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Theorem List for Metamath Proof Explorer - 11701-11800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremresubd 11701 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 11702 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 11703 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 11704 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 11705 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 11706 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 11707 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 ) ) ) )
 
Theoremcjdivd 11708 Complex conjugate distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  ( * `  ( A  /  B ) )  =  ( ( * `  A )  /  ( * `  B ) ) )
 
Theoremrered 11709 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 11710 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 11711 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 11712 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 11713 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 ) ) )
 
Theoremredivd 11714 Real part of a division. Related to remul2 11615. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   =>    |-  ( ph  ->  ( Re `  ( B  /  A ) )  =  ( ( Re `  B )  /  A ) )
 
Theoremimdivd 11715 Imaginary part of a division. Related to remul2 11615. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   =>    |-  ( ph  ->  ( Im `  ( B  /  A ) )  =  ( ( Im `  B )  /  A ) )
 
Theoremcrred 11716 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 11717 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 )
 
5.7.3  Square root; absolute value
 
Syntaxcsqr 11718 Extend class notation to include square root of a complex number.
 class  sqr
 
Syntaxcabs 11719 Extend class notation to include a function for the absolute value (modulus) of a complex number.
 class  abs
 
Definitiondf-sqr 11720* Define a function whose value is the square root of a complex number. Since  ( y ^
2 )  =  x iff  ( -u y ^
2 )  =  x, we ensure uniqueness by restricting the range to numbers with positive real part, or numbers with 0 real part and nonnegative imaginary part. A description can be found under "Principal square root of a complex number" at http://en.wikipedia.org/wiki/Square_root. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrcl 11845 for its closure, sqrval 11722 for its value, sqrth 11848 and sqsqri 11859 for its relationship to squares, and sqr11i 11868 for uniqueness. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 8-Jul-2013.)

 |- 
 sqr  =  ( x  e.  CC  |->  ( iota_ y  e. 
 CC ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y ) 
 /\  ( _i  x.  y )  e/  RR+ )
 ) )
 
Definitiondf-abs 11721 Define the function for the absolute value (modulus) of a complex number. See abscli 11878 for its closure and absval 11723 or absval2i 11880 for its value. (Contributed by NM, 27-Jul-1999.)
 |- 
 abs  =  ( x  e.  CC  |->  ( sqr `  ( x  x.  ( * `  x ) ) ) )
 
Theoremsqrval 11722* Value of square root function. (Contributed by Mario Carneiro, 8-Jul-2013.)
 |-  ( A  e.  CC  ->  ( sqr `  A )  =  ( iota_ x  e. 
 CC ( ( x ^ 2 )  =  A  /\  0  <_  ( Re `  x ) 
 /\  ( _i  x.  x )  e/  RR+ )
 ) )
 
Theoremabsval 11723 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 11724 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+ )
 
Theoremcnpart 11725 The specification of restriction to the right half-plane partitions the complex plane without 0 into two disjoint pieces, which are related by a reflection about the origin (under the map  x 
|->  -u x). (Contributed by Mario Carneiro, 8-Jul-2013.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( (
 0  <_  ( Re `  A )  /\  ( _i  x.  A )  e/  RR+ )  <->  -.  ( 0  <_  ( Re `  -u A )  /\  ( _i  x.  -u A )  e/  RR+ )
 ) )
 
Theoremsqr0lem 11726 Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  CC  /\  ( ( A ^ 2 )  =  0  /\  0  <_  ( Re `  A ) 
 /\  ( _i  x.  A )  e/  RR+ )
 ) 
 <->  A  =  0 )
 
Theoremsqr0 11727 Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( sqr `  0
 )  =  0
 
Theoremsqrlem1 11728* Lemma for 01sqrex 11735. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  S  =  { x  e.  RR+  |  ( x ^ 2 )  <_  A }   &    |-  B  =  sup ( S ,  RR ,  <  )   =>    |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  A. y  e.  S  y  <_  1 )
 
Theoremsqrlem2 11729* Lemma for 01sqrex 11735. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  S  =  { x  e.  RR+  |  ( x ^ 2 )  <_  A }   &    |-  B  =  sup ( S ,  RR ,  <  )   =>    |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  A  e.  S )
 
Theoremsqrlem3 11730* Lemma for 01sqrex 11735. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  S  =  { x  e.  RR+  |  ( x ^ 2 )  <_  A }   &    |-  B  =  sup ( S ,  RR ,  <  )   =>    |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  ( S  C_  RR  /\  S  =/=  (/)  /\  E. z  e.  RR  A. y  e.  S  y  <_  z
 ) )
 
Theoremsqrlem4 11731* Lemma for 01sqrex 11735. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  S  =  { x  e.  RR+  |  ( x ^ 2 )  <_  A }   &    |-  B  =  sup ( S ,  RR ,  <  )   =>    |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  ( B  e.  RR+  /\  B  <_  1 )
 )
 
Theoremsqrlem5 11732* Lemma for 01sqrex 11735. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  S  =  { x  e.  RR+  |  ( x ^ 2 )  <_  A }   &    |-  B  =  sup ( S ,  RR ,  <  )   &    |-  T  =  {
 y  |  E. a  e.  S  E. b  e.  S  y  =  ( a  x.  b ) }   =>    |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  ( ( T  C_  RR  /\  T  =/=  (/)  /\  E. v  e.  RR  A. u  e.  T  u  <_  v
 )  /\  ( B ^ 2 )  = 
 sup ( T ,  RR ,  <  ) ) )
 
Theoremsqrlem6 11733* Lemma for 01sqrex 11735. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  S  =  { x  e.  RR+  |  ( x ^ 2 )  <_  A }   &    |-  B  =  sup ( S ,  RR ,  <  )   &    |-  T  =  {
 y  |  E. a  e.  S  E. b  e.  S  y  =  ( a  x.  b ) }   =>    |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  ( B ^ 2
 )  <_  A )
 
Theoremsqrlem7 11734* Lemma for 01sqrex 11735. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  S  =  { x  e.  RR+  |  ( x ^ 2 )  <_  A }   &    |-  B  =  sup ( S ,  RR ,  <  )   &    |-  T  =  {
 y  |  E. a  e.  S  E. b  e.  S  y  =  ( a  x.  b ) }   =>    |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  ( B ^ 2
 )  =  A )
 
Theorem01sqrex 11735* Existence of a square root for reals in the interval  ( 0 ,  1 ]. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  ( ( A  e.  RR+  /\  A  <_  1 )  ->  E. x  e.  RR+  ( x  <_  1  /\  ( x ^ 2 )  =  A ) )
 
Theoremresqrex 11736* 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 ) )
 
Theoremsqrmo 11737* Uniqueness for the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.) (Revised by NM, 17-Jun-2017.)
 |-  ( A  e.  CC  ->  E* x  e.  CC ( ( x ^
 2 )  =  A  /\  0  <_  ( Re
 `  x )  /\  ( _i  x.  x )  e/  RR+ ) )
 
Theoremresqreu 11738* Existence and uniqueness for the real square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  E! x  e. 
 CC  ( ( x ^ 2 )  =  A  /\  0  <_  ( Re `  x ) 
 /\  ( _i  x.  x )  e/  RR+ )
 )
 
Theoremresqrcl 11739 Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( sqr `  A )  e.  RR )
 
Theoremresqrthlem 11740 Lemma for resqrth 11741. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( ( ( sqr `  A ) ^ 2 )  =  A  /\  0  <_  ( Re `  ( sqr `  A ) )  /\  ( _i  x.  ( sqr `  A ) ) 
 e/  RR+ ) )
 
Theoremresqrth 11741 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 )
 
Theoremremsqsqr 11742 Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( ( sqr `  A )  x.  ( sqr `  A ) )  =  A )
 
Theoremsqrge0 11743 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 ) )
 
Theoremsqrgt0 11744 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 ) )
 
Theoremsqrmul 11745 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 ) ) )
 
Theoremsqrle 11746 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 ) ) )
 
Theoremsqrlt 11747 Square root is strictly monotonic. Closed form of sqrlti 11873. (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 ) ) )
 
Theoremsqr11 11748 The square root function is one to one. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( sqr `  A )  =  ( sqr `  B )  <->  A  =  B ) )
 
Theoremsqr00 11749 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 )
 )
 
Theoremrpsqrcl 11750 The square root of a positive real is a postive real. (Contributed by NM, 22-Feb-2008.)
 |-  ( A  e.  RR+  ->  ( sqr `  A )  e.  RR+ )
 
Theoremsqrdiv 11751 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 ) ) )
 
Theoremsqrneglem 11752 The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( ( ( _i  x.  ( sqr `  A ) ) ^
 2 )  =  -u A  /\  0  <_  ( Re `  ( _i  x.  ( sqr `  A )
 ) )  /\  ( _i  x.  ( _i  x.  ( sqr `  A )
 ) )  e/  RR+ )
 )
 
Theoremsqrneg 11753 The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( sqr `  -u A )  =  ( _i  x.  ( sqr `  A ) ) )
 
Theoremsqrsq2 11754 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 ) ) )
 
Theoremsqrsq 11755 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 )
 
Theoremsqrmsq 11756 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 )
 
Theoremsqr1 11757 The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)
 |-  ( sqr `  1
 )  =  1
 
Theoremsqr4 11758 The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)
 |-  ( sqr `  4
 )  =  2
 
Theoremsqr9 11759 The square root of 9 is 3. (Contributed by NM, 11-May-2004.)
 |-  ( sqr `  9
 )  =  3
 
Theoremsqr2gt1lt2 11760 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 )
 
Theoremsqrm1 11761 The imaginary unit is the square root of negative 1. A lot of people like to call this the "definition" of  _i, but the definition of  sqr df-sqr 11720 has already been crafted with  _i being mentioned explicitly, and in any case it doesn't make too much sense to define a value based on a function evaluated outside its domain. A more appropriate view is to take ax-i2m1 8805 or i2 11203 as the "definition", and simply postulate the existence of a number satisfying this property. This is the approach we take here. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  _i  =  ( sqr `  -u 1 )
 
Theoremabsneg 11762 Absolute value of negative. (Contributed by NM, 27-Feb-2005.)
 |-  ( A  e.  CC  ->  ( abs `  -u A )  =  ( abs `  A ) )
 
Theoremabscl 11763 Real closure of absolute value. (Contributed by NM, 3-Oct-1999.)
 |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
 
Theoremabscj 11764 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 11765 Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.)
 |-  ( A  e.  CC  ->  ( ( abs `  A ) ^ 2 )  =  ( A  x.  ( * `  A ) ) )
 
Theoremabsvalsq2 11766 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 11767 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 11768 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 11769 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 11770 The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( abs `  0
 )  =  0
 
Theoremabsi 11771 The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
 |-  ( abs `  _i )  =  1
 
Theoremabsge0 11772 Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( A  e.  CC  ->  0  <_  ( abs `  A ) )
 
Theoremabsrpcl 11773 The absolute value of a nonzero number is a positive real. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( abs `  A )  e.  RR+ )
 
Theoremabs00 11774 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, 26-Sep-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( A  e.  CC  ->  ( ( abs `  A )  =  0  <->  A  =  0
 ) )
 
Theoremabs00ad 11775 A complex number is zero iff its absolute value is zero. Deduction form of abs00 11774. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( abs `  A )  =  0  <->  A  =  0
 ) )
 
Theoremabs00bd 11776 If a complex number is zero, its absolute value is zero. Converse of abs00d 11928. One-way deduction form of abs00 11774. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  =  0 )   =>    |-  ( ph  ->  ( abs `  A )  =  0 )
 
Theoremabsreimsq 11777 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 11778 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 11779 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 ) ) )
 
Theoremabsdiv 11780 Absolute value distributes over division. (Contributed by NM, 27-Apr-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( abs `  ( A  /  B ) )  =  ( ( abs `  A )  /  ( abs `  B ) ) )
 
Theoremabsid 11781 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 11782 The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
 |-  ( abs `  1
 )  =  1
 
Theoremabsnid 11783 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 11784 A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005.)
 |-  ( A  e.  RR  ->  A  <_  ( abs `  A ) )
 
Theoremabsor 11785 The absolute value of a real number is either that number or its negative. (Contributed by NM, 27-Feb-2005.)
 |-  ( A  e.  RR  ->  ( ( abs `  A )  =  A  \/  ( abs `  A )  =  -u A ) )
 
Theoremabsre 11786 Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)
 |-  ( A  e.  RR  ->  ( abs `  A )  =  ( sqr `  ( A ^ 2
 ) ) )
 
Theoremabsresq 11787 Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)
 |-  ( A  e.  RR  ->  ( ( abs `  A ) ^ 2 )  =  ( A ^ 2
 ) )
 
Theoremabsmod0 11788  A is divisible by  B iff its absolute value is. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A 
 mod  B )  =  0  <-> 
 ( ( abs `  A )  mod  B )  =  0 ) )
 
Theoremabsexp 11789 Absolute value of natural number exponentiation. (Contributed by NM, 5-Jan-2006.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N ) )
 
Theoremabsexpz 11790 Absolute value of integer exponentiation. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N ) )
 
Theoremabssq 11791 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 11792 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 11793 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 11794 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 ) )
 
Theoremmax0add 11795 The sum of the positive and negative part functions is the absolute value function over the reals. (Contributed by Mario Carneiro, 24-Aug-2014.)
 |-  ( A  e.  RR  ->  ( if ( 0 
 <_  A ,  A , 
 0 )  +  if ( 0  <_  -u A ,  -u A ,  0 ) )  =  ( abs `  A )
 )
 
Theoremabsz 11796 A real number is an integer iff its absolute value is an integer. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( A  e.  RR  ->  ( A  e.  ZZ  <->  ( abs `  A )  e. 
 ZZ ) )
 
Theoremnn0abscl 11797 The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( A  e.  ZZ  ->  ( abs `  A )  e.  NN0 )
 
Theoremabslt 11798 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 11799 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 )
 ) )
 
Theoremabssubne0 11800 If the absolute value of a complex number is less than a real, its difference from the real is nonzero. (Contributed by NM, 2-Nov-2007.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  RR  /\  ( abs `  A )  <  B )  ->  ( B  -  A )  =/=  0 )
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