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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1cnd 7101 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
 |-  ( ph  ->  1  e.  CC )
 
Theoremmulid1d 7102 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  1 )  =  A )
 
Theoremmulid2d 7103 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 1  x.  A )  =  A )
 
Theoremaddcld 7104 Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  +  B )  e.  CC )
 
Theoremmulcld 7105 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 )
 
Theoremmulcomd 7106 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 ) )
 
Theoremaddassd 7107 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 )
 ) )
 
Theoremmulassd 7108 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 ) ) )
 
Theoremadddid 7109 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 ) ) )
 
Theoremadddird 7110 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 ) ) )
 
Theoremadddirp1d 7111 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 ) )
 
Theoremjoinlmuladdmuld 7112 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 )
 
Theoremrecnd 7113 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremreaddcld 7114 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 )
 
Theoremremulcld 7115 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
 
Syntaxcpnf 7116 Plus infinity.
 class +oo
 
Syntaxcmnf 7117 Minus infinity.
 class -oo
 
Syntaxcxr 7118 The set of extended reals (includes plus and minus infinity).
 class  RR*
 
Syntaxclt 7119 'Less than' predicate (extended to include the extended reals).
 class  <
 
Syntaxcle 7120 Extend wff notation to include the 'less than or equal to' relation.
 class  <_
 
Definitiondf-pnf 7121 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 7122). 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 7126 and mnfnre 7127, 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
 
Definitiondf-mnf 7122 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 7127). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
 |- -oo  =  ~P +oo
 
Definitiondf-xr 7123 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 } )
 
Definitiondf-ltxr 7124* 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 ) ) )
 
Definitiondf-le 7125 Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.)
 |- 
 <_  =  ( ( RR*  X.  RR* )  \  `'  <  )
 
Theorempnfnre 7126 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
 |- +oo  e/  RR
 
Theoremmnfnre 7127 Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
 |- -oo  e/  RR
 
Theoremressxr 7128 The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
 |- 
 RR  C_  RR*
 
Theoremrexpssxrxp 7129 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* )
 
Theoremrexr 7130 A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  ->  A  e.  RR* )
 
Theorem0xr 7131 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
 |-  0  e.  RR*
 
Theoremrenepnf 7132 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 )
 
Theoremrenemnf 7133 No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( A  e.  RR  ->  A  =/= -oo )
 
Theoremrexrd 7134 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  e.  RR* )
 
Theoremrenepnfd 7135 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  =/= +oo )
 
Theoremrenemnfd 7136 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  =/= -oo )
 
Theoremrexri 7137 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
 |-  A  e.  RR   =>    |-  A  e.  RR*
 
Theoremrenfdisj 7138 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 } )  =  (/)
 
Theoremltrelxr 7139 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 <  C_  ( RR*  X.  RR* )
 
Theoremltrel 7140 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
 |- 
 Rel  <
 
Theoremlerelxr 7141 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 <_  C_  ( RR*  X.  RR* )
 
Theoremlerel 7142 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |- 
 Rel  <_
 
Theoremxrlenlt 7143 '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 ) )
 
Theoremltxrlt 7144 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"
 
Theoremaxltirr 7145 Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7054 with ordering on the extended reals. New proofs should use ltnr 7154 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
 |-  ( A  e.  RR  ->  -.  A  <  A )
 
Theoremaxltwlin 7146 Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7055 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 ) ) )
 
Theoremaxlttrn 7147 Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 7056 with ordering on the extended reals. New proofs should use lttr 7151 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 ) )
 
Theoremaxltadd 7148 Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7058 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 ) ) )
 
Theoremaxapti 7149 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7057 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 )
 
Theoremaxmulgt0 7150 The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7059 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
 
Theoremlttr 7151 Alias for axlttrn 7147, for naming consistency with lttri 7181. New proofs should generally use this instead of ax-pre-lttrn 7056. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <  B 
 /\  B  <  C )  ->  A  <  C ) )
 
Theoremmulgt0 7152 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 ) )
 
Theoremlenlt 7153 '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 ) )
 
Theoremltnr 7154 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  RR  ->  -.  A  <  A )
 
Theoremltso 7155 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
 |- 
 <  Or  RR
 
Theoremgtso 7156 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
 |-  `'  <  Or  RR
 
Theoremlttri3 7157 Tightness of real apartness. (Contributed by NM, 5-May-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B 
 <->  ( -.  A  <  B 
 /\  -.  B  <  A ) ) )
 
Theoremletri3 7158 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B 
 <->  ( A  <_  B  /\  B  <_  A )
 ) )
 
Theoremltleletr 7159 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 ) )
 
Theoremletr 7160 Transitive law. (Contributed by NM, 12-Nov-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <_  B 
 /\  B  <_  C )  ->  A  <_  C ) )
 
Theoremleid 7161 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  RR  ->  A  <_  A )
 
Theoremltne 7162 'Less than' implies not equal. See also ltap 7696 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 )
 
Theoremltnsym 7163 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B 
 ->  -.  B  <  A ) )
 
Theoremltle 7164 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B 
 ->  A  <_  B )
 )
 
Theoremlelttr 7165 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 ) )
 
Theoremltletr 7166 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 ) )
 
Theoremltnsym2 7167 '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 ) )
 
Theoremeqle 7168 Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
 |-  ( ( A  e.  RR  /\  A  =  B )  ->  A  <_  B )
 
Theoremltnri 7169 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  RR   =>    |-  -.  A  <  A
 
Theoremeqlei 7170 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 )
 
Theoremeqlei2 7171 Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
 |-  A  e.  RR   =>    |-  ( B  =  A  ->  B  <_  A )
 
Theoremgtneii 7172 'Less than' implies not equal. See also gtapii 7697 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.)
 |-  A  e.  RR   &    |-  A  <  B   =>    |-  B  =/=  A
 
Theoremltneii 7173 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  A  e.  RR   &    |-  A  <  B   =>    |-  A  =/=  B
 
Theoremlttri3i 7174 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) )
 
Theoremletri3i 7175 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  =  B  <->  ( A  <_  B  /\  B  <_  A ) )
 
Theoremltnsymi 7176 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  ->  -.  B  <  A )
 
Theoremlenlti 7177 '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 )
 
Theoremltlei 7178 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  ->  A  <_  B )
 
Theoremltleii 7179 '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
 
Theoremltnei 7180 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  ->  B  =/=  A )
 
Theoremlttri 7181 '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 )
 
Theoremlelttri 7182 '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 )
 
Theoremltletri 7183 '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 )
 
Theoremletri 7184 '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 )
 
Theoremle2tri3i 7185 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 ) )
 
Theoremmulgt0i 7186 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 ) )
 
Theoremmulgt0ii 7187 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 )
 
Theoremltnrd 7188 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  -.  A  <  A )
 
Theoremgtned 7189 'Less than' implies not equal. See also gtapd 7700 which is the same but for apartness. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  B  =/=  A )
 
Theoremltned 7190 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremlttri3d 7191 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 ) ) )
 
Theoremletri3d 7192 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 ) ) )
 
Theoremlenltd 7193 '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 ) )
 
Theoremltled 7194 '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 )
 
Theoremltnsymd 7195 '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 )
 
Theoremnltled 7196 '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 )
 
Theoremlensymd 7197 'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  -.  B  <  A )
 
Theoremmulgt0d 7198 The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <  A )   &    |-  ( ph  ->  0  <  B )   =>    |-  ( ph  ->  0  <  ( A  x.  B ) )
 
Theoremletrd 7199 Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  B 
 <_  C )   =>    |-  ( ph  ->  A  <_  C )
 
Theoremlelttrd 7200 Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  B  <  C )   =>    |-  ( ph  ->  A  <  C )
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