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Theorem List for Metamath Proof Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssxr 9101 The three (non-exclusive) possibilities implied by a subset of extended reals. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( A  C_  RR*  ->  ( A  C_  RR  \/  +oo 
 e.  A  \/  -oo  e.  A ) )
 
Theoremltxrlt 9102 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 ) )
 
5.2.3  Restate the ordering postulates with extended real "less than"
 
Theoremaxlttri 9103 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-lttri 9020 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A ) ) )
 
Theoremaxlttrn 9104 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. This restates ax-pre-lttrn 9021 with ordering on the extended reals. New proofs should use lttr 9108 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 9105 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-ltadd 9022 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 ) ) )
 
Theoremaxmulgt0 9106 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-mulgt0 9023 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 ) ) )
 
Theoremaxsup 9107* A non-empty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-sup 9024 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <  x )  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
 y  <  x  ->  E. z  e.  A  y  <  z ) ) )
 
5.2.4  Ordering on reals
 
Theoremlttr 9108 Alias for axlttrn 9104, for naming consistency with lttri 9155. New proofs should generally use this instead of ax-pre-lttrn 9021. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <  B 
 /\  B  <  C )  ->  A  <  C ) )
 
Theoremmulgt0 9109 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 9110 'Less than or equal to' expressed in terms of 'less than'. (Contributed by NM, 13-May-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  -.  B  <  A ) )
 
Theoremltnle 9111 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -.  B  <_  A )
 )
 
Theoremltso 9112 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
 |- 
 <  Or  RR
 
Theoremlttri2 9113 Consequence of trichotomy. (Contributed by NM, 9-Oct-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =/=  B  <-> 
 ( A  <  B  \/  B  <  A ) ) )
 
Theoremlttri3 9114 Trichotomy law for 'less than'. (Contributed by NM, 5-May-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B 
 <->  ( -.  A  <  B 
 /\  -.  B  <  A ) ) )
 
Theoremlttri4 9115 Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  \/  A  =  B  \/  B  <  A ) )
 
Theoremletri3 9116 Trichotomy law. (Contributed by NM, 14-May-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B 
 <->  ( A  <_  B  /\  B  <_  A )
 ) )
 
Theoremleloe 9117 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by NM, 13-May-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <-> 
 ( A  <  B  \/  A  =  B ) ) )
 
Theoremeqlelt 9118 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 ) ) )
 
Theoremltle 9119 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B 
 ->  A  <_  B )
 )
 
Theoremleltne 9120 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by NM, 27-Jul-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( A  <  B  <->  B  =/=  A ) )
 
Theoremlelttr 9121 Transitive law. (Contributed by NM, 23-May-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <_  B 
 /\  B  <  C )  ->  A  <  C ) )
 
Theoremltletr 9122 Transitive law. (Contributed by NM, 25-Aug-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <  B 
 /\  B  <_  C )  ->  A  <  C ) )
 
Theoremletr 9123 Transitive law. (Contributed by NM, 12-Nov-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <_  B 
 /\  B  <_  C )  ->  A  <_  C ) )
 
Theoremltnr 9124 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  RR  ->  -.  A  <  A )
 
Theoremleid 9125 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  RR  ->  A  <_  A )
 
Theoremltne 9126 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)
 |-  ( ( A  e.  RR  /\  A  <  B )  ->  B  =/=  A )
 
TheoremltneOLD 9127 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) 
 ->  B  =/=  A )
 
Theoremltnsym 9128 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B 
 ->  -.  B  <  A ) )
 
Theoremltnsym2 9129 '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 ) )
 
Theoremletric 9130 Trichotomy law. (Contributed by NM, 18-Aug-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  \/  B  <_  A ) )
 
Theoremltlen 9131 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 ( A  <_  B  /\  B  =/=  A ) ) )
 
Theoremeqle 9132 Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
 |-  ( ( A  e.  RR  /\  A  =  B )  ->  A  <_  B )
 
Theoremltadd2 9133 Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B ) ) )
 
Theoremne0gt0 9134 A nonzero nonnegative number is positive. (Contributed by NM, 20-Nov-2007.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( A  =/=  0 
 <->  0  <  A ) )
 
Theoremlecasei 9135 Ordering elimination by cases. (Contributed by NM, 6-Jul-2007.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  (
 ( ph  /\  A  <_  B )  ->  ps )   &    |-  (
 ( ph  /\  B  <_  A )  ->  ps )   =>    |-  ( ph  ->  ps )
 
Theoremlelttric 9136 Trichotomy law. (Contributed by NM, 4-Apr-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  \/  B  <  A ) )
 
Theoremltlecasei 9137 Ordering elimination by cases. (Contributed by NM, 1-Jul-2007.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( ph  /\  A  <  B )  ->  ps )   &    |-  (
 ( ph  /\  B  <_  A )  ->  ps )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ps )
 
Theoremltnri 9138 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  RR   =>    |-  -.  A  <  A
 
Theoremeqlei 9139 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 9140 Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
 |-  A  e.  RR   =>    |-  ( B  =  A  ->  B  <_  A )
 
Theoremgtneii 9141 'Less than' implies not equal. (Contributed by Mario Carneiro, 30-Sep-2013.)
 |-  A  e.  RR   &    |-  A  <  B   =>    |-  B  =/=  A
 
Theoremltneii 9142 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  A  e.  RR   &    |-  A  <  B   =>    |-  A  =/=  B
 
Theoremlttri2i 9143 Consequence of trichotomy. (Contributed by NM, 19-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  =/=  B  <->  ( A  <  B  \/  B  <  A ) )
 
Theoremlttri3i 9144 Consequence of trichotomy. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) )
 
Theoremletri3i 9145 Consequence of trichotomy. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  =  B  <->  ( A  <_  B  /\  B  <_  A ) )
 
Theoremleloei 9146 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <_  B  <->  ( A  <  B  \/  A  =  B )
 )
 
Theoremltleni 9147 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  <->  ( A  <_  B  /\  B  =/=  A ) )
 
Theoremltnsymi 9148 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  ->  -.  B  <  A )
 
Theoremlenlti 9149 '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 )
 
Theoremltnlei 9150 'Less than' in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  <->  -.  B  <_  A )
 
Theoremltlei 9151 '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 9152 '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 9153 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  ->  B  =/=  A )
 
Theoremletrii 9154 Trichotomy law for 'less than or equal to'. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <_  B  \/  B  <_  A )
 
Theoremlttri 9155 '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 9156 '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 9157 '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 9158 '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 9159 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 ) )
 
Theoremltadd2i 9160 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Paul Chapman, 27-Jan-2008.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B ) )
 
Theoremmulgt0i 9161 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 9162 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 9163 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  -.  A  <  A )
 
Theoremgtned 9164 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  B  =/=  A )
 
Theoremltned 9165 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremne0gt0d 9166 A nonzero nonnegative number is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  0  <  A )
 
Theoremlttrid 9167 Ordering on reals satisfies strict trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A ) ) )
 
Theoremlttri2d 9168 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  =/=  B  <->  ( A  <  B  \/  B  <  A ) ) )
 
Theoremlttri3d 9169 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
 
Theoremlttri4d 9170 Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  \/  A  =  B  \/  B  <  A ) )
 
Theoremletri3d 9171 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  =  B  <->  ( A  <_  B 
 /\  B  <_  A ) ) )
 
Theoremleloed 9172 '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  <->  ( A  <  B  \/  A  =  B ) ) )
 
Theoremeqleltd 9173 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 ) ) )
 
Theoremltlend 9174 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A  <_  B 
 /\  B  =/=  A ) ) )
 
Theoremlenltd 9175 '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 ) )
 
Theoremltnled 9176 'Less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  -.  B  <_  A ) )
 
Theoremltled 9177 '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 9178 '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 )
 
Theoremletrid 9179 Trichotomy law for 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  \/  B  <_  A ) )
 
Theoremleltned 9180 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( A  <  B  <->  B  =/=  A ) )
 
Theoremmulgt0d 9181 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 ) )
 
Theoremltadd2d 9182 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B ) ) )
 
Theoremletrd 9183 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 9184 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 )
 
Theoremltadd2dd 9185 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( C  +  A )  <  ( C  +  B ) )
 
Theoremltletrd 9186 Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  B 
 <_  C )   =>    |-  ( ph  ->  A  <  C )
 
Theoremlttrd 9187 Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  B  <  C )   =>    |-  ( ph  ->  A  <  C )
 
5.2.5  Initial properties of the complex numbers
 
Theoremmul12 9188 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C ) ) )
 
Theoremmul32 9189 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B ) )
 
Theoremmul31 9190 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  x.  C )  =  ( ( C  x.  B )  x.  A ) )
 
Theoremmul4 9191 Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  x.  B )  x.  ( C  x.  D ) )  =  ( ( A  x.  C )  x.  ( B  x.  D ) ) )
 
Theoremmuladd11 9192 A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  ( 1  +  B ) )  =  (
 ( 1  +  A )  +  ( B  +  ( A  x.  B ) ) ) )
 
Theorem1p1times 9193 Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( ( 1  +  1 )  x.  A )  =  ( A  +  A ) )
 
Theorempeano2cn 9194 A theorem for complex numbers analogous the second Peano postulate peano2nn 9968. (Contributed by NM, 17-Aug-2005.)
 |-  ( A  e.  CC  ->  ( A  +  1 )  e.  CC )
 
Theorempeano2re 9195 A theorem for reals analogous the second Peano postulate peano2nn 9968. (Contributed by NM, 5-Jul-2005.)
 |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
 
Theoremreaddcan 9196 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( C  +  A )  =  ( C  +  B )  <->  A  =  B ) )
 
Theorem00id 9197  0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( 0  +  0 )  =  0
 
Theoremmul02lem1 9198 Lemma for mul02 9200. If any real does not produce  0 when multiplied by  0, then any complex is equal to double itself. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( ( ( A  e.  RR  /\  (
 0  x.  A )  =/=  0 )  /\  B  e.  CC )  ->  B  =  ( B  +  B ) )
 
Theoremmul02lem2 9199 Lemma for mul02 9200. Zero times a real is zero. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  RR  ->  ( 0  x.  A )  =  0 )
 
Theoremmul02 9200 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. Based on ideas by Eric Schmidt. (Contributed by NM, 10-Aug-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  ( 0  x.  A )  =  0 )
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