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Theorem List for Intuitionistic Logic Explorer - 10801-10900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremabsnegd 10801 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  -u A )  =  ( abs `  A ) )
 
Theoremabscjd 10802 The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( * `  A ) )  =  ( abs `  A ) )
 
Theoremreleabsd 10803 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 Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  A )  <_  ( abs `  A )
 )
 
Theoremabsexpd 10804 Absolute value of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N ) )
 
Theoremabssubd 10805 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  =  ( abs `  ( B  -  A ) ) )
 
Theoremabsmuld 10806 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  x.  B ) )  =  ( ( abs `  A )  x.  ( abs `  B ) ) )
 
Theoremabsdivapd 10807 Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( abs `  ( A  /  B ) )  =  ( ( abs `  A )  /  ( abs `  B ) ) )
 
Theoremabstrid 10808 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  +  B ) )  <_  ( ( abs `  A )  +  ( abs `  B ) ) )
 
Theoremabs2difd 10809 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( abs `  A )  -  ( abs `  B ) )  <_  ( abs `  ( A  -  B ) ) )
 
Theoremabs2dif2d 10810 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  <_  ( ( abs `  A )  +  ( abs `  B ) ) )
 
Theoremabs2difabsd 10811 Absolute value of difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( ( abs `  A )  -  ( abs `  B )
 ) )  <_  ( abs `  ( A  -  B ) ) )
 
Theoremabs3difd 10812 Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  <_  ( ( abs `  ( A  -  C ) )  +  ( abs `  ( C  -  B ) ) ) )
 
Theoremabs3lemd 10813 Lemma involving absolute value of differences. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  ( abs `  ( A  -  C ) )  < 
 ( D  /  2
 ) )   &    |-  ( ph  ->  ( abs `  ( C  -  B ) )  < 
 ( D  /  2
 ) )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  <  D )
 
Theoremqdenre 10814* The rational numbers are dense in 
RR: any real number can be approximated with arbitrary precision by a rational number. For order theoretic density, see qbtwnre 9875. (Contributed by BJ, 15-Oct-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  E. x  e.  QQ  ( abs `  ( x  -  A ) )  <  B )
 
3.7.5  The maximum of two real numbers
 
Theoremmaxcom 10815 The maximum of two reals is commutative. Lemma 3.9 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)
 |- 
 sup ( { A ,  B } ,  RR ,  <  )  =  sup ( { B ,  A } ,  RR ,  <  )
 
Theoremmaxabsle 10816 An upper bound for  { A ,  B }. (Contributed by Jim Kingdon, 20-Dec-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  (
 ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) ) 
 /  2 ) )
 
Theoremmaxleim 10817 Value of maximum when we know which number is larger. (Contributed by Jim Kingdon, 21-Dec-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B 
 ->  sup ( { A ,  B } ,  RR ,  <  )  =  B ) )
 
Theoremmaxabslemab 10818 Lemma for maxabs 10821. A variation of maxleim 10817- that is, if we know which of two real numbers is larger, we know the maximum of the two. (Contributed by Jim Kingdon, 21-Dec-2021.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  (
 ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) ) 
 /  2 )  =  B )
 
Theoremmaxabslemlub 10819 Lemma for maxabs 10821. A least upper bound for  { A ,  B }. (Contributed by Jim Kingdon, 20-Dec-2021.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  C  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) ) 
 /  2 ) )   =>    |-  ( ph  ->  ( C  <  A  \/  C  <  B ) )
 
Theoremmaxabslemval 10820* Lemma for maxabs 10821. Value of the supremum. (Contributed by Jim Kingdon, 22-Dec-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  / 
 2 )  e.  RR  /\ 
 A. x  e.  { A ,  B }  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) ) 
 /  2 )  < 
 x  /\  A. x  e. 
 RR  ( x  < 
 ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) ) 
 /  2 )  ->  E. z  e.  { A ,  B } x  < 
 z ) ) )
 
Theoremmaxabs 10821 Maximum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 20-Dec-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) ) 
 /  2 ) )
 
Theoremmaxcl 10822 The maximum of two real numbers is a real number. (Contributed by Jim Kingdon, 22-Dec-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
 
Theoremmaxle1 10823 The maximum of two reals is no smaller than the first real. Lemma 3.10 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  sup ( { A ,  B } ,  RR ,  <  )
 )
 
Theoremmaxle2 10824 The maximum of two reals is no smaller than the second real. Lemma 3.10 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  <_  sup ( { A ,  B } ,  RR ,  <  )
 )
 
Theoremmaxleast 10825 The maximum of two reals is a least upper bound. Lemma 3.11 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 22-Dec-2021.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A 
 <_  C  /\  B  <_  C ) )  ->  sup ( { A ,  B } ,  RR ,  <  )  <_  C )
 
Theoremmaxleastb 10826 Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Jim Kingdon, 31-Jan-2022.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( sup ( { A ,  B } ,  RR ,  <  )  <_  C  <->  ( A  <_  C 
 /\  B  <_  C ) ) )
 
Theoremmaxleastlt 10827 The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  C  <  sup ( { A ,  B } ,  RR ,  <  ) ) ) 
 ->  ( C  <  A  \/  C  <  B ) )
 
Theoremmaxleb 10828 Equivalence of  <_ and being equal to the maximum of two reals. Lemma 3.12 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  sup ( { A ,  B } ,  RR ,  <  )  =  B ) )
 
Theoremdfabsmax 10829 Absolute value of a real number in terms of maximum. Definition 3.13 of [Geuvers], p. 11. (Contributed by BJ and Jim Kingdon, 21-Dec-2021.)
 |-  ( A  e.  RR  ->  ( abs `  A )  =  sup ( { A ,  -u A } ,  RR ,  <  )
 )
 
Theoremmaxltsup 10830 Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 10-Feb-2022.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( sup ( { A ,  B } ,  RR ,  <  )  <  C  <->  ( A  <  C 
 /\  B  <  C ) ) )
 
Theoremmax0addsup 10831 The sum of the positive and negative part functions is the absolute value function over the reals. (Contributed by Jim Kingdon, 30-Jan-2022.)
 |-  ( A  e.  RR  ->  ( sup ( { A ,  0 } ,  RR ,  <  )  +  sup ( { -u A ,  0 } ,  RR ,  <  ) )  =  ( abs `  A ) )
 
Theoremrexanre 10832* 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 ) ) ) )
 
Theoremrexico 10833* 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 ) ) )
 
Theoremmaxclpr 10834 The maximum of two real numbers is one of those numbers if and only if dichotomy ( A  <_  B  \/  B  <_  A) holds. For example, this can be combined with zletric 8950 if one is dealing with integers, but real number dichotomy in general does not follow from our axioms. (Contributed by Jim Kingdon, 1-Feb-2022.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( sup ( { A ,  B } ,  RR ,  <  )  e.  { A ,  B } 
 <->  ( A  <_  B  \/  B  <_  A )
 ) )
 
Theoremzmaxcl 10835 The maximum of two integers is an integer. (Contributed by Jim Kingdon, 27-Sep-2022.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  ZZ )
 
Theorem2zsupmax 10836 Two ways to express the maximum of two integers. Because order of integers is decidable, we have more flexibility than for real numbers. (Contributed by Jim Kingdon, 22-Jan-2023.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  sup ( { A ,  B } ,  RR ,  <  )  =  if ( A  <_  B ,  B ,  A )
 )
 
Theoremfimaxre2 10837* A nonempty finite set of real numbers has an upper bound. (Contributed by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro, 13-Feb-2014.)
 |-  ( ( A  C_  RR  /\  A  e.  Fin )  ->  E. x  e.  RR  A. y  e.  A  y 
 <_  x )
 
Theoremnegfi 10838* The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.)
 |-  ( ( A  C_  RR  /\  A  e.  Fin )  ->  { n  e. 
 RR  |  -u n  e.  A }  e.  Fin )
 
3.7.6  The minimum of two real numbers
 
Theoremmincom 10839 The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.)
 |- inf
 ( { A ,  B } ,  RR ,  <  )  = inf ( { B ,  A } ,  RR ,  <  )
 
Theoremminmax 10840 Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 8-Feb-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR ,  <  )  =  -u sup ( { -u A ,  -u B } ,  RR ,  <  ) )
 
Theoremmincl 10841 The minumum of two real numbers is a real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR ,  <  )  e.  RR )
 
Theoremmin1inf 10842 The minimum of two numbers is less than or equal to the first. (Contributed by Jim Kingdon, 8-Feb-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR ,  <  )  <_  A )
 
Theoremmin2inf 10843 The minimum of two numbers is less than or equal to the second. (Contributed by Jim Kingdon, 9-Feb-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR ,  <  )  <_  B )
 
Theoremlemininf 10844 Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by NM, 3-Aug-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_ inf ( { B ,  C } ,  RR ,  <  )  <->  ( A  <_  B  /\  A  <_  C ) ) )
 
Theoremltmininf 10845 Two ways of saying a number is less than the minimum of two others. (Contributed by Jim Kingdon, 10-Feb-2022.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  < inf ( { B ,  C } ,  RR ,  <  )  <->  ( A  <  B  /\  A  <  C ) ) )
 
Theoremminabs 10846 The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR ,  <  )  =  ( ( ( A  +  B )  -  ( abs `  ( A  -  B ) ) ) 
 /  2 ) )
 
Theoremminclpr 10847 The minimum of two real numbers is one of those numbers if and only if dichotomy ( A  <_  B  \/  B  <_  A) holds. For example, this can be combined with zletric 8950 if one is dealing with integers, but real number dichotomy in general does not follow from our axioms. (Contributed by Jim Kingdon, 23-May-2023.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (inf ( { A ,  B } ,  RR ,  <  )  e.  { A ,  B } 
 <->  ( A  <_  B  \/  B  <_  A )
 ) )
 
Theoremrpmincl 10848 The minumum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  -> inf ( { A ,  B } ,  RR ,  <  )  e.  RR+ )
 
Theorembdtrilem 10849 Lemma for bdtri 10850. (Contributed by Steven Nguyen and Jim Kingdon, 17-May-2023.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  ->  (
 ( abs `  ( A  -  C ) )  +  ( abs `  ( B  -  C ) ) ) 
 <_  ( C  +  ( abs `  ( ( A  +  B )  -  C ) ) ) )
 
Theorembdtri 10850 Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )  /\  C  e.  RR+ )  -> inf ( {
 ( A  +  B ) ,  C } ,  RR ,  <  )  <_  (inf ( { A ,  C } ,  RR ,  <  )  + inf ( { B ,  C } ,  RR ,  <  )
 ) )
 
Theoremmul0inf 10851 Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 10674 and mulap0bd 8279 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  =  0  <-> inf ( { ( abs `  A ) ,  ( abs `  B ) } ,  RR ,  <  )  =  0 ) )
 
3.7.7  The maximum of two extended reals
 
Theoremxrmaxleim 10852 Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  B ) )
 
Theoremxrmaxiflemcl 10853 Lemma for xrmaxif 10859. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  e.  RR* )
 
Theoremxrmaxifle 10854 An upper bound for  { A ,  B } in the extended reals. (Contributed by Jim Kingdon, 26-Apr-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  <_  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )
 
Theoremxrmaxiflemab 10855 Lemma for xrmaxif 10859. A variation of xrmaxleim 10852- that is, if we know which of two real numbers is larger, we know the maximum of the two. (Contributed by Jim Kingdon, 26-Apr-2023.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
 ) ) ) )  =  B )
 
Theoremxrmaxiflemlub 10856 Lemma for xrmaxif 10859. A least upper bound for  { A ,  B }. (Contributed by Jim Kingdon, 28-Apr-2023.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  C  <  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )   =>    |-  ( ph  ->  ( C  <  A  \/  C  <  B ) )
 
Theoremxrmaxiflemcom 10857 Lemma for xrmaxif 10859. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  sup ( { B ,  A } ,  RR ,  <  ) ) ) ) ) )
 
Theoremxrmaxiflemval 10858* Lemma for xrmaxif 10859. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
 |-  M  =  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )   =>    |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( M  e.  RR*  /\ 
 A. x  e.  { A ,  B }  -.  M  <  x  /\  A. x  e.  RR*  ( x  <  M  ->  E. z  e.  { A ,  B } x  <  z ) ) )
 
Theoremxrmaxif 10859 Maximum of two extended reals in terms of  if expressions. (Contributed by Jim Kingdon, 26-Apr-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
 ) ) ) ) )
 
Theoremxrmaxcl 10860 The maximum of two extended reals is an extended real. (Contributed by Jim Kingdon, 29-Apr-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  sup ( { A ,  B } ,  RR* ,  <  )  e.  RR* )
 
Theoremxrmax1sup 10861 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 30-Apr-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  <_  sup ( { A ,  B } ,  RR* ,  <  )
 )
 
Theoremxrmax2sup 10862 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 30-Apr-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  <_  sup ( { A ,  B } ,  RR* ,  <  )
 )
 
Theoremxrmaxrecl 10863 The maximum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 30-Apr-2023.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  sup ( { A ,  B } ,  RR ,  <  ) )
 
Theoremxrmaxleastlt 10864 The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  C  <  sup ( { A ,  B } ,  RR* ,  <  ) ) )  ->  ( C  <  A  \/  C  <  B ) )
 
Theoremxrltmaxsup 10865 The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( C  <  sup ( { A ,  B } ,  RR* ,  <  )  <->  ( C  <  A  \/  C  <  B ) ) )
 
Theoremxrmaxltsup 10866 Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( sup ( { A ,  B } ,  RR* ,  <  )  <  C  <->  ( A  <  C 
 /\  B  <  C ) ) )
 
Theoremxrmaxlesup 10867 Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 10-May-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( sup ( { A ,  B } ,  RR* ,  <  ) 
 <_  C  <->  ( A  <_  C 
 /\  B  <_  C ) ) )
 
Theoremxrmaxaddlem 10868 Lemma for xrmaxadd 10869. The case where  A is real. (Contributed by Jim Kingdon, 11-May-2023.)
 |-  ( ( A  e.  RR  /\  B  e.  RR*  /\  C  e.  RR* )  ->  sup ( { ( A +e B ) ,  ( A +e C ) } ,  RR*
 ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
 
Theoremxrmaxadd 10869 Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  sup ( { ( A +e B ) ,  ( A +e C ) } ,  RR* ,  <  )  =  ( A +e sup ( { B ,  C } ,  RR* ,  <  ) ) )
 
3.7.8  The minimum of two extended reals
 
Theoremxrnegiso 10870 Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
 |-  F  =  ( x  e.  RR*  |->  -e
 x )   =>    |-  ( F  Isom  <  ,  `'  <  ( RR* ,  RR* )  /\  `' F  =  F )
 
Theoreminfxrnegsupex 10871* The infimum of a set of extended reals  A is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 2-May-2023.)
 |-  ( ph  ->  E. x  e.  RR*  ( A. y  e.  A  -.  y  < 
 x  /\  A. y  e.  RR*  ( x  <  y  ->  E. z  e.  A  z  <  y ) ) )   &    |-  ( ph  ->  A 
 C_  RR* )   =>    |-  ( ph  -> inf ( A ,  RR* ,  <  )  =  -e sup ( { z  e.  RR*  |  -e z  e.  A } ,  RR* ,  <  ) )
 
Theoremxrnegcon1d 10872 Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   =>    |-  ( ph  ->  (  -e A  =  B  <->  -e B  =  A ) )
 
Theoremxrminmax 10873 Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> inf ( { A ,  B } ,  RR* ,  <  )  =  -e sup ( {  -e A ,  -e B } ,  RR* ,  <  ) )
 
Theoremxrmincl 10874 The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> inf ( { A ,  B } ,  RR* ,  <  )  e.  RR* )
 
Theoremxrmin1inf 10875 The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> inf ( { A ,  B } ,  RR* ,  <  ) 
 <_  A )
 
Theoremxrmin2inf 10876 The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> inf ( { A ,  B } ,  RR* ,  <  ) 
 <_  B )
 
Theoremxrmineqinf 10877 The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) (Revised by Jim Kingdon, 3-May-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  B  <_  A )  -> inf ( { A ,  B } ,  RR* ,  <  )  =  B )
 
Theoremxrltmininf 10878 Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 3-May-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( A  < inf ( { B ,  C } ,  RR* ,  <  )  <->  ( A  <  B 
 /\  A  <  C ) ) )
 
Theoremxrlemininf 10879 Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 4-May-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( A  <_ inf ( { B ,  C } ,  RR* ,  <  )  <->  ( A  <_  B 
 /\  A  <_  C ) ) )
 
Theoremxrminltinf 10880 Two ways of saying an extended real is greater than the minimum of two others. (Contributed by Jim Kingdon, 19-May-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (inf ( { B ,  C } ,  RR* ,  <  )  <  A  <->  ( B  <  A  \/  C  <  A ) ) )
 
Theoremxrminrecl 10881 The minimum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 18-May-2023.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR* ,  <  )  = inf ( { A ,  B } ,  RR ,  <  )
 )
 
Theoremxrminrpcl 10882 The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  -> inf ( { A ,  B } ,  RR* ,  <  )  e.  RR+ )
 
Theoremxrminadd 10883 Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  -> inf ( {
 ( A +e B ) ,  ( A +e C ) } ,  RR* ,  <  )  =  ( A +einf ( { B ,  C } ,  RR* ,  <  ) ) )
 
Theoremxrbdtri 10884 Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
 |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B )  /\  ( C  e.  RR*  /\  0  <  C ) )  -> inf ( { ( A +e B ) ,  C } ,  RR* ,  <  ) 
 <_  (inf ( { A ,  C } ,  RR* ,  <  ) +einf ( { B ,  C } ,  RR* ,  <  ) ) )
 
Theoremiooinsup 10885 Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* ) )  ->  (
 ( A (,) B )  i^i  ( C (,) D ) )  =  ( sup ( { A ,  C } ,  RR* ,  <  ) (,)inf ( { B ,  D } ,  RR* ,  <  )
 ) )
 
3.8  Elementary limits and convergence
 
3.8.1  Limits
 
Syntaxcli 10886 Extend class notation with convergence relation for limits.
 class  ~~>
 
Definitiondf-clim 10887* Define the limit relation for complex number sequences. See clim 10889 for its relational expression. (Contributed by NM, 28-Aug-2005.)
 |-  ~~>  =  { <. f ,  y >.  |  ( y  e. 
 CC  /\  A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ( ( f `  k )  e.  CC  /\  ( abs `  ( ( f `
  k )  -  y ) )  < 
 x ) ) }
 
Theoremclimrel 10888 The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |- 
 Rel 
 ~~>
 
Theoremclim 10889* Express the predicate: The limit of complex number sequence  F is  A, or  F converges to  A. This means that for any real  x, no matter how small, there always exists an integer 
j such that the absolute difference of any later complex number in the sequence and the limit is less than  x. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  ZZ )  ->  ( F `  k
 )  =  B )   =>    |-  ( ph  ->  ( F  ~~>  A 
 <->  ( A  e.  CC  /\ 
 A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  < 
 x ) ) ) )
 
Theoremclimcl 10890 Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( F  ~~>  A  ->  A  e.  CC )
 
Theoremclim2 10891* Express the predicate: The limit of complex number sequence  F is  A, or  F converges to  A, with more general quantifier restrictions than clim 10889. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   =>    |-  ( ph  ->  ( F 
 ~~>  A  <->  ( A  e.  CC  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  < 
 x ) ) ) )
 
Theoremclim2c 10892* Express the predicate  F converges to  A. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ph  ->  A  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  B  e.  CC )   =>    |-  ( ph  ->  ( F 
 ~~>  A  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( B  -  A ) )  <  x ) )
 
Theoremclim0 10893* Express the predicate  F converges to  0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   =>    |-  ( ph  ->  ( F 
 ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  B )  < 
 x ) ) )
 
Theoremclim0c 10894* Express the predicate  F converges to  0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  ( F  ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  B )  <  x ) )
 
Theoremclimi 10895* Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ph  ->  F  ~~>  A )   =>    |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  C ) )
 
Theoremclimi2 10896* Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ph  ->  F  ~~>  A )   =>    |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( abs `  ( B  -  A ) )  <  C )
 
Theoremclimi0 10897* Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ph  ->  F  ~~>  0 )   =>    |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( abs `  B )  <  C )
 
Theoremclimconst 10898* An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  =  A )   =>    |-  ( ph  ->  F  ~~>  A )
 
Theoremclimconst2 10899 A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ZZ>= `  M )  C_  Z   &    |-  Z  e.  _V   =>    |-  (
 ( A  e.  CC  /\  M  e.  ZZ )  ->  ( Z  X.  { A } )  ~~>  A )
 
Theoremclimz 10900 The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ZZ  X.  {
 0 } )  ~~>  0
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