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Theorem List for Metamath Proof Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxrmin2 10501 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 10502 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 10503 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 10504 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 10505 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 10506 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 10507 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 10508 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 10509 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 10510 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 10511 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 10512 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 10513 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 10514 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 10515 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 10516 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 10517 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 10518 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 10519* 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 10520* 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 10521* 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 10522* 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 10523* Lemma for qextlt 10524 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 10524* 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 10525* 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 10526* 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 10527* 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 10528 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 10529 A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  - e A  e.  _V
 
Theoremxnegpnf 10530 Minus  +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
 |-  - e  +oo  =  -oo
 
Theoremxnegmnf 10531 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 10532 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 10533 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  - e 0  =  0
 
Theoremxnegcl 10534 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  - e A  e.  RR* )
 
Theoremxnegneg 10535 Extended real version of negneg 9092. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  - e  - e A  =  A )
 
Theoremxneg11 10536 Extended real version of neg11 9093. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  - e A  =  - e B  <->  A  =  B ) )
 
Theoremxltnegi 10537 Forward direction of xltneg 10538. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  - e B  <  - e A )
 
Theoremxltneg 10538 Extended real version of ltneg 9269. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  - e B  <  - e A ) )
 
Theoremxleneg 10539 Extended real version of leneg 9272. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  - e B  <_  - e A ) )
 
Theoremxlt0neg1 10540 Extended real version of lt0neg1 9275. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  - e A ) )
 
Theoremxlt0neg2 10541 Extended real version of lt0neg2 9276. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <  A  <->  - e A  <  0 ) )
 
Theoremxle0neg1 10542 Extended real version of le0neg1 9277. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( A  <_  0  <->  0  <_  - e A ) )
 
Theoremxle0neg2 10543 Extended real version of le0neg2 9278. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <_  A  <->  - e A  <_  0 ) )
 
Theoremxaddval 10544 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 10545 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 + e : (
 RR*  X.  RR* ) --> RR*
 
Theoremxmulval 10546 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 10547 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  -oo )  ->  ( A + e  +oo )  =  +oo )
 
Theoremxaddpnf2 10548 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  -oo )  ->  (  +oo + e A )  =  +oo )
 
Theoremxaddmnf1 10549 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  ( A + e  -oo )  =  -oo )
 
Theoremxaddmnf2 10550 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  (  -oo + e A )  =  -oo )
 
Theorempnfaddmnf 10551 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 10552 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 10553 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 10554 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 10555 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 10556 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 10557 Extended real version of negid 9089. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A + e  - e A )  =  0 )
 
Theoremxaddcl 10558 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 10559 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 10560 Extended real version of addid1 8987. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A + e 0 )  =  A )
 
Theoremxaddid2 10561 Extended real version of addid2 8990. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0 + e A )  =  A )
 
Theoremxnegdi 10562 Extended real version of xnegdi 10562. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> 
 - e ( A + e B )  =  (  - e A + e  - e B ) )
 
Theoremxaddass 10563 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 10564, 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 10564 Associativity of extended real addition. See xaddass 10563 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 10565 Extended real version of pncan 9052. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A + e B ) + e  - e B )  =  A )
 
Theoremxnpcan 10566 Extended real version of npcan 9055. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A + e  - e B ) + e B )  =  A )
 
Theoremxleadd1a 10567 Extended real version of leadd1 9237; 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 10568 Commuted form of xleadd1a 10567. (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 10569 Weakened version of xleadd1a 10567 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 10570 Extended real version of ltadd1 9236. (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 10571 Extended real version of ltadd2 8919. (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 10572 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 10573 Extended real version of le2add 9251. (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 10574 Extended real version of lt2add 9254. Note that ltleadd 9252, 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 10575 Extended real version of subge0 9282. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( 0  <_  ( A + e  - e B )  <->  B  <_  A ) )
 
Theoremxposdif 10576 Extended real version of posdif 9262. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  0  <  ( B + e  - e A ) ) )
 
Theoremxlesubadd 10577 Under certain conditions, the conclusion of lesubadd 9241 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 10578 Lemma for rexmul 10585. (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 10579 Lemma for xmulneg1 10583. (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 10580 Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A x e B )  =  ( B x e A ) )
 
Theoremxmul01 10581 Extended real version of mul01 8986. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A x e 0 )  =  0 )
 
Theoremxmul02 10582 Extended real version of mul02 8985. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0 x e A )  =  0
 )
 
Theoremxmulneg1 10583 Extended real version of mulneg1 9211. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  - e A x e B )  =  - e ( A x e B ) )
 
Theoremxmulneg2 10584 Extended real version of mulneg2 9212. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A x e  - e B )  =  - e ( A x e B ) )
 
Theoremrexmul 10585 The extended real multiplication when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A x e B )  =  ( A  x.  B ) )
 
Theoremxmulf 10586 The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  x e : (
 RR*  X.  RR* ) --> RR*
 
Theoremxmulcl 10587 Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A x e B )  e.  RR* )
 
Theoremxmulpnf1 10588 Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  ( A x e  +oo )  =  +oo )
 
Theoremxmulpnf2 10589 Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  (  +oo x e A )  =  +oo )
 
Theoremxmulmnf1 10590 Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  ( A x e  -oo )  =  -oo )
 
Theoremxmulmnf2 10591 Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  (  -oo x e A )  =  -oo )
 
Theoremxmulpnf1n 10592 Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  <  0 ) 
 ->  ( A x e  +oo )  =  -oo )
 
Theoremxmulid1 10593 Extended real version of mulid1 8830. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A x e 1 )  =  A )
 
Theoremxmulid2 10594 Extended real version of mulid2 8831. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 1 x e A )  =  A )
 
Theoremxmulm1 10595 Extended real version of mulm1 9216. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( -u 1 x e A )  =  - e A )
 
Theoremxmulasslem2 10596 Lemma for xmulass 10601. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( 0  <  A  /\  A  =  -oo )  ->  ph )
 
Theoremxmulgt0 10597 Extended real version of mulgt0 8895. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  -> 
 0  <  ( A x e B ) )
 
Theoremxmulge0 10598 Extended real version of mulge0 9286. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  -> 
 0  <_  ( A x e B ) )
 
Theoremxmulasslem 10599* Lemma for xmulass 10601. (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 10600 Lemma for xmulass 10601. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B )  /\  ( C  e.  RR*  /\  0  <  C ) )  ->  ( ( A x e B ) x e C )  =  ( A x e ( B x e C ) ) )
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