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Theorem List for Intuitionistic Logic Explorer - 10901-11000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremleabsd 10901 A real number is less than or equal to its absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  <_  ( abs `  A ) )
 
Theoremabsred 10902 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( abs `  A )  =  ( sqr `  ( A ^ 2 ) ) )
 
Theoremresqrtcld 10903 The square root of a nonnegative real is a real. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( sqr `  A )  e. 
 RR )
 
Theoremsqrtmsqd 10904 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( sqr `  ( A  x.  A ) )  =  A )
 
Theoremsqrtsqd 10905 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( sqr `  ( A ^
 2 ) )  =  A )
 
Theoremsqrtge0d 10906 The square root of a nonnegative real is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  0  <_  ( sqr `  A ) )
 
Theoremabsidd 10907 A nonnegative number is its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( abs `  A )  =  A )
 
Theoremsqrtdivd 10908 Square root distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( sqr `  ( A  /  B ) )  =  ( ( sqr `  A )  /  ( sqr `  B ) ) )
 
Theoremsqrtmuld 10909 Square root distributes over multiplication. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  ( sqr `  ( A  x.  B ) )  =  ( ( sqr `  A )  x.  ( sqr `  B ) ) )
 
Theoremsqrtsq2d 10910 Relationship between square root and squares. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  (
 ( sqr `  A )  =  B  <->  A  =  ( B ^ 2 ) ) )
 
Theoremsqrtled 10911 Square root is monotonic. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( sqr `  A )  <_  ( sqr `  B ) ) )
 
Theoremsqrtltd 10912 Square root is strictly monotonic. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  ( A  <  B  <->  ( sqr `  A )  <  ( sqr `  B ) ) )
 
Theoremsqr11d 10913 The square root function is one-to-one. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   &    |-  ( ph  ->  ( sqr `  A )  =  ( sqr `  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremabsltd 10914 Absolute value and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( ( abs `  A )  <  B  <->  ( -u B  <  A  /\  A  <  B ) ) )
 
Theoremabsled 10915 Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( ( abs `  A )  <_  B  <->  ( -u B  <_  A  /\  A  <_  B ) ) )
 
Theoremabssubge0d 10916 Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( abs `  ( B  -  A ) )  =  ( B  -  A ) )
 
Theoremabssuble0d 10917 Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  =  ( B  -  A ) )
 
Theoremabsdifltd 10918 The absolute value of a difference and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( abs `  ( A  -  B ) )  <  C 
 <->  ( ( B  -  C )  <  A  /\  A  <  ( B  +  C ) ) ) )
 
Theoremabsdifled 10919 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( abs `  ( A  -  B ) )  <_  C 
 <->  ( ( B  -  C )  <_  A  /\  A  <_  ( B  +  C ) ) ) )
 
Theoremicodiamlt 10920 Two elements in a half-open interval have separation strictly less than the difference between the endpoints. (Contributed by Stefan O'Rear, 12-Sep-2014.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,) B )  /\  D  e.  ( A [,) B ) ) )  ->  ( abs `  ( C  -  D ) )  < 
 ( B  -  A ) )
 
Theoremabscld 10921 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  A )  e. 
 RR )
 
Theoremabsvalsqd 10922 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( abs `  A ) ^ 2 )  =  ( A  x.  ( * `  A ) ) )
 
Theoremabsvalsq2d 10923 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( abs `  A ) ^ 2 )  =  ( ( ( Re
 `  A ) ^
 2 )  +  (
 ( Im `  A ) ^ 2 ) ) )
 
Theoremabsge0d 10924 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  0  <_  ( abs `  A ) )
 
Theoremabsval2d 10925 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  A )  =  ( sqr `  (
 ( ( Re `  A ) ^ 2
 )  +  ( ( Im `  A ) ^ 2 ) ) ) )
 
Theoremabs00d 10926 The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  =  0 )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremabsne0d 10927 The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( abs `  A )  =/=  0 )
 
Theoremabsrpclapd 10928 The absolute value of a complex number apart from zero is a positive real. (Contributed by Jim Kingdon, 13-Aug-2021.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  ( abs `  A )  e.  RR+ )
 
Theoremabsnegd 10929 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  -u A )  =  ( abs `  A ) )
 
Theoremabscjd 10930 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 10931 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 10932 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 10933 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 10934 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 10935 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 10936 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 10937 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 10938 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 10939 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 10940 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 10941 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 10942* 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 10002. (Contributed by BJ, 15-Oct-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  E. x  e.  QQ  ( abs `  ( x  -  A ) )  <  B )
 
4.7.5  The maximum of two real numbers
 
Theoremmaxcom 10943 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 10944 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 10945 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 10946 Lemma for maxabs 10949. A variation of maxleim 10945- 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 10947 Lemma for maxabs 10949. 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 10948* Lemma for maxabs 10949. 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 10949 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 10950 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 10951 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 10952 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 10953 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 10954 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 10955 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 10956 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 10957 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 10958 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 10959 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 10960* 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 10961* 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 10962 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 9066 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 )
 ) )
 
Theoremrpmaxcl 10963 The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR+ )
 
Theoremzmaxcl 10964 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 10965 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 10966* 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 10967* 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 )
 
4.7.6  The minimum of two real numbers
 
Theoremmincom 10968 The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.)
 |- inf
 ( { A ,  B } ,  RR ,  <  )  = inf ( { B ,  A } ,  RR ,  <  )
 
Theoremminmax 10969 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 10970 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 10971 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 10972 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 10973 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 10974 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 10975 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 10976 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 9066 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 10977 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 10978 Lemma for bdtri 10979. (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 10979 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 10980 Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 10802 and mulap0bd 8386 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 ) )
 
4.7.7  The maximum of two extended reals
 
Theoremxrmaxleim 10981 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 10982 Lemma for xrmaxif 10988. 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 10983 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 10984 Lemma for xrmaxif 10988. A variation of xrmaxleim 10981- 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 10985 Lemma for xrmaxif 10988. 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 10986 Lemma for xrmaxif 10988. 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 10987* Lemma for xrmaxif 10988. 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 10988 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 10989 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 10990 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 10991 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 10992 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 10993 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 10994 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 10995 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 10996 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 10997 Lemma for xrmaxadd 10998. 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 10998 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* ,  <  ) ) )
 
4.7.8  The minimum of two extended reals
 
Theoremxrnegiso 10999 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 11000* 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* ,  <  ) )
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