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Theorem List for Metamath Proof Explorer - 11501-11600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxrmin2 11501 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 11502 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 11503 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 11504 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 11505 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 11506 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 11507 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 11508 A number is less than or equal to the maximum of it and another. See also max1ALT 11509. (Contributed by NM, 3-Apr-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  if ( A  <_  B ,  B ,  A )
 )
 
Theoremmax1ALT 11509 A number is less than or equal to the maximum of it and another. This version of max1 11508 omits the  B  e.  RR antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 11508 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 11510 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 11511 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 11512 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 11513 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 11514 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 11515 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 11516 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 11517 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 11518 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 11519* 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 11520* 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 11521* 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 11522* 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 11523* Lemma for qextlt 11524 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 11524* 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 11525* 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 11526* 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 11527* 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 11528 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 11529 A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  -e A  e.  _V
 
Theoremxnegpnf 11530 Minus +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
 |-  -e +oo  = -oo
 
Theoremxnegmnf 11531 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 11532 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 11533 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  -e 0  =  0
 
Theoremxnegcl 11534 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  -e A  e.  RR* )
 
Theoremxnegneg 11535 Extended real version of negneg 9949. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  -e  -e A  =  A )
 
Theoremxneg11 11536 Extended real version of neg11 9950. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A  =  -e B  <->  A  =  B )
 )
 
Theoremxltnegi 11537 Forward direction of xltneg 11538. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A )
 
Theoremxltneg 11538 Extended real version of ltneg 10141. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -e B  <  -e A ) )
 
Theoremxleneg 11539 Extended real version of leneg 10144. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -e B  <_  -e A ) )
 
Theoremxlt0neg1 11540 Extended real version of lt0neg1 10147. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  -e A ) )
 
Theoremxlt0neg2 11541 Extended real version of lt0neg2 10148. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <  A  <->  -e A  <  0 ) )
 
Theoremxle0neg1 11542 Extended real version of le0neg1 10149. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( A  <_  0  <->  0  <_  -e A ) )
 
Theoremxle0neg2 11543 Extended real version of le0neg2 10150. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <_  A  <->  -e A  <_  0 ) )
 
Theoremxaddval 11544 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 11545 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 +e : (
 RR*  X.  RR* ) --> RR*
 
Theoremxmulval 11546 Value of the extended real multiplication operation. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe 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 11547 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
 
Theoremxaddpnf2 11548 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( +oo +e A )  = +oo )
 
Theoremxaddmnf1 11549 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
 
Theoremxaddmnf2 11550 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( -oo +e A )  = -oo )
 
Theorempnfaddmnf 11551 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 11552 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 11553 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 11554 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 11555 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 11556 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 11557 Extended real version of negid 9946. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A +e  -e A )  =  0 )
 
Theoremxaddcl 11558 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 11559 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 11560 Extended real version of addid1 9838. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
 
Theoremxaddid2 11561 Extended real version of addid2 9841. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0 +e A )  =  A )
 
Theoremxnegdi 11562 Extended real version of xnegdi 11562. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> 
 -e ( A +e B )  =  (  -e A +e  -e B ) )
 
Theoremxaddass 11563 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 11564, 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 11564 Associativity of extended real addition. See xaddass 11563 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 11565 Extended real version of pncan 9906. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A +e B ) +e  -e B )  =  A )
 
Theoremxnpcan 11566 Extended real version of npcan 9909. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A +e  -e B ) +e B )  =  A )
 
Theoremxleadd1a 11567 Extended real version of leadd1 10109; 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 11568 Commuted form of xleadd1a 11567. (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 11569 Weakened version of xleadd1a 11567 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 11570 Extended real version of ltadd1 10108. (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 11571 Extended real version of ltadd2 9763. (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 11572 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 11573 Extended real version of le2add 10123. (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 11574 Extended real version of lt2add 10126. Note that ltleadd 10124, 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 11575 Extended real version of subge0 10154. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( 0  <_  ( A +e  -e B )  <->  B  <_  A ) )
 
Theoremxposdif 11576 Extended real version of posdif 10134. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) )
 
Theoremxlesubadd 11577 Under certain conditions, the conclusion of lesubadd 10113 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 11578 Lemma for rexmul 11585. (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 )
 
Theoremxmullem2 11579 Lemma for xmulneg1 11583. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( ( ( 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 )
 ) ) ) )
 
Theoremxmulcom 11580 Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe B )  =  ( B xe A ) )
 
Theoremxmul01 11581 Extended real version of mul01 9837. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A xe 0 )  =  0 )
 
Theoremxmul02 11582 Extended real version of mul02 9836. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0 xe A )  =  0
 )
 
Theoremxmulneg1 11583 Extended real version of mulneg1 10082. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A xe B )  =  -e ( A xe B ) )
 
Theoremxmulneg2 11584 Extended real version of mulneg2 10083. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe  -e B )  =  -e ( A xe B ) )
 
Theoremrexmul 11585 The extended real multiplication when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A xe B )  =  ( A  x.  B ) )
 
Theoremxmulf 11586 The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  xe : (
 RR*  X.  RR* ) --> RR*
 
Theoremxmulcl 11587 Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe B )  e.  RR* )
 
Theoremxmulpnf1 11588 Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  ( A xe +oo )  = +oo )
 
Theoremxmulpnf2 11589 Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  ( +oo xe A )  = +oo )
 
Theoremxmulmnf1 11590 Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  ( A xe -oo )  = -oo )
 
Theoremxmulmnf2 11591 Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  ( -oo xe A )  = -oo )
 
Theoremxmulpnf1n 11592 Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  <  0 ) 
 ->  ( A xe +oo )  = -oo )
 
Theoremxmulid1 11593 Extended real version of mulid1 9665. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A xe 1 )  =  A )
 
Theoremxmulid2 11594 Extended real version of mulid2 9666. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 1 xe A )  =  A )
 
Theoremxmulm1 11595 Extended real version of mulm1 10087. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( -u 1 xe A )  =  -e A )
 
Theoremxmulasslem2 11596 Lemma for xmulass 11601. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( 0  <  A  /\  A  = -oo )  ->  ph )
 
Theoremxmulgt0 11597 Extended real version of mulgt0 9736. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  -> 
 0  <  ( A xe B ) )
 
Theoremxmulge0 11598 Extended real version of mulge0 10159. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  -> 
 0  <_  ( A xe B ) )
 
Theoremxmulasslem 11599* Lemma for xmulass 11601. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( x  =  D  ->  ( ps  <->  X  =  Y ) )   &    |-  ( x  =  -e D  ->  ( ps 
 <->  E  =  F ) )   &    |-  ( ph  ->  X  e.  RR* )   &    |-  ( ph  ->  Y  e.  RR* )   &    |-  ( ph  ->  D  e.  RR* )   &    |-  ( ( ph  /\  ( x  e.  RR*  /\  0  <  x ) )  ->  ps )   &    |-  ( ph  ->  ( x  =  0  ->  ps )
 )   &    |-  ( ph  ->  E  =  -e X )   &    |-  ( ph  ->  F  =  -e Y )   =>    |-  ( ph  ->  X  =  Y )
 
Theoremxmulasslem3 11600 Lemma for xmulass 11601. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B )  /\  ( C  e.  RR*  /\  0  <  C ) )  ->  ( ( A xe B ) xe C )  =  ( A xe ( B xe C ) ) )
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