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Theorem List for Intuitionistic Logic Explorer - 11201-11300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremabsdifled 11201 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 11202 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 11203 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  A )  e. 
 RR )
 
Theoremabsvalsqd 11204 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 11205 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 11206 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  0  <_  ( abs `  A ) )
 
Theoremabsval2d 11207 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 11208 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 11209 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 11210 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 11211 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  -u A )  =  ( abs `  A ) )
 
Theoremabscjd 11212 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 11213 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 11214 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 11215 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 11216 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 11217 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 11218 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 11219 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 11220 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 11221 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 11222 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 11223 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 11224* 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 10270. (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 11225 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 11226 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 11227 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 11228 Lemma for maxabs 11231. A variation of maxleim 11227- 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 11229 Lemma for maxabs 11231. 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 11230* Lemma for maxabs 11231. 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 11231 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 11232 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 11233 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 11234 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 11235 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 11236 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 11237 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 11238 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 11239 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 11240 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 11241 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 11242* 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 11243* 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 11244 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 9310 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 11245 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 11246 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 11247 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 11248* 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 11249* 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 11250 The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.)
 |- inf
 ( { A ,  B } ,  RR ,  <  )  = inf ( { B ,  A } ,  RR ,  <  )
 
Theoremminmax 11251 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 11252 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 11253 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 11254 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 11255 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 11256 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 11257 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 11258 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 9310 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 11259 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 11260 Lemma for bdtri 11261. (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 11261 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 11262 Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11084 and mulap0bd 8627 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 11263 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 11264 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 11265 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 11266 Lemma for xrmaxif 11272. 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 11267 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 11268 Lemma for xrmaxif 11272. A variation of xrmaxleim 11265- 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 11269 Lemma for xrmaxif 11272. 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 11270 Lemma for xrmaxif 11272. 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 11271* Lemma for xrmaxif 11272. 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 11272 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 11273 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 11274 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 11275 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 11276 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 11277 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 11278 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 11279 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 11280 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 11281 Lemma for xrmaxadd 11282. 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 11282 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 11283 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 11284* 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 11285 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 11286 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 11287 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 11288 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 11289 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 11290 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 11291 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 11292 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 11293 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 11294 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 11295 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 11296 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 11297 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 11298 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* ,  <  )
 ) )
 
4.8  Elementary limits and convergence
 
4.8.1  Limits
 
Syntaxcli 11299 Extend class notation with convergence relation for limits.
 class  ~~>
 
Definitiondf-clim 11300* Define the limit relation for complex number sequences. See clim 11302 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 ) ) }
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