P. 72 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	remulcl 7101	Alias for ax-mulrcl 7075, for naming consistency with remulcli 7133. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
Theorem	mulcom 7102	Alias for ax-mulcom 7077, for naming consistency with mulcomi 7125. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
Theorem	addass 7103	Alias for ax-addass 7078, for naming consistency with addassi 7127. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
Theorem	mulass 7104	Alias for ax-mulass 7079, for naming consistency with mulassi 7128. (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 7105	Alias for ax-distr 7080, for naming consistency with adddii 7129. (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 7106	A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
|- ( A e. RR -> A e. CC )
Theorem	reex 7107	The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.)
|- RR e. _V
Theorem	reelprrecn 7108	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 7109	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 7110	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 7111	0 is a complex number. (Contributed by NM, 19-Feb-2005.)
|- 0 e. CC
Theorem	0cnd 7112	0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 0 e. CC )
Theorem	c0ex 7113	0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 0 e. _V
Theorem	1ex 7114	1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 1 e. _V
Theorem	cnre 7115*	Alias for ax-cnre 7087, 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 7116	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 7117	Identity law for multiplication. Note: see mulid1 7116 for commuted version. (Contributed by NM, 8-Oct-1999.)
|- ( A e. CC -> ( 1 x. A ) = A )
Theorem	1re 7118	1 is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.)
|- 1 e. RR
Theorem	0re 7119	0 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
|- 0 e. RR
Theorem	0red 7120	0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 0 e. RR )
Theorem	mulid1i 7121	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
|- A e. CC   =>    |- ( A x. 1 ) = A
Theorem	mulid2i 7122	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
|- A e. CC   =>    |- ( 1 x. A ) = A
Theorem	addcli 7123	Closure law for addition. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A + B ) e. CC
Theorem	mulcli 7124	Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A x. B ) e. CC
Theorem	mulcomi 7125	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A x. B ) = ( B x. A )
Theorem	mulcomli 7126	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 7127	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 7128	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 7129	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 7130	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 7131	A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
|- A e. RR   =>    |- A e. CC
Theorem	readdcli 7132	Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
|- A e. RR   &    |- B e. RR   =>    |- ( A + B ) e. RR
Theorem	remulcli 7133	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 7134	1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. RR )
Theorem	1cnd 7135	1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. CC )
Theorem	mulid1d 7136	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. 1 ) = A )
Theorem	mulid2d 7137	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 1 x. A ) = A )
Theorem	addcld 7138	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 7139	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 7140	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 7141	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 7142	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 7143	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 7144	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 7145	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 7146	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 7147	Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A e. CC )
Theorem	readdcld 7148	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 7149	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 )
3.2.2  Infinity and the extended real number system
Syntax	cpnf 7150	Plus infinity.
class +oo
Syntax	cmnf 7151	Minus infinity.
class -oo
Syntax	cxr 7152	The set of extended reals (includes plus and minus infinity).
class RR*
Syntax	clt 7153	'Less than' predicate (extended to include the extended reals).
class <
Syntax	cle 7154	Extend wff notation to include the 'less than or equal to' relation.
class <_
Definition	df-pnf 7155	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 7156). 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 7160 and mnfnre 7161, 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 7156	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 7161). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
|- -oo = ~P +oo
Definition	df-xr 7157	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 7158*	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 7159	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 7160	Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
|- +oo e/ RR
Theorem	mnfnre 7161	Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
|- -oo e/ RR
Theorem	ressxr 7162	The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
|- RR C_ RR*
Theorem	rexpssxrxp 7163	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 7164	A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
|- ( A e. RR -> A e. RR* )
Theorem	0xr 7165	Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
|- 0 e. RR*
Theorem	renepnf 7166	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 7167	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 7168	A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A e. RR* )
Theorem	renepnfd 7169	No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A =/= +oo )
Theorem	renemnfd 7170	No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A =/= -oo )
Theorem	rexri 7171	A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
|- A e. RR   =>    |- A e. RR*
Theorem	renfdisj 7172	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 7173	'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
|- < C_ ( RR* X. RR* )
Theorem	ltrel 7174	'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
|- Rel <
Theorem	lerelxr 7175	'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
|- <_ C_ ( RR* X. RR* )
Theorem	lerel 7176	'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- Rel <_
Theorem	xrlenlt 7177	'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 7178	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 ) )
3.2.3  Restate the ordering postulates with extended real "less than"
Theorem	axltirr 7179	Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7088 with ordering on the extended reals. New proofs should use ltnr 7188 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
|- ( A e. RR -> -. A < A )
Theorem	axltwlin 7180	Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7089 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 7181	Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 7090 with ordering on the extended reals. New proofs should use lttr 7185 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 7182	Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7092 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 7183	Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7091 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 7184	The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7093 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 ) ) )
3.2.4  Ordering on reals
Theorem	lttr 7185	Alias for axlttrn 7181, for naming consistency with lttri 7215. New proofs should generally use this instead of ax-pre-lttrn 7090. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
Theorem	mulgt0 7186	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 7187	'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 7188	'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
|- ( A e. RR -> -. A < A )
Theorem	ltso 7189	'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
|- < Or RR
Theorem	gtso 7190	'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
|- `' < Or RR
Theorem	lttri3 7191	Tightness of real apartness. (Contributed by NM, 5-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
Theorem	letri3 7192	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
Theorem	ltleletr 7193	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 7194	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 7195	'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
|- ( A e. RR -> A <_ A )
Theorem	ltne 7196	'Less than' implies not equal. See also ltap 7731 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 7197	'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -. B < A ) )
Theorem	ltle 7198	'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 7199	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 7200	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 ) )