P. 81 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8001-8100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	1red 8001	1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. RR )
Theorem	1cnd 8002	1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. CC )
Theorem	mulridd 8003	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. 1 ) = A )
Theorem	mullidd 8004	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 1 x. A ) = A )
Theorem	mulid2d 8005	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 1 x. A ) = A )
Theorem	addcld 8006	Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( A + B ) e. CC )
Theorem	mulcld 8007	Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( A x. B ) e. CC )
Theorem	mulcomd 8008	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 8009	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 8010	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 8011	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 8012	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 8013	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 8014	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 8015	Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A e. CC )
Theorem	readdcld 8016	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 8017	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 8018	Plus infinity.
class +oo
Syntax	cmnf 8019	Minus infinity.
class -oo
Syntax	cxr 8020	The set of extended reals (includes plus and minus infinity).
class RR*
Syntax	clt 8021	'Less than' predicate (extended to include the extended reals).
class <
Syntax	cle 8022	Extend wff notation to include the 'less than or equal to' relation.
class <_
Definition	df-pnf 8023	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 8024). 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 8028 and mnfnre 8029, 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 8024	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 8029). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
|- -oo = ~P +oo
Definition	df-xr 8025	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 8026*	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 8027	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 8028	Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
|- +oo e/ RR
Theorem	mnfnre 8029	Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
|- -oo e/ RR
Theorem	ressxr 8030	The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
|- RR C_ RR*
Theorem	rexpssxrxp 8031	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 8032	A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
|- ( A e. RR -> A e. RR* )
Theorem	0xr 8033	Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
|- 0 e. RR*
Theorem	renepnf 8034	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 8035	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 8036	A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A e. RR* )
Theorem	renepnfd 8037	No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A =/= +oo )
Theorem	renemnfd 8038	No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A =/= -oo )
Theorem	pnfxr 8039	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 8040	Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- +oo e. _V
Theorem	pnfnemnf 8041	Plus and minus infinity are different elements of RR*. (Contributed by NM, 14-Oct-2005.)
|- +oo =/= -oo
Theorem	mnfnepnf 8042	Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -oo =/= +oo
Theorem	mnfxr 8043	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 8044	A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
|- A e. RR   =>    |- A e. RR*
Theorem	1xr 8045	1 is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
|- 1 e. RR*
Theorem	renfdisj 8046	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 8047	'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
|- < C_ ( RR* X. RR* )
Theorem	ltrel 8048	'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
|- Rel <
Theorem	lerelxr 8049	'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
|- <_ C_ ( RR* X. RR* )
Theorem	lerel 8050	'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- Rel <_
Theorem	xrlenlt 8051	'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 8052	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 8053	Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7952 with ordering on the extended reals. New proofs should use ltnr 8063 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
|- ( A e. RR -> -. A < A )
Theorem	axltwlin 8054	Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7953 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 8055	Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 7954 with ordering on the extended reals. New proofs should use lttr 8060 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 8056	Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7956 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 8057	Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7955 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 8058	The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7957 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 8059*	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 7961 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 8060	Alias for axlttrn 8055, for naming consistency with lttri 8091. New proofs should generally use this instead of ax-pre-lttrn 7954. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
Theorem	mulgt0 8061	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 8062	'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 8063	'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
|- ( A e. RR -> -. A < A )
Theorem	ltso 8064	'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
|- < Or RR
Theorem	gtso 8065	'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
|- `' < Or RR
Theorem	lttri3 8066	Tightness of real apartness. (Contributed by NM, 5-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
Theorem	letri3 8067	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
Theorem	ltleletr 8068	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 8069	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 8070	'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
|- ( A e. RR -> A <_ A )
Theorem	ltne 8071	'Less than' implies not equal. See also ltap 8619 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 8072	'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -. B < A ) )
Theorem	eqlelt 8073	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 8074	'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 8075	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 8076	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 8077	'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 8078	Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
|- ( ( A e. RR /\ A = B ) -> A <_ B )
Theorem	ltnri 8079	'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
|- A e. RR   =>    |- -. A < A
Theorem	eqlei 8080	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 8081	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 8082	'Less than' implies not equal. See also gtapii 8620 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.)
|- A e. RR   &    |- A < B   =>    |- B =/= A
Theorem	ltneii 8083	'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
|- A e. RR   &    |- A < B   =>    |- A =/= B
Theorem	lttri3i 8084	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A = B <-> ( -. A < B /\ -. B < A ) )
Theorem	letri3i 8085	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A = B <-> ( A <_ B /\ B <_ A ) )
Theorem	ltnsymi 8086	'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> -. B < A )
Theorem	lenlti 8087	'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 8088	'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 8089	'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 8090	'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> B =/= A )
Theorem	lttri 8091	'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 8092	'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 8093	'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 8094	'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 8095	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 8096	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 8097	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 8098	'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> -. A < A )
Theorem	gtned 8099	'Less than' implies not equal. See also gtapd 8623 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 8100	'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> A =/= B )