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Theorem List for Metamath Proof Explorer - 9701-9800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremletri3 9701 Trichotomy law. (Contributed by NM, 14-May-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B 
 <->  ( A  <_  B  /\  B  <_  A )
 ) )
 
Theoremleloe 9702 '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 9703 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 9704 '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 9705 '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 9706 Transitive law. (Contributed by NM, 23-May-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <_  B 
 /\  B  <  C )  ->  A  <  C ) )
 
Theoremltletr 9707 Transitive law. (Contributed by NM, 25-Aug-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <  B 
 /\  B  <_  C )  ->  A  <  C ) )
 
Theoremltleletr 9708 Transitive law, weaker form of ltletr 9707. (Contributed by AV, 14-Oct-2018.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <  B 
 /\  B  <_  C )  ->  A  <_  C ) )
 
Theoremletr 9709 Transitive law. (Contributed by NM, 12-Nov-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <_  B 
 /\  B  <_  C )  ->  A  <_  C ) )
 
Theoremltnr 9710 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  RR  ->  -.  A  <  A )
 
Theoremleid 9711 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  RR  ->  A  <_  A )
 
Theoremltne 9712 '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 9713 '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 9714 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B 
 ->  -.  B  <  A ) )
 
Theoremltnsym2 9715 '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 9716 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 9717 '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 9718 Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
 |-  ( ( A  e.  RR  /\  A  =  B )  ->  A  <_  B )
 
Theoremeqled 9719 Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  A 
 <_  B )
 
Theoremltadd2 9720 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 9721 A nonzero nonnegative number is positive. (Contributed by NM, 20-Nov-2007.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( A  =/=  0 
 <->  0  <  A ) )
 
Theoremlecasei 9722 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 9723 Trichotomy law. (Contributed by NM, 4-Apr-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  \/  B  <  A ) )
 
Theoremltlecasei 9724 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 9725 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  RR   =>    |-  -.  A  <  A
 
Theoremeqlei 9726 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 9727 Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
 |-  A  e.  RR   =>    |-  ( B  =  A  ->  B  <_  A )
 
Theoremgtneii 9728 'Less than' implies not equal. (Contributed by Mario Carneiro, 30-Sep-2013.)
 |-  A  e.  RR   &    |-  A  <  B   =>    |-  B  =/=  A
 
Theoremltneii 9729 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  A  e.  RR   &    |-  A  <  B   =>    |-  A  =/=  B
 
Theoremlttri2i 9730 Consequence of trichotomy. (Contributed by NM, 19-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  =/=  B  <->  ( A  <  B  \/  B  <  A ) )
 
Theoremlttri3i 9731 Consequence of trichotomy. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) )
 
Theoremletri3i 9732 Consequence of trichotomy. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  =  B  <->  ( A  <_  B  /\  B  <_  A ) )
 
Theoremleloei 9733 '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 9734 '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 9735 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  ->  -.  B  <  A )
 
Theoremlenlti 9736 '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 9737 '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 9738 '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 9739 '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 9740 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  ->  B  =/=  A )
 
Theoremletrii 9741 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 9742 '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 9743 '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 9744 '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 9745 '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 9746 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 9747 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.) (Proof shortened by OpenAI, 25-Mar-2020.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B ) )
 
Theoremltadd2iOLD 9748 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Paul Chapman, 27-Jan-2008.) Obsolete version of ltadd2i 9747 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B ) )
 
Theoremmulgt0i 9749 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 9750 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 9751 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  -.  A  <  A )
 
Theoremgtned 9752 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  B  =/=  A )
 
Theoremltned 9753 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremne0gt0d 9754 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 9755 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 9756 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 9757 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 9758 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 9759 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 9760 '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 9761 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 9762 '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 9763 '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 9764 '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 9765 '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 9766 '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 9767 '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 9768 '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 )
 
Theoremletrid 9769 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 9770 '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 9771 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 9772 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 9773 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 9774 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 9775 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 9776 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 9777 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 )
 
Theoremdedekind 9778* The Dedekind cut theorem. This theorem, which may be used to replace ax-pre-sup 9600 with appropriate adjustments, states that, if  A completely preceeds  B, then there is some number separating the two of them. (Contributed by Scott Fenton, 13-Jun-2013.)
 |-  ( ( A  C_  RR  /\  B  C_  RR  /\ 
 A. x  e.  A  A. y  e.  B  x  <  y )  ->  E. z  e.  RR  A. x  e.  A  A. y  e.  B  ( x  <_  z  /\  z  <_  y
 ) )
 
Theoremdedekindle 9779* The Dedekind cut theorem, with the hypothesis weakened to only require non-strict less than. (Contributed by Scott Fenton, 2-Jul-2013.)
 |-  ( ( A  C_  RR  /\  B  C_  RR  /\ 
 A. x  e.  A  A. y  e.  B  x  <_  y )  ->  E. z  e.  RR  A. x  e.  A  A. y  e.  B  ( x  <_  z  /\  z  <_  y
 ) )
 
5.2.5  Initial properties of the complex numbers
 
Theoremmul12 9780 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 9781 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 9782 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 9783 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 9784 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 9785 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 9786 A theorem for complex numbers analogous the second Peano postulate peano2nn 10588. (Contributed by NM, 17-Aug-2005.)
 |-  ( A  e.  CC  ->  ( A  +  1 )  e.  CC )
 
Theorempeano2re 9787 A theorem for reals analogous the second Peano postulate peano2nn 10588. (Contributed by NM, 5-Jul-2005.)
 |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
 
Theoremreaddcan 9788 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 9789  0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( 0  +  0 )  =  0
 
Theoremmul02lem1 9790 Lemma for mul02 9792. 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 9791 Lemma for mul02 9792. Zero times a real is zero. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  RR  ->  ( 0  x.  A )  =  0 )
 
Theoremmul02 9792 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 )
 
Theoremmul01 9793 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  ( A  x.  0
 )  =  0 )
 
Theoremaddid1 9794  0 is an additive identity. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
 
Theoremcnegex 9795* Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  E. x  e.  CC  ( A  +  x )  =  0 )
 
Theoremcnegex2 9796* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  E. x  e.  CC  ( x  +  A )  =  0 )
 
Theoremaddid2 9797  0 is a left identity for addition. This used to be one of our complex number axioms, until it was discovered that it was dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  ( 0  +  A )  =  A )
 
Theoremaddcan 9798 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  =  ( A  +  C )  <->  B  =  C ) )
 
Theoremaddcan2 9799 Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  C )  =  ( B  +  C )  <->  A  =  B ) )
 
Theoremaddcom 9800 Addition commutes. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  =  ( B  +  A )
 )
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