P. 80 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mulcl 7901	Alias for ax-mulcl 7872, for naming consistency with mulcli 7925. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
Theorem	remulcl 7902	Alias for ax-mulrcl 7873, for naming consistency with remulcli 7934. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
Theorem	mulcom 7903	Alias for ax-mulcom 7875, for naming consistency with mulcomi 7926. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
Theorem	addass 7904	Alias for ax-addass 7876, for naming consistency with addassi 7928. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
Theorem	mulass 7905	Alias for ax-mulass 7877, for naming consistency with mulassi 7929. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
Theorem	adddi 7906	Alias for ax-distr 7878, for naming consistency with adddii 7930. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) )
Theorem	recn 7907	A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
|- ( A e. RR -> A e. CC )
Theorem	reex 7908	The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.)
|- RR e. _V
Theorem	reelprrecn 7909	Reals are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- RR e. { RR , CC }
Theorem	cnelprrecn 7910	Complex numbers are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- CC e. { RR , CC }
Theorem	adddir 7911	Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )
Theorem	0cn 7912	0 is a complex number. (Contributed by NM, 19-Feb-2005.)
|- 0 e. CC
Theorem	0cnd 7913	0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 0 e. CC )
Theorem	c0ex 7914	0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 0 e. _V
Theorem	1ex 7915	1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 1 e. _V
Theorem	cnre 7916*	Alias for ax-cnre 7885, for naming consistency. (Contributed by NM, 3-Jan-2013.)
|- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
Theorem	mulid1 7917	1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( A e. CC -> ( A x. 1 ) = A )
Theorem	mulid2 7918	Identity law for multiplication. Note: see mulid1 7917 for commuted version. (Contributed by NM, 8-Oct-1999.)
|- ( A e. CC -> ( 1 x. A ) = A )
Theorem	1re 7919	1 is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.)
|- 1 e. RR
Theorem	0re 7920	0 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
|- 0 e. RR
Theorem	0red 7921	0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 0 e. RR )
Theorem	mulid1i 7922	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
|- A e. CC   =>    |- ( A x. 1 ) = A
Theorem	mulid2i 7923	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
|- A e. CC   =>    |- ( 1 x. A ) = A
Theorem	addcli 7924	Closure law for addition. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A + B ) e. CC
Theorem	mulcli 7925	Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A x. B ) e. CC
Theorem	mulcomi 7926	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A x. B ) = ( B x. A )
Theorem	mulcomli 7927	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   &    |- ( A x. B ) = C   =>    |- ( B x. A ) = C
Theorem	addassi 7928	Associative law for addition. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( ( A + B ) + C ) = ( A + ( B + C ) )
Theorem	mulassi 7929	Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( ( A x. B ) x. C ) = ( A x. ( B x. C ) )
Theorem	adddii 7930	Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) )
Theorem	adddiri 7931	Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) )
Theorem	recni 7932	A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
|- A e. RR   =>    |- A e. CC
Theorem	readdcli 7933	Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
|- A e. RR   &    |- B e. RR   =>    |- ( A + B ) e. RR
Theorem	remulcli 7934	Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
|- A e. RR   &    |- B e. RR   =>    |- ( A x. B ) e. RR
Theorem	1red 7935	1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. RR )
Theorem	1cnd 7936	1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. CC )
Theorem	mulid1d 7937	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. 1 ) = A )
Theorem	mulid2d 7938	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 1 x. A ) = A )
Theorem	addcld 7939	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 7940	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 7941	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 7942	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 7943	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 7944	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 7945	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 7946	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 7947	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 7948	Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A e. CC )
Theorem	readdcld 7949	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 7950	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 7951	Plus infinity.
class +oo
Syntax	cmnf 7952	Minus infinity.
class -oo
Syntax	cxr 7953	The set of extended reals (includes plus and minus infinity).
class RR*
Syntax	clt 7954	'Less than' predicate (extended to include the extended reals).
class <
Syntax	cle 7955	Extend wff notation to include the 'less than or equal to' relation.
class <_
Definition	df-pnf 7956	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 7957). 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 7961 and mnfnre 7962, 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 7957	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 7962). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
|- -oo = ~P +oo
Definition	df-xr 7958	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 7959*	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 7960	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 7961	Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
|- +oo e/ RR
Theorem	mnfnre 7962	Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
|- -oo e/ RR
Theorem	ressxr 7963	The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
|- RR C_ RR*
Theorem	rexpssxrxp 7964	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 7965	A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
|- ( A e. RR -> A e. RR* )
Theorem	0xr 7966	Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
|- 0 e. RR*
Theorem	renepnf 7967	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 7968	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 7969	A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A e. RR* )
Theorem	renepnfd 7970	No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A =/= +oo )
Theorem	renemnfd 7971	No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A =/= -oo )
Theorem	pnfxr 7972	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 7973	Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- +oo e. _V
Theorem	pnfnemnf 7974	Plus and minus infinity are different elements of RR*. (Contributed by NM, 14-Oct-2005.)
|- +oo =/= -oo
Theorem	mnfnepnf 7975	Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -oo =/= +oo
Theorem	mnfxr 7976	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 7977	A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
|- A e. RR   =>    |- A e. RR*
Theorem	1xr 7978	1 is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
|- 1 e. RR*
Theorem	renfdisj 7979	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 7980	'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
|- < C_ ( RR* X. RR* )
Theorem	ltrel 7981	'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
|- Rel <
Theorem	lerelxr 7982	'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
|- <_ C_ ( RR* X. RR* )
Theorem	lerel 7983	'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- Rel <_
Theorem	xrlenlt 7984	'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 7985	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 7986	Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7886 with ordering on the extended reals. New proofs should use ltnr 7996 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
|- ( A e. RR -> -. A < A )
Theorem	axltwlin 7987	Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7887 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 7988	Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 7888 with ordering on the extended reals. New proofs should use lttr 7993 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 7989	Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7890 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 7990	Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7889 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 7991	The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7891 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 7992*	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 7895 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 7993	Alias for axlttrn 7988, for naming consistency with lttri 8024. New proofs should generally use this instead of ax-pre-lttrn 7888. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
Theorem	mulgt0 7994	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 7995	'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 7996	'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
|- ( A e. RR -> -. A < A )
Theorem	ltso 7997	'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
|- < Or RR
Theorem	gtso 7998	'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
|- `' < Or RR
Theorem	lttri3 7999	Tightness of real apartness. (Contributed by NM, 5-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
Theorem	letri3 8000	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )