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Theorem List for Intuitionistic Logic Explorer - 11201-11300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsqrtltd 11201 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 11202 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 11203 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 11204 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 11205 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 11206 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 11207 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 11208 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 11209 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 11210 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  A )  e. 
 RR )
 
Theoremabsvalsqd 11211 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 11212 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 11213 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  0  <_  ( abs `  A ) )
 
Theoremabsval2d 11214 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 11215 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 11216 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 11217 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 11218 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  -u A )  =  ( abs `  A ) )
 
Theoremabscjd 11219 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 11220 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 11221 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 11222 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 11223 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 11224 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 11225 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 11226 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 11227 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 11228 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 11229 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 11230 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 11231* 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 10277. (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 11232 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 11233 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 11234 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 11235 Lemma for maxabs 11238. A variation of maxleim 11234- 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 11236 Lemma for maxabs 11238. 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 11237* Lemma for maxabs 11238. 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 11238 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 11239 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 11240 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 11241 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 11242 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 11243 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 11244 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 11245 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 11246 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 11247 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 11248 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 11249* 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 11250* 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 11251 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 9317 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 11252 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 11253 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 11254 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 11255* 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 11256* 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 11257 The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.)
 |- inf
 ( { A ,  B } ,  RR ,  <  )  = inf ( { B ,  A } ,  RR ,  <  )
 
Theoremminmax 11258 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 11259 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 11260 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 11261 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 11262 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 11263 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 11264 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 11265 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 9317 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 11266 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 11267 Lemma for bdtri 11268. (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 11268 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 11269 Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11091 and mulap0bd 8634 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 ) )
 
Theoremmingeb 11270 Equivalence of  <_ and being equal to the minimum of two reals. (Contributed by Jim Kingdon, 14-Oct-2024.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <-> inf ( { A ,  B } ,  RR ,  <  )  =  A ) )
 
Theorem2zinfmin 11271 Two ways to express the minimum of two integers. Because order of integers is decidable, we have more flexibility than for real numbers. (Contributed by Jim Kingdon, 14-Oct-2024.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  -> inf ( { A ,  B } ,  RR ,  <  )  =  if ( A  <_  B ,  A ,  B )
 )
 
4.7.7  The maximum of two extended reals
 
Theoremxrmaxleim 11272 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 11273 Lemma for xrmaxif 11279. 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 11274 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 11275 Lemma for xrmaxif 11279. A variation of xrmaxleim 11272- 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 11276 Lemma for xrmaxif 11279. 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 11277 Lemma for xrmaxif 11279. 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 11278* Lemma for xrmaxif 11279. 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 11279 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 11280 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 11281 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 11282 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 11283 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 11284 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 11285 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 11286 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 11287 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 11288 Lemma for xrmaxadd 11289. 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 11289 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 11290 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 11291* 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 11292 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 11293 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 11294 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 11295 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 11296 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 11297 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 11298 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 11299 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 11300 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 ) ) )
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