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Theorem List for Metamath Proof Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlt2mul2divd 10401 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  x.  B )  <  ( C  x.  D )  <->  ( A  /  D )  <  ( C 
 /  B ) ) )
 
5.5.2  Infinity and the extended real number system (cont.)
 
Syntaxcxne 10402 Extend class notation to include the negative of an extended real.
 class  - e A
 
Syntaxcxad 10403 Extend class notation to include addition of extended reals.
 class  + e
 
Syntaxcxmu 10404 Extend class notation to include multiplication of extended reals.
 class  x e
 
Definitiondf-xneg 10405 Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.)
 |-  - e A  =  if ( A  =  +oo , 
 -oo ,  if ( A  =  -oo ,  +oo ,  -u A ) )
 
Definitiondf-xadd 10406* Define addition over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |- 
 + e  =  ( x  e.  RR* ,  y  e.  RR*  |->  if ( x  = 
 +oo ,  if (
 y  =  -oo , 
 0 ,  +oo ) ,  if ( x  = 
 -oo ,  if (
 y  =  +oo , 
 0 ,  -oo ) ,  if ( y  = 
 +oo ,  +oo ,  if ( y  =  -oo , 
 -oo ,  ( x  +  y ) ) ) ) ) )
 
Definitiondf-xmul 10407* Define multiplication over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  x e  =  ( x  e.  RR* ,  y  e.  RR*  |->  if ( ( x  =  0  \/  y  =  0 ) ,  0 ,  if (
 ( ( ( 0  <  y  /\  x  =  +oo )  \/  (
 y  <  0  /\  x  =  -oo ) )  \/  ( ( 0  <  x  /\  y  =  +oo )  \/  ( x  <  0  /\  y  =  -oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  y  /\  x  =  -oo )  \/  ( y  <  0  /\  x  =  +oo ) )  \/  (
 ( 0  <  x  /\  y  =  -oo )  \/  ( x  < 
 0  /\  y  =  +oo ) ) ) , 
 -oo ,  ( x  x.  y ) ) ) ) )
 
Theorempnfxr 10408 Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)
 |- 
 +oo  e.  RR*
 
Theoremmnfxr 10409 Minus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |- 
 -oo  e.  RR*
 
Theoremltxr 10410 The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B )  \/  ( A  =  -oo  /\  B  =  +oo ) )  \/  ( ( A  e.  RR  /\  B  =  +oo )  \/  ( A  =  -oo  /\  B  e.  RR ) ) ) ) )
 
Theoremelxr 10411 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo ) )
 
Theorempnfnemnf 10412 Plus and minus infinity are distinguished elements of  RR*. (Contributed by NM, 14-Oct-2005.)
 |- 
 +oo  =/=  -oo
 
Theoremxrnemnf 10413 An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  -oo )  <->  ( A  e.  RR  \/  A  =  +oo ) )
 
Theoremxrnepnf 10414 An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  <->  ( A  e.  RR  \/  A  =  -oo ) )
 
Theoremxrltnr 10415 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR*  ->  -.  A  <  A )
 
Theoremltpnf 10416 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  ->  A  <  +oo )
 
Theoremmnflt 10417 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  ->  -oo  <  A )
 
Theoremmnfltpnf 10418 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
 |- 
 -oo  <  +oo
 
Theoremmnfltxr 10419 Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)
 |-  ( ( A  e.  RR  \/  A  =  +oo )  ->  -oo  <  A )
 
Theorempnfnlt 10420 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
 |-  ( A  e.  RR*  ->  -.  +oo  <  A )
 
Theoremnltmnf 10421 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
 |-  ( A  e.  RR*  ->  -.  A  <  -oo )
 
Theorempnfge 10422 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
 |-  ( A  e.  RR*  ->  A  <_  +oo )
 
Theoremmnfle 10423 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
 |-  ( A  e.  RR*  ->  -oo  <_  A )
 
Theoremxrltnsym 10424 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  -.  B  <  A ) )
 
Theoremxrltnsym2 10425 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -.  ( A  <  B 
 /\  B  <  A ) )
 
Theoremxrlttri 10426 Ordering on the extended reals satisfies strict trichotomy. (Contributed by NM, 14-Oct-2005.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A ) ) )
 
Theoremxrlttr 10427 Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A  <  B  /\  B  <  C ) 
 ->  A  <  C ) )
 
Theoremxrltso 10428 'Less than' is a strict ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)
 |- 
 <  Or  RR*
 
Theoremxrlttri2 10429 Trichotomy law for 'less than' for extended reals. (Contributed by NM, 10-Dec-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =/=  B  <->  ( A  <  B  \/  B  <  A ) ) )
 
Theoremxrlttri3 10430 Trichotomy law for 'less than' for extended reals. (Contributed by NM, 9-Feb-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
 
Theoremxrleloe 10431 'Less than or equal' expressed in terms of 'less than' or 'equals', for extended reals. (Contributed by NM, 19-Jan-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  ( A  <  B  \/  A  =  B )
 ) )
 
Theoremxrleltne 10432 'Less than or equal to' implies 'less than' is not 'equals', for extended reals. (Contributed by NM, 9-Feb-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( A  <  B  <->  B  =/=  A ) )
 
Theoremxrltlen 10433 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  ( A  <_  B  /\  B  =/=  A ) ) )
 
Theoremdfle2 10434 Alternative definition of 'less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 6-Nov-2015.)
 |- 
 <_  =  (  <  u.  (  _I  |`  RR* )
 )
 
Theoremdflt2 10435 Alternative definition of 'less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.)
 |- 
 <  =  (  <_  \  _I  )
 
Theoremxrltle 10436 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  A  <_  B )
 )
 
Theoremxrleid 10437 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
 |-  ( A  e.  RR*  ->  A  <_  A )
 
Theoremxrletri 10438 Trichotomy law for extended reals. (Contributed by NM, 7-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  \/  B  <_  A )
 )
 
Theoremxrletri3 10439 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( A  <_  B  /\  B  <_  A ) ) )
 
Theoremxrlelttr 10440 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A  <_  B  /\  B  <  C ) 
 ->  A  <  C ) )
 
Theoremxrltletr 10441 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A  <  B  /\  B  <_  C )  ->  A  <  C ) )
 
Theoremxrletr 10442 Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A  <_  B  /\  B  <_  C )  ->  A  <_  C )
 )
 
Theoremxrlttrd 10443 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  B  <  C )   =>    |-  ( ph  ->  A  <  C )
 
Theoremxrlelttrd 10444 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  B  <  C )   =>    |-  ( ph  ->  A  <  C )
 
Theoremxrltletrd 10445 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  B  <_  C )   =>    |-  ( ph  ->  A  <  C )
 
Theoremxrletrd 10446 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  B 
 <_  C )   =>    |-  ( ph  ->  A  <_  C )
 
Theoremxrltne 10447 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  B  =/=  A )
 
Theoremnltpnft 10448 An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)
 |-  ( A  e.  RR*  ->  ( A  =  +oo  <->  -.  A  <  +oo ) )
 
Theoremngtmnft 10449 An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)
 |-  ( A  e.  RR*  ->  ( A  =  -oo  <->  -.  -oo 
 <  A ) )
 
Theoremxrrebnd 10450 An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)
 |-  ( A  e.  RR*  ->  ( A  e.  RR  <->  (  -oo  <  A  /\  A  <  +oo ) ) )
 
Theoremxrre 10451 A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  (  -oo  <  A  /\  A  <_  B ) )  ->  A  e.  RR )
 
Theoremxrre2 10452 An extended real between two others is real. (Contributed by NM, 6-Feb-2007.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <  C ) )  ->  B  e.  RR )
 
Theoremge0gtmnf 10453 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <_  A ) 
 ->  -oo  <  A )
 
Theoremge0nemnf 10454 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <_  A ) 
 ->  A  =/=  -oo )
 
Theoremxrrege0 10455 A nonnegative extended real that is less than a real bound is real. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( 0 
 <_  A  /\  A  <_  B ) )  ->  A  e.  RR )
 
Theoremxrmax1 10456 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  <_  if ( A  <_  B ,  B ,  A ) )
 
Theoremxrmax2 10457 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  <_  if ( A  <_  B ,  B ,  A ) )
 
Theoremxrmin1 10458 The minimum of two extended reals is less than or equal to one of them. (Contributed by NM, 7-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( A  <_  B ,  A ,  B )  <_  A )
 
Theoremxrmin2 10459 The minimum of two extended reals is less than or equal to one of them. (Contributed by NM, 7-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( A  <_  B ,  A ,  B )  <_  B )
 
Theoremxrmaxeq 10460 The maximum of two extended reals is equal to the first if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  B  <_  A )  ->  if ( A  <_  B ,  B ,  A )  =  A )
 
Theoremxrmineq 10461 The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  B  <_  A )  ->  if ( A  <_  B ,  A ,  B )  =  B )
 
Theoremxrmaxlt 10462 Two ways of saying the maximum of two extended reals is less than a third. (Contributed by NM, 7-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( if ( A  <_  B ,  B ,  A )  <  C  <->  ( A  <  C 
 /\  B  <  C ) ) )
 
Theoremxrltmin 10463 Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( A  <  if ( B 
 <_  C ,  B ,  C )  <->  ( A  <  B 
 /\  A  <  C ) ) )
 
Theoremxrmaxle 10464 Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( if ( A  <_  B ,  B ,  A ) 
 <_  C  <->  ( A  <_  C 
 /\  B  <_  C ) ) )
 
Theoremxrlemin 10465 Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( A  <_  if ( B 
 <_  C ,  B ,  C )  <->  ( A  <_  B 
 /\  A  <_  C ) ) )
 
Theoremmax1 10466 A number is less than or equal to the maximum of it and another. (Contributed by NM, 3-Apr-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  if ( A  <_  B ,  B ,  A )
 )
 
Theoremmax1ALT 10467 A number is less than or equal to the maximum of it and another. (Contributed by NM, 3-Apr-2005.)
 |-  ( A  e.  RR  ->  A  <_  if ( A  <_  B ,  B ,  A ) )
 
Theoremmax2 10468 A number is less than or equal to the maximum of it and another. (Contributed by NM, 3-Apr-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  <_  if ( A  <_  B ,  B ,  A )
 )
 
Theoremmin1 10469 The minimum of two numbers is less than or equal to the first. (Contributed by NM, 3-Aug-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  if ( A 
 <_  B ,  A ,  B )  <_  A )
 
Theoremmin2 10470 The minimum of two numbers is less than or equal to the second. (Contributed by NM, 3-Aug-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  if ( A 
 <_  B ,  A ,  B )  <_  B )
 
Theoremmaxle 10471 Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by NM, 29-Sep-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( if ( A 
 <_  B ,  B ,  A )  <_  C  <->  ( A  <_  C 
 /\  B  <_  C ) ) )
 
Theoremlemin 10472 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  <_  if ( B  <_  C ,  B ,  C )  <->  ( A  <_  B  /\  A  <_  C ) ) )
 
Theoremmaxlt 10473 Two ways of saying the maximum of two numbers is less than a third. (Contributed by NM, 3-Aug-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( if ( A 
 <_  B ,  B ,  A )  <  C  <->  ( A  <  C 
 /\  B  <  C ) ) )
 
Theoremltmin 10474 Two ways of saying a number is less than the minimum of two others. (Contributed by NM, 1-Sep-2006.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  if ( B  <_  C ,  B ,  C )  <->  ( A  <  B  /\  A  <  C ) ) )
 
Theoremmax0sub 10475 Decompose a real number into positive and negative parts. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  ( A  e.  RR  ->  ( if ( 0 
 <_  A ,  A , 
 0 )  -  if ( 0  <_  -u A ,  -u A ,  0 ) )  =  A )
 
Theoremifle 10476 An if statement transforms an implication into an inequality of terms. (Contributed by Mario Carneiro, 31-Aug-2014.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B ) )
 
Theoremz2ge 10477* There exists an integer greater than or equal to any two others. (Contributed by NM, 28-Aug-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  E. k  e.  ZZ  ( M  <_  k  /\  N  <_  k ) )
 
Theoremqbtwnre 10478* The rational numbers are dense in 
RR: any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.) (Proof shortened by Mario Carneiro, 13-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) 
 ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
 
Theoremqbtwnxr 10479* The rational numbers are dense in  RR*: any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  E. x  e.  QQ  ( A  <  x 
 /\  x  <  B ) )
 
Theoremqsqueeze 10480* If a nonnegative real is less than any positive rational, it is zero. (Contributed by NM, 6-Feb-2007.)
 |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  QQ  ( 0  <  x  ->  A  <  x ) )  ->  A  =  0 )
 
Theoremqextltlem 10481* Lemma for qextlt 10482 and qextle . (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( -.  ( x  <  A 
 <->  x  <  B ) 
 /\  -.  ( x  <_  A  <->  x  <_  B ) ) ) )
 
Theoremqextlt 10482* An extensionality-like property for extended real ordering. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  A. x  e.  QQ  ( x  <  A  <->  x  <  B ) ) )
 
Theoremqextle 10483* An extensionality-like property for extended real ordering. (Contributed by Mario Carneiro, 3-Oct-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  A. x  e.  QQ  ( x  <_  A  <->  x  <_  B ) ) )
 
Theoremxralrple 10484* Show that  A is less than  B by showing that there is no positive bound on the difference. (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A  <_  B  <->  A. x  e.  RR+  A  <_  ( B  +  x ) ) )
 
Theoremalrple 10485* Show that  A is less than  B by showing that there is no positive bound on the difference. (Contributed by Mario Carneiro, 12-Jun-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  A. x  e.  RR+  A  <_  ( B  +  x ) ) )
 
Theoremxnegeq 10486 Equality of two extended numbers with  - e in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  =  B  -> 
 - e A  =  - e B )
 
Theoremxnegex 10487 A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  - e A  e.  _V
 
Theoremxnegpnf 10488 Minus  +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
 |-  - e  +oo  =  -oo
 
Theoremxnegmnf 10489 Minus  -oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
 |-  - e  -oo  =  +oo
 
Theoremrexneg 10490 Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR  -> 
 - e A  =  -u A )
 
Theoremxneg0 10491 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  - e 0  =  0
 
Theoremxnegcl 10492 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  - e A  e.  RR* )
 
Theoremxnegneg 10493 Extended real version of negneg 9051. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  - e  - e A  =  A )
 
Theoremxneg11 10494 Extended real version of neg11 9052. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  - e A  =  - e B  <->  A  =  B ) )
 
Theoremxltnegi 10495 Forward direction of xltneg 10496. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  - e B  <  - e A )
 
Theoremxltneg 10496 Extended real version of ltneg 9228. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  - e B  <  - e A ) )
 
Theoremxleneg 10497 Extended real version of leneg 9231. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  - e B  <_  - e A ) )
 
Theoremxlt0neg1 10498 Extended real version of lt0neg1 9234. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  - e A ) )
 
Theoremxlt0neg2 10499 Extended real version of lt0neg2 9235. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <  A  <->  - e A  <  0 ) )
 
Theoremxle0neg1 10500 Extended real version of le0neg1 9236. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( A  <_  0  <->  0  <_  - e A ) )
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