P. 103 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10201-10300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xrltled 10201	'Less than' implies 'less than or equal to' for extended reals. Deduction form of xrltle 10200. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> A < B )   =>    |- ( ph -> A <_ B )
Theorem	xrleid 10202	'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
|- ( A e. RR* -> A <_ A )
Theorem	xrleidd 10203	'Less than or equal to' is reflexive for extended reals. Deduction form of xrleid 10202. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> A e. RR* )   =>    |- ( ph -> A <_ A )
Theorem	xnn0dcle 10204	Decidability of <_ for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
|- ( ( A e. NN0* /\ B e. NN0* ) -> DECID A <_ B )
Theorem	xnn0letri 10205	Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
|- ( ( A e. NN0* /\ B e. NN0* ) -> ( A <_ B \/ B <_ A ) )
Theorem	xrletri3 10206	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 10207	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 10208	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 10209	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 10210	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 10211	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 10212	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 10213	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 10214	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 10215	'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 10216	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 10217	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 10218	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 10219	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 10220	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 10221	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 10222	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 10223	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 10224	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 10225	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 10226	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 10227	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 10228*	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 10229	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 10230	Minus +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
|- -e +oo = -oo
Theorem	xnegmnf 10231	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 10232	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 10233	The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- -e 0 = 0
Theorem	xnegcl 10234	Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> -e A e. RR* )
Theorem	xnegneg 10235	Extended real version of negneg 8576. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> -e -e A = A )
Theorem	xneg11 10236	Extended real version of neg11 8577. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( -e A = -e B <-> A = B ) )
Theorem	xltnegi 10237	Forward direction of xltneg 10238. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> -e B < -e A )
Theorem	xltneg 10238	Extended real version of ltneg 8790. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -e B < -e A ) )
Theorem	xleneg 10239	Extended real version of leneg 8793. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -e B <_ -e A ) )
Theorem	xlt0neg1 10240	Extended real version of lt0neg1 8796. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( A < 0 <-> 0 < -e A ) )
Theorem	xlt0neg2 10241	Extended real version of lt0neg2 8797. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( 0 < A <-> -e A < 0 ) )
Theorem	xle0neg1 10242	Extended real version of le0neg1 8798. (Contributed by Mario Carneiro, 9-Sep-2015.)
|- ( A e. RR* -> ( A <_ 0 <-> 0 <_ -e A ) )
Theorem	xle0neg2 10243	Extended real version of le0neg2 8799. (Contributed by Mario Carneiro, 9-Sep-2015.)
|- ( A e. RR* -> ( 0 <_ A <-> -e A <_ 0 ) )
Theorem	xrpnfdc 10244	An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
|- ( A e. RR* -> DECID A = +oo )
Theorem	xrmnfdc 10245	An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
|- ( A e. RR* -> DECID A = -oo )
Theorem	xaddf 10246	The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- +e : ( RR* X. RR* ) --> RR*
Theorem	xaddval 10247	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 10248	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 10249	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 10250	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 10251	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 10252	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 10253	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 10254	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 10255	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 10256	The extended real addition operation when both arguments are real. Deduction version of rexadd 10254. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A +e B ) = ( A + B ) )
Theorem	xnegcld 10257	Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR* )   =>    |- ( ph -> -e A e. RR* )
Theorem	xrex 10258	The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
|- RR* e. _V
Theorem	xaddnemnf 10259	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 10260	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 10261	Extended real version of negid 8573. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( A +e -e A ) = 0 )
Theorem	xaddcl 10262	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 10263	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 10264	Extended real version of addrid 8464. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( A +e 0 ) = A )
Theorem	xaddid2 10265	Extended real version of addlid 8465. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( A e. RR* -> ( 0 +e A ) = A )
Theorem	xaddid1d 10266	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 10267	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 10268	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 10269	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 10270	Extended real version of negdi 8583. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> -e ( A +e B ) = ( -e A +e -e B ) )
Theorem	xaddass 10271	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 10272, 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 10272	Associativity of extended real addition. See xaddass 10271 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 10273	Extended real version of pncan 8532. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -e B ) = A )
Theorem	xnpcan 10274	Extended real version of npcan 8535. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e -e B ) +e B ) = A )
Theorem	xleadd1a 10275	Extended real version of leadd1 8758; 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 10276	Commuted form of xleadd1a 10275. (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 10277	Weakened version of xleadd1a 10275 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 10278	Extended real version of ltadd1 8757. (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 10279	Extended real version of ltadd2 8747. (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 10280	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 10281	Extended real version of le2add 8772. (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 10282	Extended real version of lt2add 8773. Note that ltleadd 8774, 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 10283	Extended real version of subge0 8803. (Contributed by Mario Carneiro, 24-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( 0 <_ ( A +e -e B ) <-> B <_ A ) )
Theorem	xposdif 10284	Extended real version of posdif 8783. (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 10285	Under certain conditions, the conclusion of lesubadd 8762 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 10286	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 10287	Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 8495. (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 10288	Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 10287. (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 10289	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 ) ) )
4.5.3  Real number intervals
Syntax	cioo 10290	Extend class notation with the set of open intervals of extended reals.
class (,)
Syntax	cioc 10291	Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
Syntax	cico 10292	Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
Syntax	cicc 10293	Extend class notation with the set of closed intervals of extended reals.
class [,]
Definition	df-ioo 10294*	Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
|- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
Definition	df-ioc 10295*	Define the set of open-below, closed-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
|- (,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z <_ y ) } )
Definition	df-ico 10296*	Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
|- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
Definition	df-icc 10297*	Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
|- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
Theorem	ixxval 10298*	Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)
|- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )   =>    |- ( ( A e. RR* /\ B e. RR* ) -> ( A O B ) = { z e. RR* | ( A R z /\ z S B ) } )
Theorem	elixx1 10299*	Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)
|- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )   =>    |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <-> ( C e. RR* /\ A R C /\ C S B ) ) )
Theorem	ixxf 10300*	The set of intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
|- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )   =>    |- O : ( RR* X. RR* ) --> ~P RR*