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Theorem List for Metamath Proof Explorer - 11601-11700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrennim 11601 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 11602 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 11603 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 11604 Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( sqr `  0
 )  =  0
 
Theoremsqrlem1 11605* Lemma for 01sqrex 11612. (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 11606* Lemma for 01sqrex 11612. (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 11607* Lemma for 01sqrex 11612. (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 11608* Lemma for 01sqrex 11612. (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 11609* Lemma for 01sqrex 11612. (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 11610* Lemma for 01sqrex 11612. (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 11611* Lemma for 01sqrex 11612. (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 11612* 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 11613* 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 11614* Uniqueness for the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( A  e.  CC  ->  E* x ( x  e.  CC  /\  (
 ( x ^ 2
 )  =  A  /\  0  <_  ( Re `  x )  /\  ( _i 
 x.  x )  e/  RR+ ) ) )
 
Theoremresqreu 11615* 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 11616 Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( sqr `  A )  e.  RR )
 
Theoremresqrthlem 11617 Lemma for resqrth 11618. (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 11618 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 11619 Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( ( sqr `  A )  x.  ( sqr `  A ) )  =  A )
 
Theoremsqrge0 11620 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 11621 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 11622 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 11623 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 11624 Square root is strictly monotonic. Closed form of sqrlti 11750. (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 11625 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 11626 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 11627 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 11628 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 11629 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 11630 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 11631 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 11632 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 11633 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 11634 The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)
 |-  ( sqr `  1
 )  =  1
 
Theoremsqr4 11635 The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)
 |-  ( sqr `  4
 )  =  2
 
Theoremsqr9 11636 The square root of 9 is 3. (Contributed by NM, 11-May-2004.)
 |-  ( sqr `  9
 )  =  3
 
Theoremsqr2gt1lt2 11637 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 11638 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 11597 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 8685 or i2 11081 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 11639 Absolute value of negative. (Contributed by NM, 27-Feb-2005.)
 |-  ( A  e.  CC  ->  ( abs `  -u A )  =  ( abs `  A ) )
 
Theoremabscl 11640 Real closure of absolute value. (Contributed by NM, 3-Oct-1999.)
 |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
 
Theoremabscj 11641 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 11642 Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.)
 |-  ( A  e.  CC  ->  ( ( abs `  A ) ^ 2 )  =  ( A  x.  ( * `  A ) ) )
 
Theoremabsvalsq2 11643 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 11644 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 11645 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 11646 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 11647 The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( abs `  0
 )  =  0
 
Theoremabsi 11648 The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
 |-  ( abs `  _i )  =  1
 
Theoremabsge0 11649 Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( A  e.  CC  ->  0  <_  ( abs `  A ) )
 
Theoremabsrpcl 11650 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 11651 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 11652 A complex number is zero iff its absolute value is zero. Deduction form of abs00 11651. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( abs `  A )  =  0  <->  A  =  0
 ) )
 
Theoremabs00bd 11653 If a complex number is zero, its absolute value is zero. Converse of abs00d 11805. One-way deduction form of abs00 11651. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  =  0 )   =>    |-  ( ph  ->  ( abs `  A )  =  0 )
 
Theoremabsreimsq 11654 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 11655 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 11656 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 11657 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 11658 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 11659 The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
 |-  ( abs `  1
 )  =  1
 
Theoremabsnid 11660 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 11661 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 11662 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 11663 Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)
 |-  ( A  e.  RR  ->  ( abs `  A )  =  ( sqr `  ( A ^ 2
 ) ) )
 
Theoremabsresq 11664 Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)
 |-  ( A  e.  RR  ->  ( ( abs `  A ) ^ 2 )  =  ( A ^ 2
 ) )
 
Theoremabsmod0 11665  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 11666 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 11667 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 11668 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 11669 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 11670 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 11671 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 11672 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 11673 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 11674 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 11675 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 11676 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 11677 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 )
 
Theoremabsdiflt 11678 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 11679 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 11680 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 11681 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 11682 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 11683 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 11684 The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)
 |-  ( A  e.  CC  ->  ( abs `  ( abs `  A ) )  =  ( abs `  A ) )
 
Theoremabsgt0 11685 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 11686 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 11687 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 11688 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 11689 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 11690 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 11691 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 11692 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 11693 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 11694 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 11695 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 11696* 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 11697* 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 11698 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 11699 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 11700 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 ) ) )
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