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Theorem List for Intuitionistic Logic Explorer - 9501-9600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmul2lt0np 9501 The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  0
 )   &    |-  ( ph  ->  0  <  B )   =>    |-  ( ph  ->  ( A  x.  B )  < 
 0 )
 
Theoremmul2lt0pn 9502 The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  0
 )   &    |-  ( ph  ->  0  <  B )   =>    |-  ( ph  ->  ( B  x.  A )  < 
 0 )
 
Theoremlt2mul2divd 9503 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 ) ) )
 
Theoremnnledivrp 9504 Division of a positive integer by a positive number is less than or equal to the integer iff the number is greater than or equal to 1. (Contributed by AV, 19-Jun-2021.)
 |-  ( ( A  e.  NN  /\  B  e.  RR+ )  ->  ( 1  <_  B 
 <->  ( A  /  B )  <_  A ) )
 
Theoremnn0ledivnn 9505 Division of a nonnegative integer by a positive integer is less than or equal to the integer. (Contributed by AV, 19-Jun-2021.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( A  /  B )  <_  A )
 
Theoremaddlelt 9506 If the sum of a real number and a positive real number is less than or equal to a third real number, the first real number is less than the third real number. (Contributed by AV, 1-Jul-2021.)
 |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  ( ( M  +  A )  <_  N  ->  M  <  N ) )
 
4.5.2  Infinity and the extended real number system (cont.)
 
Syntaxcxne 9507 Extend class notation to include the negative of an extended real.
 class  -e A
 
Syntaxcxad 9508 Extend class notation to include addition of extended reals.
 class  +e
 
Syntaxcxmu 9509 Extend class notation to include multiplication of extended reals.
 class  xe
 
Definitiondf-xneg 9510 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 9511* 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 9512* 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 9513 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 9514 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
 
Theoremxrnemnf 9515 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 9516 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 9517 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR*  ->  -.  A  <  A )
 
Theoremltpnf 9518 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  ->  A  < +oo )
 
Theorem0ltpnf 9519 Zero is less than plus infinity (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  < +oo
 
Theoremmnflt 9520 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  -> -oo  <  A )
 
Theoremmnflt0 9521 Minus infinity is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |- -oo  <  0
 
Theoremmnfltpnf 9522 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
 |- -oo  < +oo
 
Theoremmnfltxr 9523 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 9524 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
 |-  ( A  e.  RR*  ->  -. +oo  <  A )
 
Theoremnltmnf 9525 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
 |-  ( A  e.  RR*  ->  -.  A  < -oo )
 
Theorempnfge 9526 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
 |-  ( A  e.  RR*  ->  A  <_ +oo )
 
Theorem0lepnf 9527 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  <_ +oo
 
Theoremnn0pnfge0 9528 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 9529 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
 |-  ( A  e.  RR*  -> -oo  <_  A )
 
Theoremxrltnsym 9530 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 9531 '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 9532 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 9533 'Less than' is a weakly linear ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)
 |- 
 <  Or  RR*
 
Theoremxrlttri3 9534 Extended real version of lttri3 7808. (Contributed by NM, 9-Feb-2006.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
 
Theoremxrltle 9535 '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 9536 'Less than' implies 'less than or equal to' for extended reals. Deduction form of xrltle 9535. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  A 
 <_  B )
 
Theoremxrleid 9537 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
 |-  ( A  e.  RR*  ->  A  <_  A )
 
Theoremxrleidd 9538 'Less than or equal to' is reflexive for extended reals. Deduction form of xrleid 9537. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
 |-  ( ph  ->  A  e.  RR* )   =>    |-  ( ph  ->  A  <_  A )
 
Theoremxrletri3 9539 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( A  <_  B  /\  B  <_  A ) ) )
 
Theoremxrlelttr 9540 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 9541 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 9542 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 9543 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 9544 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 9545 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 9546 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 9547 '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 9548 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 9549 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 9550 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 9551 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 9552 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 9553 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 9554 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 9555 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 9556 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 9557 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <_  A ) 
 -> -oo  <  A )
 
Theoremge0nemnf 9558 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  0  <_  A ) 
 ->  A  =/= -oo )
 
Theoremxrrege0 9559 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 9560* 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 9561 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 9562 Minus +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
 |-  -e +oo  = -oo
 
Theoremxnegmnf 9563 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 9564 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 9565 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  -e 0  =  0
 
Theoremxnegcl 9566 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  -e A  e.  RR* )
 
Theoremxnegneg 9567 Extended real version of negneg 7976. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  -e  -e A  =  A )
 
Theoremxneg11 9568 Extended real version of neg11 7977. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A  =  -e B  <->  A  =  B )
 )
 
Theoremxltnegi 9569 Forward direction of xltneg 9570. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A )
 
Theoremxltneg 9570 Extended real version of ltneg 8188. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -e B  <  -e A ) )
 
Theoremxleneg 9571 Extended real version of leneg 8191. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -e B  <_  -e A ) )
 
Theoremxlt0neg1 9572 Extended real version of lt0neg1 8194. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  -e A ) )
 
Theoremxlt0neg2 9573 Extended real version of lt0neg2 8195. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <  A  <->  -e A  <  0 ) )
 
Theoremxle0neg1 9574 Extended real version of le0neg1 8196. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( A  <_  0  <->  0  <_  -e A ) )
 
Theoremxle0neg2 9575 Extended real version of le0neg2 8197. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( A  e.  RR*  ->  ( 0  <_  A  <->  -e A  <_  0 ) )
 
Theoremxrpnfdc 9576 An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
 |-  ( A  e.  RR*  -> DECID  A  = +oo )
 
Theoremxrmnfdc 9577 An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
 |-  ( A  e.  RR*  -> DECID  A  = -oo )
 
Theoremxaddf 9578 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |- 
 +e : (
 RR*  X.  RR* ) --> RR*
 
Theoremxaddval 9579 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 9580 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
 
Theoremxaddpnf2 9581 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( +oo +e A )  = +oo )
 
Theoremxaddmnf1 9582 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
 
Theoremxaddmnf2 9583 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( -oo +e A )  = -oo )
 
Theorempnfaddmnf 9584 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 9585 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 9586 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 9587 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 9588 The extended real addition operation when both arguments are real. Deduction version of rexadd 9586. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A +e B )  =  ( A  +  B ) )
 
Theoremxnegcld 9589 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR* )   =>    |-  ( ph  ->  -e A  e.  RR* )
 
Theoremxrex 9590 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
 |-  RR*  e.  _V
 
Theoremxaddnemnf 9591 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 9592 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 9593 Extended real version of negid 7973. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A +e  -e A )  =  0 )
 
Theoremxaddcl 9594 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 9595 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 9596 Extended real version of addid1 7864. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
 
Theoremxaddid2 9597 Extended real version of addid2 7865. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( A  e.  RR*  ->  ( 0 +e A )  =  A )
 
Theoremxaddid1d 9598  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 9599 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 9600 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 ) )
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