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Theorem List for Metamath Proof Explorer - 10701-10800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltdiv1dd 10701 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( A  /  C )  < 
 ( B  /  C ) )
 
Theoremlediv1dd 10702 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( A  /  C )  <_  ( B  /  C ) )
 
Theoremlediv12ad 10703 Comparison of ratio of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  C 
 <_  D )   =>    |-  ( ph  ->  ( A  /  D )  <_  ( B  /  C ) )
 
Theoremltdiv23d 10704 Swap denominator with other side of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  ( A  /  B )  <  C )   =>    |-  ( ph  ->  ( A  /  C )  <  B )
 
Theoremlediv23d 10705 Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  ( A  /  B ) 
 <_  C )   =>    |-  ( ph  ->  ( A  /  C )  <_  B )
 
Theoremlt2mul2divd 10706 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 10707 Extend class notation to include the negative of an extended real.
 class  - e A
 
Syntaxcxad 10708 Extend class notation to include addition of extended reals.
 class  + e
 
Syntaxcxmu 10709 Extend class notation to include multiplication of extended reals.
 class  x e
 
Definitiondf-xneg 10710 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 10711* 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 10712* 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 10713 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 10714 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 10715 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 10716 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo ) )
 
Theorempnfnemnf 10717 Plus and minus infinity are distinguished elements of  RR*. (Contributed by NM, 14-Oct-2005.)
 |- 
 +oo  =/=  -oo
 
Theoremxrnemnf 10718 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 10719 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 10720 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR*  ->  -.  A  <  A )
 
Theoremltpnf 10721 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  ->  A  <  +oo )
 
Theoremmnflt 10722 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  ->  -oo  <  A )
 
Theoremmnfltpnf 10723 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
 |- 
 -oo  <  +oo
 
Theoremmnfltxr 10724 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 10725 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
 |-  ( A  e.  RR*  ->  -.  +oo  <  A )
 
Theoremnltmnf 10726 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
 |-  ( A  e.  RR*  ->  -.  A  <  -oo )
 
Theorempnfge 10727 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
 |-  ( A  e.  RR*  ->  A  <_  +oo )
 
Theoremnn0pnfge0 10728 If a number is a nonnegative integer or positive infinity, it is greater than or equal to 0. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
 |-  ( ( N  e.  NN0 
 \/  N  =  +oo )  ->  0  <_  N )
 
Theoremmnfle 10729 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
 |-  ( A  e.  RR*  ->  -oo  <_  A )
 
Theoremxrltnsym 10730 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 10731 '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 10732 Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 9064 or axlttri 9147. (Contributed by NM, 14-Oct-2005.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A ) ) )
 
Theoremxrlttr 10733 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 10734 'Less than' is a strict ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)
 |- 
 <  Or  RR*
 
Theoremxrlttri2 10735 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 10736 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 10737 '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 10738 '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 10739 '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 10740 Alternative definition of 'less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 6-Nov-2015.)
 |- 
 <_  =  (  <  u.  (  _I  |`  RR* )
 )
 
Theoremdflt2 10741 Alternative definition of 'less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.)
 |- 
 <  =  (  <_  \  _I  )
 
Theoremxrltle 10742 '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 10743 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
 |-  ( A  e.  RR*  ->  A  <_  A )
 
Theoremxrletri 10744 Trichotomy law for extended reals. (Contributed by NM, 7-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  \/  B  <_  A )
 )
 
Theoremxrletri3 10745 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( A  <_  B  /\  B  <_  A ) ) )
 
Theoremxrlelttr 10746 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 10747 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 10748 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 10749 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 10750 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 10751 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 10752 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 10753 '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 10754 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 10755 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 10756 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 10757 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 10758 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 )
 
Theoremxrre3 10759 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B 
 <_  A  /\  A  <  +oo ) )  ->  A  e.  RR )
 
Theoremge0gtmnf 10760 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <_  A ) 
 ->  -oo  <  A )
 
Theoremge0nemnf 10761 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <_  A ) 
 ->  A  =/=  -oo )
 
Theoremxrrege0 10762 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 10763 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 10764 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 10765 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 10766 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 10767 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 10768 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 10769 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 10770 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 10771 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 10772 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 10773 A number is less than or equal to the maximum of it and another. See also max1ALT 10774. (Contributed by NM, 3-Apr-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  if ( A  <_  B ,  B ,  A )
 )
 
Theoremmax1ALT 10774 A number is less than or equal to the maximum of it and another. This version of max1 10773 omits the  B  e.  RR antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 10773 is preferred. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 3-Apr-2005.)
 |-  ( A  e.  RR  ->  A  <_  if ( A  <_  B ,  B ,  A ) )
 
Theoremmax2 10775 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 10776 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 10777 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 10778 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 10779 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 10780 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 10781 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 10782 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 10783 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 10784* 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 10785* 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 10786* 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 10787* 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 10788* Lemma for qextlt 10789 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 10789* 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 10790* 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 10791* 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 10792* 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 10793 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 10794 A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  - e A  e.  _V
 
Theoremxnegpnf 10795 Minus  +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
 |-  - e  +oo  =  -oo
 
Theoremxnegmnf 10796 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 10797 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 10798 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  - e 0  =  0
 
Theoremxnegcl 10799 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  - e A  e.  RR* )
 
Theoremxnegneg 10800 Extended real version of negneg 9351. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  - e  - e A  =  A )
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