `
  k )  +  ( 1  /  n ) )  /\  ( F `
  k )  < 
 ( ( F `  n )  +  (
 1  /  n )
 ) ) )   =>    |-  ( ph  ->  A. n  e.  NN  A. k  e.  NN  ( n  <RR  k  ->  (
 ( F `  n )  <RR  ( ( F `
  k )  +  ( iota_ r  e.  RR  ( n  x.  r
 )  =  1 ) )  /\  ( F `
  k )  <RR  ( ( F `  n )  +  ( iota_ r  e. 
 RR  ( n  x.  r )  =  1
 ) ) ) ) )
 
Theoremcaucvgre 10639* Convergence of real sequences.

A Cauchy sequence (as defined here, which has a rate of convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within  1  /  n of the nth term.

(Contributed by Jim Kingdon, 19-Jul-2021.)

 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n ) ( ( F `  n )  <  ( ( F `
  k )  +  ( 1  /  n ) )  /\  ( F `
  k )  < 
 ( ( F `  n )  +  (
 1  /  n )
 ) ) )   =>    |-  ( ph  ->  E. y  e.  RR  A. x  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
 ( ( F `  i )  <  ( y  +  x )  /\  y  <  ( ( F `
  i )  +  x ) ) )
 
Theoremcvg1nlemcxze 10640 Lemma for cvg1n 10644. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.)
 |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  X  e.  RR+ )   &    |-  ( ph  ->  Z  e.  NN )   &    |-  ( ph  ->  E  e.  NN )   &    |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  ( ( ( ( C  x.  2 )  /  X )  /  Z )  +  A )  <  E )   =>    |-  ( ph  ->  ( C  /  ( E  x.  Z ) )  < 
 ( X  /  2
 ) )
 
Theoremcvg1nlemf 10641* Lemma for cvg1n 10644. The modified sequence  G is a sequence. (Contributed by Jim Kingdon, 1-Aug-2021.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( ( F `  n )  <  ( ( F `  k )  +  ( C  /  n ) )  /\  ( F `  k )  <  ( ( F `
  n )  +  ( C  /  n ) ) ) )   &    |-  G  =  ( j  e.  NN  |->  ( F `  ( j  x.  Z ) ) )   &    |-  ( ph  ->  Z  e.  NN )   &    |-  ( ph  ->  C  <  Z )   =>    |-  ( ph  ->  G : NN --> RR )
 
Theoremcvg1nlemcau 10642* Lemma for cvg1n 10644. By selecting spaced out terms for the modified sequence  G, the terms are within  1  /  n (without the constant  C). (Contributed by Jim Kingdon, 1-Aug-2021.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( ( F `  n )  <  ( ( F `  k )  +  ( C  /  n ) )  /\  ( F `  k )  <  ( ( F `
  n )  +  ( C  /  n ) ) ) )   &    |-  G  =  ( j  e.  NN  |->  ( F `  ( j  x.  Z ) ) )   &    |-  ( ph  ->  Z  e.  NN )   &    |-  ( ph  ->  C  <  Z )   =>    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( ( G `  n )  <  ( ( G `  k )  +  ( 1  /  n ) )  /\  ( G `  k )  <  ( ( G `
  n )  +  ( 1  /  n ) ) ) )
 
Theoremcvg1nlemres 10643* Lemma for cvg1n 10644. The original sequence  F has a limit (turns out it is the same as the limit of the modified sequence  G). (Contributed by Jim Kingdon, 1-Aug-2021.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( ( F `  n )  <  ( ( F `  k )  +  ( C  /  n ) )  /\  ( F `  k )  <  ( ( F `
  n )  +  ( C  /  n ) ) ) )   &    |-  G  =  ( j  e.  NN  |->  ( F `  ( j  x.  Z ) ) )   &    |-  ( ph  ->  Z  e.  NN )   &    |-  ( ph  ->  C  <  Z )   =>    |-  ( ph  ->  E. y  e.  RR  A. x  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j ) ( ( F `  i )  <  ( y  +  x )  /\  y  < 
 ( ( F `  i )  +  x ) ) )
 
Theoremcvg1n 10644* Convergence of real sequences.

This is a version of caucvgre 10639 with a constant multiplier  C on the rate of convergence. That is, all terms after the nth term must be within  C  /  n of the nth term.

(Contributed by Jim Kingdon, 1-Aug-2021.)

 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( ( F `  n )  <  ( ( F `  k )  +  ( C  /  n ) )  /\  ( F `  k )  <  ( ( F `
  n )  +  ( C  /  n ) ) ) )   =>    |-  ( ph  ->  E. y  e.  RR  A. x  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j ) ( ( F `  i )  <  ( y  +  x )  /\  y  < 
 ( ( F `  i )  +  x ) ) )
 
Theoremuzin2 10645 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 10646* 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 ) )
 
Theoremrexfiuz 10647* 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 10648* Restrict 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 10649* 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 10650* A version of 19.29 1580 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 10651* A version of r19.2m 3413 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 )
 
Theoremrecvguniqlem 10652 Lemma for recvguniq 10653. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  A  <  (
 ( F `  K )  +  ( ( A  -  B )  / 
 2 ) ) )   &    |-  ( ph  ->  ( F `  K )  <  ( B  +  ( ( A  -  B )  / 
 2 ) ) )   =>    |-  ( ph  -> F.  )
 
Theoremrecvguniq 10653* Limits are unique. (Contributed by Jim Kingdon, 7-Aug-2021.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  L  e.  RR )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e. 
 NN  A. k  e.  ( ZZ>=
 `  j ) ( ( F `  k
 )  <  ( L  +  x )  /\  L  <  ( ( F `  k )  +  x ) ) )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e. 
 NN  A. k  e.  ( ZZ>=
 `  j ) ( ( F `  k
 )  <  ( M  +  x )  /\  M  <  ( ( F `  k )  +  x ) ) )   =>    |-  ( ph  ->  L  =  M )
 
3.7.4  Square root; absolute value
 
Syntaxcsqrt 10654 Extend class notation to include square root of a complex number.
 class  sqr
 
Syntaxcabs 10655 Extend class notation to include a function for the absolute value (modulus) of a complex number.
 class  abs
 
Definitiondf-rsqrt 10656* Define a function whose value is the square root of a nonnegative real number.

Defining the square root for complex numbers has one difficult part: choosing between the two roots. The usual way to define a principal square root for all complex numbers relies on excluded middle or something similar. But in the case of a nonnegative real number, we don't have the complications presented for general complex numbers, and we can choose the nonnegative root.

(Contributed by Jim Kingdon, 23-Aug-2020.)

 |- 
 sqr  =  ( x  e.  RR  |->  ( iota_ y  e. 
 RR  ( ( y ^ 2 )  =  x  /\  0  <_  y ) ) )
 
Definitiondf-abs 10657 Define the function for the absolute value (modulus) of a complex number. (Contributed by NM, 27-Jul-1999.)
 |- 
 abs  =  ( x  e.  CC  |->  ( sqr `  ( x  x.  ( * `  x ) ) ) )
 
Theoremsqrtrval 10658* Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.)
 |-  ( A  e.  RR  ->  ( sqr `  A )  =  ( iota_ x  e. 
 RR  ( ( x ^ 2 )  =  A  /\  0  <_  x ) ) )
 
Theoremabsval 10659 The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( A  e.  CC  ->  ( abs `  A )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )
 
Theoremrennim 10660 A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.)
 |-  ( A  e.  RR  ->  ( _i  x.  A )  e/  RR+ )
 
Theoremsqrt0rlem 10661 Lemma for sqrt0 10662. (Contributed by Jim Kingdon, 26-Aug-2020.)
 |-  ( ( A  e.  RR  /\  ( ( A ^ 2 )  =  0  /\  0  <_  A ) )  <->  A  =  0
 )
 
Theoremsqrt0 10662 Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( sqr `  0
 )  =  0
 
Theoremresqrexlem1arp 10663 Lemma for resqrex 10684.  1  +  A is a positive real (expressed in a way that will help apply seqf 10121 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 10664* Lemma for resqrex 10684. Applying the recursion rule yields a positive real (expressed in a way that will help apply seqf 10121 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 10665* Lemma for resqrex 10684. 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 10666* Lemma for resqrex 10684. 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 10667* Lemma for resqrex 10684. 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 10668* Lemma for resqrex 10684. 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 10669* Lemma for resqrex 10684. 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 10670* Lemma for resqrex 10684. 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 10671* Lemma for resqrex 10684. 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 10672* Lemma for resqrex 10684. 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 10673* Lemma for resqrex 10684. 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 10674* Lemma for resqrex 10684. 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 10675* Lemma for resqrex 10684. 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 10676* Lemma for resqrex 10684. 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 10677* Lemma for resqrex 10684. 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 10678* Lemma for resqrex 10684. 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 10679* Lemma for resqrex 10684. 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 10680* Lemma for resqrex 10684. 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 10681* Lemma for resqrex 10684. 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 10682* Lemma for resqrex 10684. 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 10683* Lemma for resqrex 10684. 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 10684* 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 10685* 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 10686* 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 10687 Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( sqr `  A )  e.  RR )
 
Theoremrersqrtthlem 10688 Lemma for resqrtth 10689. (Contributed by Jim Kingdon, 10-Aug-2021.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( ( ( sqr `  A ) ^ 2 )  =  A  /\  0  <_  ( sqr `  A )
 ) )
 
Theoremresqrtth 10689 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 10690 Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( ( sqr `  A )  x.  ( sqr `  A ) )  =  A )
 
Theoremsqrtge0 10691 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 10692 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 10693 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 10694 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 10695 Square root is strictly monotonic. Closed form of sqrtlti 10795. (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 10696 Analogue to sqrt11 10697 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 10697 The square root function is one-to-one. Also see sqrt11ap 10696 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 10698 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 10699 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 10700 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 ) ) )
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