P. 101 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10001-10100   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-xadd 10001*	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 ) ) ) ) ) )
Definition	df-xmul 10002*	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 ) ) ) ) )
Theorem	ltxr 10003	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 ) ) ) ) )
Theorem	elxr 10004	Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
|- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
Theorem	xrnemnf 10005	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 ) )
Theorem	xrnepnf 10006	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 ) )
Theorem	xrltnr 10007	The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
|- ( A e. RR* -> -. A < A )
Theorem	ltpnf 10008	Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
|- ( A e. RR -> A < +oo )
Theorem	ltpnfd 10009	Any (finite) real is less than plus infinity. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A < +oo )
Theorem	0ltpnf 10010	Zero is less than plus infinity (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 < +oo
Theorem	mnflt 10011	Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
|- ( A e. RR -> -oo < A )
Theorem	mnflt0 10012	Minus infinity is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -oo < 0
Theorem	mnfltpnf 10013	Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
|- -oo < +oo
Theorem	mnfltxr 10014	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 )
Theorem	pnfnlt 10015	No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
|- ( A e. RR* -> -. +oo < A )
Theorem	nltmnf 10016	No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
|- ( A e. RR* -> -. A < -oo )
Theorem	pnfge 10017	Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
|- ( A e. RR* -> A <_ +oo )
Theorem	0lepnf 10018	0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 <_ +oo
Theorem	nn0pnfge0 10019	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 )
Theorem	mnfle 10020	Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
|- ( A e. RR* -> -oo <_ A )
Theorem	xrltnsym 10021	Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> -. B < A ) )
Theorem	xrltnsym2 10022	'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 ) )
Theorem	xrlttr 10023	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 ) )
Theorem	xrltso 10024	'Less than' is a weakly linear ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)
|- < Or RR*
Theorem	xrlttri3 10025	Extended real version of lttri3 8252. (Contributed by NM, 9-Feb-2006.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
Theorem	xrltle 10026	'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 ) )
Theorem	xrltled 10027	'Less than' implies 'less than or equal to' for extended reals. Deduction form of xrltle 10026. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> A < B )   =>    |- ( ph -> A <_ B )
Theorem	xrleid 10028	'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
|- ( A e. RR* -> A <_ A )
Theorem	xrleidd 10029	'Less than or equal to' is reflexive for extended reals. Deduction form of xrleid 10028. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> A e. RR* )   =>    |- ( ph -> A <_ A )
Theorem	xnn0dcle 10030	Decidability of <_ for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
|- ( ( A e. NN0* /\ B e. NN0* ) -> DECID A <_ B )
Theorem	xnn0letri 10031	Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
|- ( ( A e. NN0* /\ B e. NN0* ) -> ( A <_ B \/ B <_ A ) )
Theorem	xrletri3 10032	Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
Theorem	xrletrid 10033	Trichotomy law for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> A <_ B )   &    |- ( ph -> B <_ A )   =>    |- ( ph -> A = B )
Theorem	xrlelttr 10034	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 ) )
Theorem	xrltletr 10035	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 ) )
Theorem	xrletr 10036	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 ) )
Theorem	xrlttrd 10037	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 )
Theorem	xrlelttrd 10038	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 )
Theorem	xrltletrd 10039	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 )
Theorem	xrletrd 10040	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 )
Theorem	xrltne 10041	'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B =/= A )
Theorem	nltpnft 10042	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 ) )
Theorem	npnflt 10043	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 ) )
Theorem	xgepnf 10044	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 ) )
Theorem	ngtmnft 10045	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 ) )
Theorem	nmnfgt 10046	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 ) )
Theorem	xrrebnd 10047	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 ) ) )
Theorem	xrre 10048	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 )
Theorem	xrre2 10049	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 )
Theorem	xrre3 10050	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 )
Theorem	ge0gtmnf 10051	A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ 0 <_ A ) -> -oo < A )
Theorem	ge0nemnf 10052	A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ 0 <_ A ) -> A =/= -oo )
Theorem	xrrege0 10053	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 )
Theorem	z2ge 10054*	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 ) )
Theorem	xnegeq 10055	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 )
Theorem	xnegpnf 10056	Minus +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
|- -e +oo = -oo
Theorem	xnegmnf 10057	Minus -oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
|- -e -oo = +oo
Theorem	rexneg 10058	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 )
Theorem	xneg0 10059	The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- -e 0 = 0
Theorem	xnegcl 10060	Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> -e A e. RR* )
Theorem	xnegneg 10061	Extended real version of negneg 8422. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> -e -e A = A )
Theorem	xneg11 10062	Extended real version of neg11 8423. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( -e A = -e B <-> A = B ) )
Theorem	xltnegi 10063	Forward direction of xltneg 10064. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> -e B < -e A )
Theorem	xltneg 10064	Extended real version of ltneg 8635. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -e B < -e A ) )
Theorem	xleneg 10065	Extended real version of leneg 8638. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -e B <_ -e A ) )
Theorem	xlt0neg1 10066	Extended real version of lt0neg1 8641. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( A < 0 <-> 0 < -e A ) )
Theorem	xlt0neg2 10067	Extended real version of lt0neg2 8642. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( 0 < A <-> -e A < 0 ) )
Theorem	xle0neg1 10068	Extended real version of le0neg1 8643. (Contributed by Mario Carneiro, 9-Sep-2015.)
|- ( A e. RR* -> ( A <_ 0 <-> 0 <_ -e A ) )
Theorem	xle0neg2 10069	Extended real version of le0neg2 8644. (Contributed by Mario Carneiro, 9-Sep-2015.)
|- ( A e. RR* -> ( 0 <_ A <-> -e A <_ 0 ) )
Theorem	xrpnfdc 10070	An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
|- ( A e. RR* -> DECID A = +oo )
Theorem	xrmnfdc 10071	An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
|- ( A e. RR* -> DECID A = -oo )
Theorem	xaddf 10072	The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- +e : ( RR* X. RR* ) --> RR*
Theorem	xaddval 10073	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 ) ) ) ) ) )
Theorem	xaddpnf1 10074	Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
Theorem	xaddpnf2 10075	Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ A =/= -oo ) -> ( +oo +e A ) = +oo )
Theorem	xaddmnf1 10076	Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
Theorem	xaddmnf2 10077	Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ A =/= +oo ) -> ( -oo +e A ) = -oo )
Theorem	pnfaddmnf 10078	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
Theorem	mnfaddpnf 10079	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
Theorem	rexadd 10080	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 ) )
Theorem	rexsub 10081	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 ) )
Theorem	rexaddd 10082	The extended real addition operation when both arguments are real. Deduction version of rexadd 10080. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A +e B ) = ( A + B ) )
Theorem	xnegcld 10083	Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR* )   =>    |- ( ph -> -e A e. RR* )
Theorem	xrex 10084	The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
|- RR* e. _V
Theorem	xaddnemnf 10085	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 )
Theorem	xaddnepnf 10086	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 )
Theorem	xnegid 10087	Extended real version of negid 8419. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( A +e -e A ) = 0 )
Theorem	xaddcl 10088	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* )
Theorem	xaddcom 10089	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 ) )
Theorem	xaddid1 10090	Extended real version of addrid 8310. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( A +e 0 ) = A )
Theorem	xaddid2 10091	Extended real version of addlid 8311. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( 0 +e A ) = A )
Theorem	xaddid1d 10092	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 )
Theorem	xnn0lenn0nn0 10093	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 )
Theorem	xnn0le2is012 10094	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 ) )
Theorem	xnn0xadd0 10095	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 ) ) )
Theorem	xnegdi 10096	Extended real version of negdi 8429. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
Theorem	xaddass 10097	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 10098, 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 ) ) )
Theorem	xaddass2 10098	Associativity of extended real addition. See xaddass 10097 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 ) ) )
Theorem	xpncan 10099	Extended real version of pncan 8378. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -e B ) = A )
Theorem	xnpcan 10100	Extended real version of npcan 8381. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e -e B ) +e B ) = A )