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Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrecnd 7801 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremreaddcld 7802 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 7803 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
 
Syntaxcpnf 7804 Plus infinity.
 class +oo
 
Syntaxcmnf 7805 Minus infinity.
 class -oo
 
Syntaxcxr 7806 The set of extended reals (includes plus and minus infinity).
 class  RR*
 
Syntaxclt 7807 'Less than' predicate (extended to include the extended reals).
 class  <
 
Syntaxcle 7808 Extend wff notation to include the 'less than or equal to' relation.
 class  <_
 
Definitiondf-pnf 7809 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 7810). 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 7814 and mnfnre 7815, 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 7810 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 7815). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
 |- -oo  =  ~P +oo
 
Definitiondf-xr 7811 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 7812* 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 7813 Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.)
 |- 
 <_  =  ( ( RR*  X.  RR* )  \  `'  <  )
 
Theorempnfnre 7814 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
 |- +oo  e/  RR
 
Theoremmnfnre 7815 Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
 |- -oo  e/  RR
 
Theoremressxr 7816 The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
 |- 
 RR  C_  RR*
 
Theoremrexpssxrxp 7817 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 7818 A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  ->  A  e.  RR* )
 
Theorem0xr 7819 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
 |-  0  e.  RR*
 
Theoremrenepnf 7820 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 7821 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 7822 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  e.  RR* )
 
Theoremrenepnfd 7823 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  =/= +oo )
 
Theoremrenemnfd 7824 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  =/= -oo )
 
Theorempnfxr 7825 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*
 
Theorempnfex 7826 Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |- +oo  e.  _V
 
Theorempnfnemnf 7827 Plus and minus infinity are different elements of  RR*. (Contributed by NM, 14-Oct-2005.)
 |- +oo  =/= -oo
 
Theoremmnfnepnf 7828 Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |- -oo  =/= +oo
 
Theoremmnfxr 7829 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*
 
Theoremrexri 7830 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
 |-  A  e.  RR   =>    |-  A  e.  RR*
 
Theoremrenfdisj 7831 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 7832 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 <  C_  ( RR*  X.  RR* )
 
Theoremltrel 7833 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
 |- 
 Rel  <
 
Theoremlerelxr 7834 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 <_  C_  ( RR*  X.  RR* )
 
Theoremlerel 7835 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |- 
 Rel  <_
 
Theoremxrlenlt 7836 '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 7837 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"
 
Theoremaxltirr 7838 Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7739 with ordering on the extended reals. New proofs should use ltnr 7848 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
 |-  ( A  e.  RR  ->  -.  A  <  A )
 
Theoremaxltwlin 7839 Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7740 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 7840 Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 7741 with ordering on the extended reals. New proofs should use lttr 7845 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 7841 Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7743 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 7842 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7742 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 7843 The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7744 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 ) ) )
 
Theoremaxsuploc 7844* 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 7748 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
 
Theoremlttr 7845 Alias for axlttrn 7840, for naming consistency with lttri 7875. New proofs should generally use this instead of ax-pre-lttrn 7741. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <  B 
 /\  B  <  C )  ->  A  <  C ) )
 
Theoremmulgt0 7846 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 7847 '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 7848 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  RR  ->  -.  A  <  A )
 
Theoremltso 7849 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
 |- 
 <  Or  RR
 
Theoremgtso 7850 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
 |-  `'  <  Or  RR
 
Theoremlttri3 7851 Tightness of real apartness. (Contributed by NM, 5-May-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B 
 <->  ( -.  A  <  B 
 /\  -.  B  <  A ) ) )
 
Theoremletri3 7852 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B 
 <->  ( A  <_  B  /\  B  <_  A )
 ) )
 
Theoremltleletr 7853 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 7854 Transitive law. (Contributed by NM, 12-Nov-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <_  B 
 /\  B  <_  C )  ->  A  <_  C ) )
 
Theoremleid 7855 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  RR  ->  A  <_  A )
 
Theoremltne 7856 'Less than' implies not equal. See also ltap 8402 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 7857 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B 
 ->  -.  B  <  A ) )
 
Theoremltle 7858 '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 7859 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 7860 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 7861 '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 7862 Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
 |-  ( ( A  e.  RR  /\  A  =  B )  ->  A  <_  B )
 
Theoremltnri 7863 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  RR   =>    |-  -.  A  <  A
 
Theoremeqlei 7864 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 7865 Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
 |-  A  e.  RR   =>    |-  ( B  =  A  ->  B  <_  A )
 
Theoremgtneii 7866 'Less than' implies not equal. See also gtapii 8403 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.)
 |-  A  e.  RR   &    |-  A  <  B   =>    |-  B  =/=  A
 
Theoremltneii 7867 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  A  e.  RR   &    |-  A  <  B   =>    |-  A  =/=  B
 
Theoremlttri3i 7868 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) )
 
Theoremletri3i 7869 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  =  B  <->  ( A  <_  B  /\  B  <_  A ) )
 
Theoremltnsymi 7870 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  ->  -.  B  <  A )
 
Theoremlenlti 7871 '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 7872 '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 7873 '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 7874 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  ->  B  =/=  A )
 
Theoremlttri 7875 '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 7876 '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 7877 '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 7878 '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 7879 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 7880 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 7881 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 7882 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  -.  A  <  A )
 
Theoremgtned 7883 'Less than' implies not equal. See also gtapd 8406 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 7884 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremlttri3d 7885 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 7886 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 7887 '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 7888 '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 7889 '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 7890 '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 7891 '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 7892 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 7893 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 7894 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 )
 
Theoremlttrd 7895 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 )
 
Theorem0lt1 7896 0 is less than 1. Theorem I.21 of [Apostol] p. 20. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 17-Jan-1997.)
 |-  0  <  1
 
Theoremltntri 7897 Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy,  A  <  B  \/  A  =  B  \/  B  <  A. Does that mean there is a pair of real numbers where none of those hold (that is, where we can refute each of those three relationships)? Actually, no, as shown here. This is another example of distinguishing between being unable to prove something, or being able to refute it. (Contributed by Jim Kingdon, 13-Aug-2023.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( -.  A  <  B  /\  -.  A  =  B  /\  -.  B  <  A ) )
 
4.2.5  Initial properties of the complex numbers
 
Theoremmul12 7898 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 7899 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 7900 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 ) )
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