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Theorem List for Metamath Proof Explorer - 13301-13400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsqrtgt0 13301 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 13302 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 13303 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 13304 Square root is strictly monotonic. Closed form of sqrtlti 13431. (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 ) ) )
 
Theoremsqrt11 13305 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 ) )
 
Theoremsqrt00 13306 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 13307 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 13308 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 ) ) )
 
Theoremsqrtneglem 13309 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+ )
 )
 
Theoremsqrtneg 13310 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 ) ) )
 
Theoremsqrtsq2 13311 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 13312 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 13313 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 13314 The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)
 |-  ( sqr `  1
 )  =  1
 
Theoremsqrt4 13315 The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)
 |-  ( sqr `  4
 )  =  2
 
Theoremsqrt9 13316 The square root of 9 is 3. (Contributed by NM, 11-May-2004.)
 |-  ( sqr `  9
 )  =  3
 
Theoremsqrt2gt1lt2 13317 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 )
 
Theoremsqrtm1 13318 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-sqrt 13277 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 9606 or i2 12372 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 13319 Absolute value of negative. (Contributed by NM, 27-Feb-2005.)
 |-  ( A  e.  CC  ->  ( abs `  -u A )  =  ( abs `  A ) )
 
Theoremabscl 13320 Real closure of absolute value. (Contributed by NM, 3-Oct-1999.)
 |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
 
Theoremabscj 13321 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 13322 Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.)
 |-  ( A  e.  CC  ->  ( ( abs `  A ) ^ 2 )  =  ( A  x.  ( * `  A ) ) )
 
Theoremabsvalsq2 13323 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 13324 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 13325 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 13326 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 13327 The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( abs `  0
 )  =  0
 
Theoremabsi 13328 The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
 |-  ( abs `  _i )  =  1
 
Theoremabsge0 13329 Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( A  e.  CC  ->  0  <_  ( abs `  A ) )
 
Theoremabsrpcl 13330 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 13331 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 13332 A complex number is zero iff its absolute value is zero. Deduction form of abs00 13331. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( abs `  A )  =  0  <->  A  =  0
 ) )
 
Theoremabs00bd 13333 If a complex number is zero, its absolute value is zero. Converse of abs00d 13486. One-way deduction form of abs00 13331. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  =  0 )   =>    |-  ( ph  ->  ( abs `  A )  =  0 )
 
Theoremabsreimsq 13334 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 13335 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 13336 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 13337 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 13338 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 13339 The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
 |-  ( abs `  1
 )  =  1
 
Theoremabsnid 13340 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 13341 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 13342 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 13343 Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)
 |-  ( A  e.  RR  ->  ( abs `  A )  =  ( sqr `  ( A ^ 2
 ) ) )
 
Theoremabsresq 13344 Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)
 |-  ( A  e.  RR  ->  ( ( abs `  A ) ^ 2 )  =  ( A ^ 2
 ) )
 
Theoremabsmod0 13345  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 13346 Absolute value of positive integer exponentiation. (Contributed by NM, 5-Jan-2006.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N ) )
 
Theoremabsexpz 13347 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 13348 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 13349 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 13350 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 13351 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 13352 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 13353 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 13354 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 )
 
Theoremzabscl 13355 The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)
 |-  ( A  e.  ZZ  ->  ( abs `  A )  e.  ZZ )
 
Theoremabslt 13356 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 13357 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 13358 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 13359 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 13360 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 13361 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 13362 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 13363 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 13364 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 13365 The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)
 |-  ( A  e.  CC  ->  ( abs `  ( abs `  A ) )  =  ( abs `  A ) )
 
Theoremabsgt0 13366 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 13367 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 13368 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 13369 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 13370 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 13371 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 13372 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 13373 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 13374 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 13375 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 13376 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 13377* 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 13378* 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 13379 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 13380 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 13381 Lemma involving absolute values. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  ( (
 ( * `  ( A  /  ( abs `  A ) ) )  x.  A )  +  (
 ( A  /  ( abs `  A ) )  x.  ( * `  A ) ) )  =  ( 2  x.  ( abs `  A ) ) )
 
Theoremrddif 13382 The difference between a real number and its nearest integer is less than or equal to one half. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Sep-2015.)
 |-  ( A  e.  RR  ->  ( abs `  (
 ( |_ `  ( A  +  ( 1  /  2 ) ) )  -  A ) )  <_  ( 1  /  2 ) )
 
Theoremabsrdbnd 13383 Bound on the absolute value of a real number rounded to the nearest integer. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Sep-2015.)
 |-  ( A  e.  RR  ->  ( abs `  ( |_ `  ( A  +  ( 1  /  2
 ) ) ) ) 
 <_  ( ( |_ `  ( abs `  A ) )  +  1 ) )
 
Theoremfzomaxdiflem 13384 Lemma for fzomaxdif 13385. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( ( ( A  e.  ( C..^ D )  /\  B  e.  ( C..^ D ) )  /\  A  <_  B )  ->  ( abs `  ( B  -  A ) )  e.  ( 0..^ ( D  -  C ) ) )
 
Theoremfzomaxdif 13385 A bound on the separation of two points in a half-open range. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( ( A  e.  ( C..^ D )  /\  B  e.  ( C..^ D ) )  ->  ( abs `  ( A  -  B ) )  e.  ( 0..^ ( D  -  C ) ) )
 
Theoremuzin2 13386 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 13387* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)
 |-  ( E. j  e. 
 ZZ  A. k  e.  ( ZZ>=
 `  j ) (
 ph  /\  ps )  <->  ( E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ph  /\  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ps ) )
 
Theoremrexanre 13388* Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.)
 |-  ( A  C_  RR  ->  ( E. j  e. 
 RR  A. k  e.  A  ( j  <_  k  ->  ( ph  /\  ps )
 ) 
 <->  ( E. j  e. 
 RR  A. k  e.  A  ( j  <_  k  ->  ph )  /\  E. j  e.  RR  A. k  e.  A  ( j  <_  k  ->  ps ) ) ) )
 
Theoremrexfiuz 13389* 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 13390* 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 13391* 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 13392* A version of 19.29 1730 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 13393* A version of r19.2z 3892 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ph  ->  E. k  e.  Z  ph )
 
Theoremrexuzre 13394* Convert an upper real quantifier to an upper integer quantifier. (Contributed by Mario Carneiro, 7-May-2016.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ph 
 <-> 
 E. j  e.  RR  A. k  e.  Z  ( j  <_  k  ->  ph ) ) )
 
Theoremrexico 13395* Restrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  ( ( A  C_  RR  /\  B  e.  RR )  ->  ( E. j  e.  ( B [,) +oo ) A. k  e.  A  ( j  <_  k  ->  ph )  <->  E. j  e.  RR  A. k  e.  A  ( j  <_  k  ->  ph ) ) )
 
Theoremcau3lem 13396* Lemma for cau3 13397. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 1-May-2014.)
 |-  Z  C_  ZZ   &    |-  ( ta  ->  ps )   &    |-  ( ( F `
  k )  =  ( F `  j
 )  ->  ( ps  <->  ch ) )   &    |-  ( ( F `
  k )  =  ( F `  m )  ->  ( ps  <->  th ) )   &    |-  (
 ( ph  /\  ch  /\  ps )  ->  ( G `  ( ( F `  j ) D ( F `  k ) ) )  =  ( G `  ( ( F `  k ) D ( F `  j ) ) ) )   &    |-  ( ( ph  /\ 
 th  /\  ch )  ->  ( G `  (
 ( F `  m ) D ( F `  j ) ) )  =  ( G `  ( ( F `  j ) D ( F `  m ) ) ) )   &    |-  (
 ( ph  /\  ( ps 
 /\  th )  /\  ( ch  /\  x  e.  RR ) )  ->  ( ( ( G `  (
 ( F `  k
 ) D ( F `
  j ) ) )  <  ( x 
 /  2 )  /\  ( G `  ( ( F `  j ) D ( F `  m ) ) )  <  ( x  / 
 2 ) )  ->  ( G `  ( ( F `  k ) D ( F `  m ) ) )  <  x ) )   =>    |-  ( ph  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( ta  /\  ( G `  ( ( F `
  k ) D ( F `  j
 ) ) )  < 
 x )  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( ta 
 /\  A. m  e.  ( ZZ>=
 `  k ) ( G `  ( ( F `  k ) D ( F `  m ) ) )  <  x ) ) )
 
Theoremcau3 13397* Convert between three-quantifier and four-quantifier versions of the Cauchy criterion. (In particular, the four-quantifier version has no occurrence of  j in the assertion, so it can be used with rexanuz 13387 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( ( F `  k )  e.  CC  /\  ( abs `  ( ( F `
  k )  -  ( F `  j ) ) )  <  x ) 
 <-> 
 A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( ( F `  k )  e.  CC  /\  A. m  e.  ( ZZ>= `  k ) ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x ) )
 
Theoremcau4 13398* Change the base of a Cauchy criterion. (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  N )   =>    |-  ( N  e.  Z  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( ( F `  k )  e.  CC  /\  ( abs `  (
 ( F `  k
 )  -  ( F `
  j ) ) )  <  x )  <->  A. x  e.  RR+  E. j  e.  W  A. k  e.  ( ZZ>= `  j )
 ( ( F `  k )  e.  CC  /\  ( abs `  (
 ( F `  k
 )  -  ( F `
  j ) ) )  <  x ) ) )
 
Theoremcaubnd2 13399* A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( ( F `  k )  e.  CC  /\  ( abs `  ( ( F `
  k )  -  ( F `  j ) ) )  <  x )  ->  E. y  e.  RR  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( F `  k
 ) )  <  y
 )
 
Theoremcaubnd 13400* A Cauchy sequence of complex numbers is bounded. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 14-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( A. k  e.  Z  ( F `  k )  e.  CC  /\ 
 A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( F `  k )  -  ( F `  j ) ) )  <  x ) 
 ->  E. y  e.  RR  A. k  e.  Z  ( abs `  ( F `  k ) )  < 
 y )
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