P. 83 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8201-8300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mulcomd 8201	Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( A x. B ) = ( B x. A ) )
Theorem	addassd 8202	Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
Theorem	mulassd 8203	Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
Theorem	adddid 8204	Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) )
Theorem	adddird 8205	Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )
Theorem	adddirp1d 8206	Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( ( A + 1 ) x. B ) = ( ( A x. B ) + B ) )
Theorem	joinlmuladdmuld 8207	Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> ( ( A x. B ) + ( C x. B ) ) = D )   =>    |- ( ph -> ( ( A + C ) x. B ) = D )
Theorem	recnd 8208	Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A e. CC )
Theorem	readdcld 8209	Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A + B ) e. RR )
Theorem	remulcld 8210	Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A x. B ) e. RR )
4.2.2  Infinity and the extended real number system
Syntax	cpnf 8211	Plus infinity.
class +oo
Syntax	cmnf 8212	Minus infinity.
class -oo
Syntax	cxr 8213	The set of extended reals (includes plus and minus infinity).
class RR*
Syntax	clt 8214	'Less than' predicate (extended to include the extended reals).
class <
Syntax	cle 8215	Extend wff notation to include the 'less than or equal to' relation.
class <_
Definition	df-pnf 8216	Define plus infinity. Note that the definition is arbitrary, requiring only that +oo be a set not in RR and different from -oo (df-mnf 8217). We use ~P U. CC to make it independent of the construction of CC, and Cantor's Theorem will show that it is different from any member of CC and therefore RR. See pnfnre 8221 and mnfnre 8222, and we'll also be able to prove +oo =/= -oo. A simpler possibility is to define +oo as CC and -oo as { CC }, but that approach requires the Axiom of Regularity to show that +oo and -oo are different from each other and from all members of RR. (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
|- +oo = ~P U. CC
Definition	df-mnf 8217	Define minus infinity as the power set of plus infinity. Note that the definition is arbitrary, requiring only that -oo be a set not in RR and different from +oo (see mnfnre 8222). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
|- -oo = ~P +oo
Definition	df-xr 8218	Define the set of extended reals that includes plus and minus infinity. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 13-Oct-2005.)
|- RR* = ( RR u. { +oo , -oo } )
Definition	df-ltxr 8219*	Define 'less than' on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. Note that in our postulates for complex numbers, <RR is primitive and not necessarily a relation on RR. (Contributed by NM, 13-Oct-2005.)
|- < = ( { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } u. ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) )
Definition	df-le 8220	Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.)
|- <_ = ( ( RR* X. RR* ) \ `' < )
Theorem	pnfnre 8221	Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
|- +oo e/ RR
Theorem	mnfnre 8222	Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
|- -oo e/ RR
Theorem	ressxr 8223	The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
|- RR C_ RR*
Theorem	rexpssxrxp 8224	The Cartesian product of standard reals are a subset of the Cartesian product of extended reals (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( RR X. RR ) C_ ( RR* X. RR* )
Theorem	rexr 8225	A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
|- ( A e. RR -> A e. RR* )
Theorem	0xr 8226	Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
|- 0 e. RR*
Theorem	renepnf 8227	No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( A e. RR -> A =/= +oo )
Theorem	renemnf 8228	No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( A e. RR -> A =/= -oo )
Theorem	rexrd 8229	A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A e. RR* )
Theorem	renepnfd 8230	No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A =/= +oo )
Theorem	renemnfd 8231	No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A =/= -oo )
Theorem	pnfxr 8232	Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)
|- +oo e. RR*
Theorem	pnfex 8233	Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- +oo e. _V
Theorem	pnfnemnf 8234	Plus and minus infinity are different elements of RR*. (Contributed by NM, 14-Oct-2005.)
|- +oo =/= -oo
Theorem	mnfnepnf 8235	Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -oo =/= +oo
Theorem	mnfxr 8236	Minus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- -oo e. RR*
Theorem	rexri 8237	A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
|- A e. RR   =>    |- A e. RR*
Theorem	1xr 8238	1 is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
|- 1 e. RR*
Theorem	renfdisj 8239	The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( RR i^i { +oo , -oo } ) = (/)
Theorem	ltrelxr 8240	'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
|- < C_ ( RR* X. RR* )
Theorem	ltrel 8241	'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
|- Rel <
Theorem	lerelxr 8242	'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
|- <_ C_ ( RR* X. RR* )
Theorem	lerel 8243	'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- Rel <_
Theorem	xrlenlt 8244	'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
Theorem	ltxrlt 8245	The standard less-than <RR and the extended real less-than < are identical when restricted to the non-extended reals RR. (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
4.2.3  Restate the ordering postulates with extended real "less than"
Theorem	axltirr 8246	Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 8144 with ordering on the extended reals. New proofs should use ltnr 8256 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
|- ( A e. RR -> -. A < A )
Theorem	axltwlin 8247	Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 8145 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( A < C \/ C < B ) ) )
Theorem	axlttrn 8248	Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 8146 with ordering on the extended reals. New proofs should use lttr 8253 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
Theorem	axltadd 8249	Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 8148 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( C + A ) < ( C + B ) ) )
Theorem	axapti 8250	Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 8147 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)
|- ( ( A e. RR /\ B e. RR /\ -. ( A < B \/ B < A ) ) -> A = B )
Theorem	axmulgt0 8251	The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 8149 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) ) )
Theorem	axsuploc 8252*	An inhabited, bounded-above, located set of reals has a supremum. Axiom for real and complex numbers, derived from ZF set theory. (This restates ax-pre-suploc 8153 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.)
|- ( ( ( A C_ RR /\ E. x x e. A ) /\ ( E. x e. RR A. y e. A y < x /\ A. x e. RR A. y e. RR ( x < y -> ( E. z e. A x < z \/ A. z e. A z < y ) ) ) ) -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )
4.2.4  Ordering on reals
Theorem	lttr 8253	Alias for axlttrn 8248, for naming consistency with lttri 8284. New proofs should generally use this instead of ax-pre-lttrn 8146. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
Theorem	mulgt0 8254	The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.)
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( A x. B ) )
Theorem	lenlt 8255	'Less than or equal to' expressed in terms of 'less than'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) )
Theorem	ltnr 8256	'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
|- ( A e. RR -> -. A < A )
Theorem	ltso 8257	'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
|- < Or RR
Theorem	gtso 8258	'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
|- `' < Or RR
Theorem	lttri3 8259	Tightness of real apartness. (Contributed by NM, 5-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
Theorem	letri3 8260	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
Theorem	ltleletr 8261	Transitive law, weaker form of ( A < B /\ B <_ C ) -> A < C. (Contributed by AV, 14-Oct-2018.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B <_ C ) -> A <_ C ) )
Theorem	letr 8262	Transitive law. (Contributed by NM, 12-Nov-1999.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <_ B /\ B <_ C ) -> A <_ C ) )
Theorem	leid 8263	'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
|- ( A e. RR -> A <_ A )
Theorem	ltne 8264	'Less than' implies not equal. See also ltap 8813 which is the same but for apartness. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)
|- ( ( A e. RR /\ A < B ) -> B =/= A )
Theorem	ltnsym 8265	'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -. B < A ) )
Theorem	eqlelt 8266	Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( A <_ B /\ -. A < B ) ) )
Theorem	ltle 8267	'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B -> A <_ B ) )
Theorem	lelttr 8268	Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 23-May-1999.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <_ B /\ B < C ) -> A < C ) )
Theorem	ltletr 8269	Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B <_ C ) -> A < C ) )
Theorem	ltnsym2 8270	'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( ( A e. RR /\ B e. RR ) -> -. ( A < B /\ B < A ) )
Theorem	eqle 8271	Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
|- ( ( A e. RR /\ A = B ) -> A <_ B )
Theorem	ltnri 8272	'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
|- A e. RR   =>    |- -. A < A
Theorem	eqlei 8273	Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.) (Revised by Alexander van der Vekens, 20-Mar-2018.)
|- A e. RR   =>    |- ( A = B -> A <_ B )
Theorem	eqlei2 8274	Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
|- A e. RR   =>    |- ( B = A -> B <_ A )
Theorem	gtneii 8275	'Less than' implies not equal. See also gtapii 8814 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.)
|- A e. RR   &    |- A < B   =>    |- B =/= A
Theorem	ltneii 8276	'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
|- A e. RR   &    |- A < B   =>    |- A =/= B
Theorem	lttri3i 8277	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A = B <-> ( -. A < B /\ -. B < A ) )
Theorem	letri3i 8278	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A = B <-> ( A <_ B /\ B <_ A ) )
Theorem	ltnsymi 8279	'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> -. B < A )
Theorem	lenlti 8280	'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A <_ B <-> -. B < A )
Theorem	ltlei 8281	'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> A <_ B )
Theorem	ltleii 8282	'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.)
|- A e. RR   &    |- B e. RR   &    |- A < B   =>    |- A <_ B
Theorem	ltnei 8283	'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> B =/= A )
Theorem	lttri 8284	'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A < B /\ B < C ) -> A < C )
Theorem	lelttri 8285	'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A <_ B /\ B < C ) -> A < C )
Theorem	ltletri 8286	'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A < B /\ B <_ C ) -> A < C )
Theorem	letri 8287	'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A <_ B /\ B <_ C ) -> A <_ C )
Theorem	le2tri3i 8288	Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A <_ B /\ B <_ C /\ C <_ A ) <-> ( A = B /\ B = C /\ C = A ) )
Theorem	mulgt0i 8289	The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) )
Theorem	mulgt0ii 8290	The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- 0 < A   &    |- 0 < B   =>    |- 0 < ( A x. B )
Theorem	ltnrd 8291	'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> -. A < A )
Theorem	gtned 8292	'Less than' implies not equal. See also gtapd 8817 which is the same but for apartness. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> B =/= A )
Theorem	ltned 8293	'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> A =/= B )
Theorem	lttri3d 8294	Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
Theorem	letri3d 8295	Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
Theorem	eqleltd 8296	Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A = B <-> ( A <_ B /\ -. A < B ) ) )
Theorem	lenltd 8297	'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A <_ B <-> -. B < A ) )
Theorem	ltled 8298	'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> A <_ B )
Theorem	ltnsymd 8299	'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> -. B < A )
Theorem	nltled 8300	'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> -. B < A )   =>    |- ( ph -> A <_ B )