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Theorem List for Metamath Proof Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxrletri 10501 Trichotomy law for extended reals. (Contributed by NM, 7-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  \/  B  <_  A )
 )
 
Theoremxrletri3 10502 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( A  <_  B  /\  B  <_  A ) ) )
 
Theoremxrlelttr 10503 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 10504 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 10505 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 10506 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 10507 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 10508 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 10509 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 10510 '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 10511 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 10512 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 10513 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 10514 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 10515 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 10516 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 10517 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <_  A ) 
 ->  -oo  <  A )
 
Theoremge0nemnf 10518 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <_  A ) 
 ->  A  =/=  -oo )
 
Theoremxrrege0 10519 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 10520 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 10521 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 10522 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 10523 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 10524 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 10525 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 10526 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 10527 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 10528 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 10529 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 10530 A number is less than or equal to the maximum of it and another. See also max1ALT 10531. (Contributed by NM, 3-Apr-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  if ( A  <_  B ,  B ,  A )
 )
 
Theoremmax1ALT 10531 A number is less than or equal to the maximum of it and another. This version of max1 10530 omits the  B  e.  RR antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 10530 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 10532 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 10533 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 10534 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 10535 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 10536 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 10537 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 10538 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 10539 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 10540 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 10541* 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 10542* 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 10543* 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 10544* 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 10545* Lemma for qextlt 10546 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 10546* 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 10547* 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 10548* 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 10549* 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 10550 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 10551 A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  - e A  e.  _V
 
Theoremxnegpnf 10552 Minus  +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
 |-  - e  +oo  =  -oo
 
Theoremxnegmnf 10553 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 10554 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 10555 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  - e 0  =  0
 
Theoremxnegcl 10556 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  - e A  e.  RR* )
 
Theoremxnegneg 10557 Extended real version of negneg 9113. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  - e  - e A  =  A )
 
Theoremxneg11 10558 Extended real version of neg11 9114. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  - e A  =  - e B  <->  A  =  B ) )
 
Theoremxltnegi 10559 Forward direction of xltneg 10560. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  - e B  <  - e A )
 
Theoremxltneg 10560 Extended real version of ltneg 9290. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  - e B  <  - e A ) )
 
Theoremxleneg 10561 Extended real version of leneg 9293. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  - e B  <_  - e A ) )
 
Theoremxlt0neg1 10562 Extended real version of lt0neg1 9296. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  - e A ) )
 
Theoremxlt0neg2 10563 Extended real version of lt0neg2 9297. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <  A  <->  - e A  <  0 ) )
 
Theoremxle0neg1 10564 Extended real version of le0neg1 9298. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( A  <_  0  <->  0  <_  - e A ) )
 
Theoremxle0neg2 10565 Extended real version of le0neg2 9299. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <_  A  <->  - e A  <_  0 ) )
 
Theoremxaddval 10566 Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A + e B )  =  if ( A  =  +oo ,  if ( B  =  -oo ,  0 ,  +oo ) ,  if ( A  =  -oo ,  if ( B  =  +oo ,  0 ,  -oo ) ,  if ( B  =  +oo ,  +oo ,  if ( B  =  -oo , 
 -oo ,  ( A  +  B ) ) ) ) ) )
 
Theoremxaddf 10567 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 + e : (
 RR*  X.  RR* ) --> RR*
 
Theoremxmulval 10568 Value of the extended real multiplication operation. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A x e B )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo )
 )  \/  ( ( 0  <  A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo )
 ) ) ,  +oo ,  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo )
 )  \/  ( ( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  =  +oo )
 ) ) ,  -oo ,  ( A  x.  B ) ) ) ) )
 
Theoremxaddpnf1 10569 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  -oo )  ->  ( A + e  +oo )  =  +oo )
 
Theoremxaddpnf2 10570 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  -oo )  ->  (  +oo + e A )  =  +oo )
 
Theoremxaddmnf1 10571 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  ( A + e  -oo )  =  -oo )
 
Theoremxaddmnf2 10572 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  (  -oo + e A )  =  -oo )
 
Theorempnfaddmnf 10573 Addition of positive and negative infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  (  +oo + e  -oo )  =  0
 
Theoremmnfaddpnf 10574 Addition of negative and positive infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  (  -oo + e  +oo )  =  0
 
Theoremrexadd 10575 The extended real addition operation when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A + e B )  =  ( A  +  B ) )
 
Theoremrexsub 10576 Extended real subtraction when both arguments are real. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A + e  - e B )  =  ( A  -  B ) )
 
Theoremxaddnemnf 10577 Closure of extended real addition in the subset  RR*  /  {  -oo }. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  A  =/=  -oo )  /\  ( B  e.  RR*  /\  B  =/=  -oo ) )  ->  ( A + e B )  =/=  -oo )
 
Theoremxaddnepnf 10578 Closure of extended real addition in the subset  RR*  /  {  +oo }. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  A  =/=  +oo )  /\  ( B  e.  RR*  /\  B  =/=  +oo ) )  ->  ( A + e B )  =/=  +oo )
 
Theoremxnegid 10579 Extended real version of negid 9110. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A + e  - e A )  =  0 )
 
Theoremxaddcl 10580 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A + e B )  e.  RR* )
 
Theoremxaddcom 10581 The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A + e B )  =  ( B + e A ) )
 
Theoremxaddid1 10582 Extended real version of addid1 9008. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A + e 0 )  =  A )
 
Theoremxaddid2 10583 Extended real version of addid2 9011. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0 + e A )  =  A )
 
Theoremxnegdi 10584 Extended real version of xnegdi 10584. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> 
 - e ( A + e B )  =  (  - e A + e  - e B ) )
 
Theoremxaddass 10585 Associativity of extended real addition. The correct condition here is "it is not the case that both  +oo and  -oo appear as one of  A ,  B ,  C, i.e.  -.  {  +oo , 
-oo }  C_  { A ,  B ,  C }", but this condition is difficult to work with, so we break the theorem into two parts: this one, where  -oo is not present in  A ,  B ,  C, and xaddass2 10586, where  +oo is not present. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  A  =/=  -oo )  /\  ( B  e.  RR*  /\  B  =/=  -oo )  /\  ( C  e.  RR*  /\  C  =/=  -oo ) )  ->  (
 ( A + e B ) + e C )  =  ( A + e ( B + e C ) ) )
 
Theoremxaddass2 10586 Associativity of extended real addition. See xaddass 10585 for notes on the hypotheses. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  A  =/=  +oo )  /\  ( B  e.  RR*  /\  B  =/=  +oo )  /\  ( C  e.  RR*  /\  C  =/=  +oo ) )  ->  (
 ( A + e B ) + e C )  =  ( A + e ( B + e C ) ) )
 
Theoremxpncan 10587 Extended real version of pncan 9073. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A + e B ) + e  - e B )  =  A )
 
Theoremxnpcan 10588 Extended real version of npcan 9076. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A + e  - e B ) + e B )  =  A )
 
Theoremxleadd1a 10589 Extended real version of leadd1 9258; note that the converse implication is not true, unlike the real version (for example  0  <  1 but  ( 1 + e  +oo )  <_ 
( 0 + e  +oo )). (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  ( A + e C )  <_  ( B + e C ) )
 
Theoremxleadd2a 10590 Commuted form of xleadd1a 10589. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  ( C + e A )  <_  ( C + e B ) )
 
Theoremxleadd1 10591 Weakened version of xleadd1a 10589 under which the reverse implication is true. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( A + e C )  <_  ( B + e C ) ) )
 
Theoremxltadd1 10592 Extended real version of ltadd1 9257. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  <  B  <->  ( A + e C )  <  ( B + e C ) ) )
 
Theoremxltadd2 10593 Extended real version of ltadd2 8940. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  <  B  <->  ( C + e A )  <  ( C + e B ) ) )
 
Theoremxaddge0 10594 The sum of nonnegative extended reals is nonnegative. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( 0 
 <_  A  /\  0  <_  B ) )  -> 
 0  <_  ( A + e B ) )
 
Theoremxle2add 10595 Extended real version of le2add 9272. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* ) )  ->  (
 ( A  <_  C  /\  B  <_  D )  ->  ( A + e B )  <_  ( C + e D ) ) )
 
Theoremxlt2add 10596 Extended real version of lt2add 9275. Note that ltleadd 9273, which has weaker assumptions, is not true for the extended reals (since  0  +  +oo  <  1  +  +oo fails). (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* ) )  ->  (
 ( A  <  C  /\  B  <  D ) 
 ->  ( A + e B )  <  ( C + e D ) ) )
 
Theoremxsubge0 10597 Extended real version of subge0 9303. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( 0  <_  ( A + e  - e B )  <->  B  <_  A ) )
 
Theoremxposdif 10598 Extended real version of posdif 9283. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  0  <  ( B + e  - e A ) ) )
 
Theoremxlesubadd 10599 Under certain conditions, the conclusion of lesubadd 9262 is true even in the extended reals. (Contributed by Mario Carneiro, 4-Sep-2015.)
 |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( 0  <_  A  /\  B  =/=  -oo  /\  0  <_  C ) )  ->  ( ( A + e  - e B ) 
 <_  C  <->  A  <_  ( C + e B ) ) )
 
Theoremxmullem 10600 Lemma for rexmul 10607. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  /\  -.  ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  ( ( 0  <  A  /\  B  =  +oo )  \/  ( A  <  0 
 /\  B  =  -oo ) ) ) ) 
 /\  -.  ( (
 ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo )
 )  \/  ( ( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  =  +oo )
 ) ) )  ->  A  e.  RR )
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