P. 97 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9601-9700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mnfltpnf 9601	Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
|- -oo < +oo
Theorem	mnfltxr 9602	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 9603	No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
|- ( A e. RR* -> -. +oo < A )
Theorem	nltmnf 9604	No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
|- ( A e. RR* -> -. A < -oo )
Theorem	pnfge 9605	Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
|- ( A e. RR* -> A <_ +oo )
Theorem	0lepnf 9606	0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 <_ +oo
Theorem	nn0pnfge0 9607	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 9608	Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
|- ( A e. RR* -> -oo <_ A )
Theorem	xrltnsym 9609	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 9610	'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 9611	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 9612	'Less than' is a weakly linear ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)
|- < Or RR*
Theorem	xrlttri3 9613	Extended real version of lttri3 7868. (Contributed by NM, 9-Feb-2006.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
Theorem	xrltle 9614	'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 9615	'Less than' implies 'less than or equal to' for extended reals. Deduction form of xrltle 9614. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> A < B )   =>    |- ( ph -> A <_ B )
Theorem	xrleid 9616	'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
|- ( A e. RR* -> A <_ A )
Theorem	xrleidd 9617	'Less than or equal to' is reflexive for extended reals. Deduction form of xrleid 9616. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> A e. RR* )   =>    |- ( ph -> A <_ A )
Theorem	xrletri3 9618	Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
Theorem	xrlelttr 9619	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 9620	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 9621	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 9622	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 9623	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 9624	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 9625	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 9626	'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 9627	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 9628	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 9629	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 9630	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 9631	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 9632	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 9633	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 9634	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 9635	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 9636	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 9637	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 9638	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 9639*	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 9640	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 9641	Minus +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
|- -e +oo = -oo
Theorem	xnegmnf 9642	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 9643	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 9644	The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- -e 0 = 0
Theorem	xnegcl 9645	Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> -e A e. RR* )
Theorem	xnegneg 9646	Extended real version of negneg 8036. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> -e -e A = A )
Theorem	xneg11 9647	Extended real version of neg11 8037. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( -e A = -e B <-> A = B ) )
Theorem	xltnegi 9648	Forward direction of xltneg 9649. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> -e B < -e A )
Theorem	xltneg 9649	Extended real version of ltneg 8248. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -e B < -e A ) )
Theorem	xleneg 9650	Extended real version of leneg 8251. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -e B <_ -e A ) )
Theorem	xlt0neg1 9651	Extended real version of lt0neg1 8254. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( A < 0 <-> 0 < -e A ) )
Theorem	xlt0neg2 9652	Extended real version of lt0neg2 8255. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( 0 < A <-> -e A < 0 ) )
Theorem	xle0neg1 9653	Extended real version of le0neg1 8256. (Contributed by Mario Carneiro, 9-Sep-2015.)
|- ( A e. RR* -> ( A <_ 0 <-> 0 <_ -e A ) )
Theorem	xle0neg2 9654	Extended real version of le0neg2 8257. (Contributed by Mario Carneiro, 9-Sep-2015.)
|- ( A e. RR* -> ( 0 <_ A <-> -e A <_ 0 ) )
Theorem	xrpnfdc 9655	An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
|- ( A e. RR* -> DECID A = +oo )
Theorem	xrmnfdc 9656	An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
|- ( A e. RR* -> DECID A = -oo )
Theorem	xaddf 9657	The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- +e : ( RR* X. RR* ) --> RR*
Theorem	xaddval 9658	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 9659	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 9660	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 9661	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 9662	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 9663	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 9664	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 9665	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 9666	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 9667	The extended real addition operation when both arguments are real. Deduction version of rexadd 9665. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A +e B ) = ( A + B ) )
Theorem	xnegcld 9668	Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR* )   =>    |- ( ph -> -e A e. RR* )
Theorem	xrex 9669	The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
|- RR* e. _V
Theorem	xaddnemnf 9670	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 9671	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 9672	Extended real version of negid 8033. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( A +e -e A ) = 0 )
Theorem	xaddcl 9673	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 9674	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 9675	Extended real version of addid1 7924. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( A +e 0 ) = A )
Theorem	xaddid2 9676	Extended real version of addid2 7925. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( 0 +e A ) = A )
Theorem	xaddid1d 9677	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 9678	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 9679	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 9680	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 9681	Extended real version of negdi 8043. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
Theorem	xaddass 9682	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 9683, 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 9683	Associativity of extended real addition. See xaddass 9682 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 9684	Extended real version of pncan 7992. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -e B ) = A )
Theorem	xnpcan 9685	Extended real version of npcan 7995. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e -e B ) +e B ) = A )
Theorem	xleadd1a 9686	Extended real version of leadd1 8216; 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 ) )
Theorem	xleadd2a 9687	Commuted form of xleadd1a 9686. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( C +e A ) <_ ( C +e B ) )
Theorem	xleadd1 9688	Weakened version of xleadd1a 9686 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 ) ) )
Theorem	xltadd1 9689	Extended real version of ltadd1 8215. (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 ) ) )
Theorem	xltadd2 9690	Extended real version of ltadd2 8205. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) -> ( A < B <-> ( C +e A ) < ( C +e B ) ) )
Theorem	xaddge0 9691	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 ) )
Theorem	xle2add 9692	Extended real version of le2add 8230. (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 ) ) )
Theorem	xlt2add 9693	Extended real version of lt2add 8231. Note that ltleadd 8232, 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 ) ) )
Theorem	xsubge0 9694	Extended real version of subge0 8261. (Contributed by Mario Carneiro, 24-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )
Theorem	xposdif 9695	Extended real version of posdif 8241. (Contributed by Mario Carneiro, 24-Aug-2015.) (Revised by Jim Kingdon, 17-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
Theorem	xlesubadd 9696	Under certain conditions, the conclusion of lesubadd 8220 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 ) ) )
Theorem	xaddcld 9697	The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   =>    |- ( ph -> ( A +e B ) e. RR* )
Theorem	xadd4d 9698	Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 7955. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
|- ( ph -> ( A e. RR* /\ A =/= -oo ) )   &    |- ( ph -> ( B e. RR* /\ B =/= -oo ) )   &    |- ( ph -> ( C e. RR* /\ C =/= -oo ) )   &    |- ( ph -> ( D e. RR* /\ D =/= -oo ) )   =>    |- ( ph -> ( ( A +e B ) +e ( C +e D ) ) = ( ( A +e C ) +e ( B +e D ) ) )
Theorem	xnn0add4d 9699	Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 9698. (Contributed by AV, 12-Dec-2020.)
|- ( ph -> A e. NN0* )   &    |- ( ph -> B e. NN0* )   &    |- ( ph -> C e. NN0* )   &    |- ( ph -> D e. NN0* )   =>    |- ( ph -> ( ( A +e B ) +e ( C +e D ) ) = ( ( A +e C ) +e ( B +e D ) ) )
Theorem	xleaddadd 9700	Cancelling a factor of two in <_ (expressed as addition rather than as a factor to avoid extended real multiplication). (Contributed by Jim Kingdon, 18-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> ( A +e A ) <_ ( B +e B ) ) )