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Theorem List for Intuitionistic Logic Explorer - 9401-9500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-xadd 9401* 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 9402* Define multiplication over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  xe  =  ( 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 ) ) ) ) )
 
Theoremltxr 9403 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 9404 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
 
Theoremxrnemnf 9405 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 9406 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 9407 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR*  ->  -.  A  <  A )
 
Theoremltpnf 9408 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  ->  A  < +oo )
 
Theorem0ltpnf 9409 Zero is less than plus infinity (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  < +oo
 
Theoremmnflt 9410 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  -> -oo  <  A )
 
Theoremmnflt0 9411 Minus infinity is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |- -oo  <  0
 
Theoremmnfltpnf 9412 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
 |- -oo  < +oo
 
Theoremmnfltxr 9413 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 9414 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
 |-  ( A  e.  RR*  ->  -. +oo  <  A )
 
Theoremnltmnf 9415 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
 |-  ( A  e.  RR*  ->  -.  A  < -oo )
 
Theorempnfge 9416 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
 |-  ( A  e.  RR*  ->  A  <_ +oo )
 
Theorem0lepnf 9417 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  <_ +oo
 
Theoremnn0pnfge0 9418 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 9419 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
 |-  ( A  e.  RR*  -> -oo  <_  A )
 
Theoremxrltnsym 9420 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 9421 '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 ) )
 
Theoremxrlttr 9422 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 9423 'Less than' is a weakly linear ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)
 |- 
 <  Or  RR*
 
Theoremxrlttri3 9424 Extended real version of lttri3 7715. (Contributed by NM, 9-Feb-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
 
Theoremxrltle 9425 '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 )
 )
 
Theoremxrltled 9426 'Less than' implies 'less than or equal to' for extended reals. Deduction form of xrltle 9425. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  A 
 <_  B )
 
Theoremxrleid 9427 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
 |-  ( A  e.  RR*  ->  A  <_  A )
 
Theoremxrleidd 9428 'Less than or equal to' is reflexive for extended reals. Deduction form of xrleid 9427. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
 |-  ( ph  ->  A  e.  RR* )   =>    |-  ( ph  ->  A  <_  A )
 
Theoremxrletri3 9429 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( A  <_  B  /\  B  <_  A ) ) )
 
Theoremxrlelttr 9430 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 9431 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 9432 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 9433 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 9434 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 9435 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 9436 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 9437 '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 9438 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 ) )
 
Theoremnpnflt 9439 An extended real is less than plus infinity iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
 |-  ( A  e.  RR*  ->  ( A  < +oo  <->  A  =/= +oo )
 )
 
Theoremxgepnf 9440 An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  ( A  e.  RR*  ->  ( +oo  <_  A  <->  A  = +oo ) )
 
Theoremngtmnft 9441 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 ) )
 
Theoremnmnfgt 9442 An extended real is greater than minus infinite iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
 |-  ( A  e.  RR*  ->  ( -oo  <  A  <->  A  =/= -oo )
 )
 
Theoremxrrebnd 9443 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 9444 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 9445 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 9446 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 9447 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <_  A ) 
 -> -oo  <  A )
 
Theoremge0nemnf 9448 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <_  A ) 
 ->  A  =/= -oo )
 
Theoremxrrege0 9449 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 )
 
Theoremz2ge 9450* 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 ) )
 
Theoremxnegeq 9451 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 )
 
Theoremxnegpnf 9452 Minus +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
 |-  -e +oo  = -oo
 
Theoremxnegmnf 9453 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 9454 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 9455 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  -e 0  =  0
 
Theoremxnegcl 9456 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  -e A  e.  RR* )
 
Theoremxnegneg 9457 Extended real version of negneg 7883. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  -e  -e A  =  A )
 
Theoremxneg11 9458 Extended real version of neg11 7884. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A  =  -e B  <->  A  =  B )
 )
 
Theoremxltnegi 9459 Forward direction of xltneg 9460. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A )
 
Theoremxltneg 9460 Extended real version of ltneg 8091. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -e B  <  -e A ) )
 
Theoremxleneg 9461 Extended real version of leneg 8094. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -e B  <_  -e A ) )
 
Theoremxlt0neg1 9462 Extended real version of lt0neg1 8097. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  -e A ) )
 
Theoremxlt0neg2 9463 Extended real version of lt0neg2 8098. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <  A  <->  -e A  <  0 ) )
 
Theoremxle0neg1 9464 Extended real version of le0neg1 8099. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( A  <_  0  <->  0  <_  -e A ) )
 
Theoremxle0neg2 9465 Extended real version of le0neg2 8100. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <_  A  <->  -e A  <_  0 ) )
 
Theoremxrpnfdc 9466 An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
 |-  ( A  e.  RR*  -> DECID  A  = +oo )
 
Theoremxrmnfdc 9467 An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
 |-  ( A  e.  RR*  -> DECID  A  = -oo )
 
Theoremxaddf 9468 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 +e : (
 RR*  X.  RR* ) --> RR*
 
Theoremxaddval 9469 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 ) ) ) ) ) )
 
Theoremxaddpnf1 9470 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
 
Theoremxaddpnf2 9471 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( +oo +e A )  = +oo )
 
Theoremxaddmnf1 9472 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
 
Theoremxaddmnf2 9473 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( -oo +e A )  = -oo )
 
Theorempnfaddmnf 9474 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 9475 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 9476 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 9477 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 ) )
 
Theoremrexaddd 9478 The extended real addition operation when both arguments are real. Deduction version of rexadd 9476. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A +e B )  =  ( A  +  B ) )
 
Theoremxnegcld 9479 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR* )   =>    |-  ( ph  ->  -e A  e.  RR* )
 
Theoremxrex 9480 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
 |-  RR*  e.  _V
 
Theoremxaddnemnf 9481 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 9482 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 9483 Extended real version of negid 7880. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A +e  -e A )  =  0 )
 
Theoremxaddcl 9484 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 9485 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 9486 Extended real version of addid1 7771. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
 
Theoremxaddid2 9487 Extended real version of addid2 7772. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0 +e A )  =  A )
 
Theoremxaddid1d 9488  0 is a right identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  RR* )   =>    |-  ( ph  ->  ( A +e 0 )  =  A )
 
Theoremxnn0lenn0nn0 9489 An extended nonnegative integer which is less than or equal to a nonnegative integer is a nonnegative integer. (Contributed by AV, 24-Nov-2021.)
 |-  ( ( M  e. NN0*  /\  N  e.  NN0  /\  M  <_  N )  ->  M  e.  NN0 )
 
Theoremxnn0le2is012 9490 An extended nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 24-Nov-2021.)
 |-  ( ( N  e. NN0*  /\  N  <_  2 )  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) )
 
Theoremxnn0xadd0 9491 The sum of two extended nonnegative integers is  0 iff each of the two extended nonnegative integers is 
0. (Contributed by AV, 14-Dec-2020.)
 |-  ( ( A  e. NN0*  /\  B  e. NN0* )  ->  ( ( A +e B )  =  0  <-> 
 ( A  =  0 
 /\  B  =  0 ) ) )
 
Theoremxnegdi 9492 Extended real version of negdi 7890. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> 
 -e ( A +e B )  =  (  -e A +e  -e B ) )
 
Theoremxaddass 9493 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 9494, 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 9494 Associativity of extended real addition. See xaddass 9493 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 9495 Extended real version of pncan 7839. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A +e B ) +e  -e B )  =  A )
 
Theoremxnpcan 9496 Extended real version of npcan 7842. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( A +e  -e B ) +e B )  =  A )
 
Theoremxleadd1a 9497 Extended real version of leadd1 8059; 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 9498 Commuted form of xleadd1a 9497. (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 9499 Weakened version of xleadd1a 9497 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 9500 Extended real version of ltadd1 8058. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Jim Kingdon, 16-Apr-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  <  B  <->  ( A +e C )  <  ( B +e C ) ) )
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