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Theorem List for Metamath Proof Explorer - 8801-8900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Axiomax-i2m1 8801 i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 12 of 22 for real and complex numbers, justified by theorem axi2m1 8777. (Contributed by NM, 29-Jan-1995.)
 |-  ( ( _i  x.  _i )  +  1
 )  =  0
 
Axiomax-1ne0 8802 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by theorem ax1ne0 8778. (Contributed by NM, 29-Jan-1995.)
 |-  1  =/=  0
 
Axiomax-1rid 8803  1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by theorem ax1rid 8779. Weakened from the original axiom in the form of statement in mulid1 8831, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.)
 |-  ( A  e.  RR  ->  ( A  x.  1
 )  =  A )
 
Axiomax-rnegex 8804* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by theorem axrnegex 8780. (Contributed by Eric Schmidt, 21-May-2007.)
 |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
 
Axiomax-rrecex 8805* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 8781. (Contributed by Eric Schmidt, 11-Apr-2007.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  E. x  e.  RR  ( A  x.  x )  =  1
 )
 
Axiomax-cnre 8806* A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom 17 of 22 for real and complex numbers, justified by theorem axcnre 8782. For naming consistency, use cnre 8830 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.)
 |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
 
Axiomax-pre-lttri 8807 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri 8783. Note: The more general version for extended reals is axlttri 8890. Normally new proofs would use xrlttri 10469. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <->  -.  ( A  =  B  \/  B  <RR  A ) ) )
 
Axiomax-pre-lttrn 8808 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by theorem axpre-lttrn 8784. Note: The more general version for extended reals is axlttrn 8891. Normally new proofs would use lttr 8895. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <RR  B 
 /\  B  <RR  C ) 
 ->  A  <RR  C ) )
 
Axiomax-pre-ltadd 8809 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by theorem axpre-ltadd 8785. Normally new proofs would use axltadd 8892. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( C  +  A ) 
 <RR  ( C  +  B ) ) )
 
Axiomax-pre-mulgt0 8810 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by theorem axpre-mulgt0 8786. Normally new proofs would use axmulgt0 8893. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0 
 <RR  A  /\  0  <RR  B )  ->  0  <RR  ( A  x.  B ) ) )
 
Axiomax-pre-sup 8811* A non-empty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by theorem axpre-sup 8787. Note: Normally new proofs would use axsup 8894. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <RR  x ) 
 ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  (
 y  <RR  x  ->  E. z  e.  A  y  <RR  z ) ) )
 
Axiomax-addf 8812 Addition is an operation on the complex numbers. This deprecated axiom is provided for historical compatibility but is not a bona fide axiom for complex numbers (independent of set theory) since it cannot be interpreted as a first- or second-order statement (see http://us.metamath.org/downloads/schmidt-cnaxioms.pdf). . It may be deleted in the future and should be avoided for new theorems. Instead, the less specific addcl 8815 should be used. Note that uses of ax-addf 8812 can be eliminated by using the defined operation  ( x  e.  CC ,  y  e.  CC  |->  ( x  +  y ) ) in place of  +, from which this axiom (with the defined operation in place of  +) follows as a theorem.

This axiom is justified by theorem axaddf 8763. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

 |- 
 +  : ( CC 
 X.  CC ) --> CC
 
Axiomax-mulf 8813 Multiplication is an operation on the complex numbers. This deprecated axiom is provided for historical compatibility but is not a bona fide axiom for complex numbers (independent of set theory) since it cannot be interpreted as a first- or second-order statement (see http://us.metamath.org/downloads/schmidt-cnaxioms.pdf). . It may be deleted in the future and should be avoided for new theorems. Instead, the less specific ax-mulcl 8795 should be used. Note that uses of ax-mulf 8813 can be eliminated by using the defined operation  ( x  e.  CC ,  y  e.  CC  |->  ( x  x.  y ) ) in place of  x., from which this axiom (with the defined operation in place of  x.) follows as a theorem.

This axiom is justified by theorem axmulf 8764. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

 |- 
 x.  : ( CC 
 X.  CC ) --> CC
 
5.2  Derive the basic properties from the field axioms
 
5.2.1  Some deductions from the field axioms for complex numbers
 
Theoremcnex 8814 Alias for ax-cnex 8789. See also cnexALT 10346. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |- 
 CC  e.  _V
 
Theoremaddcl 8815 Alias for ax-addcl 8793, for naming consistency with addcli 8837. Use this theorem instead of ax-addcl 8793 or axaddcl 8769. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Theoremreaddcl 8816 Alias for ax-addrcl 8794, for naming consistency with readdcli 8846. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Theoremmulcl 8817 Alias for ax-mulcl 8795, for naming consistency with mulcli 8838. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Theoremremulcl 8818 Alias for ax-mulrcl 8796, for naming consistency with remulcli 8847. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Theoremmulcom 8819 Alias for ax-mulcom 8797, for naming consistency with mulcomi 8839. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Theoremaddass 8820 Alias for ax-addass 8798, for naming consistency with addassi 8841. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) ) )
 
Theoremmulass 8821 Alias for ax-mulass 8799, for naming consistency with mulassi 8842. (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 ) ) )
 
Theoremadddi 8822 Alias for ax-distr 8800, for naming consistency with adddii 8843. (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 ) ) )
 
Theoremrecn 8823 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
 |-  ( A  e.  RR  ->  A  e.  CC )
 
Theoremreex 8824 The real numbers form a set. See also reexALT 10344. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |- 
 RR  e.  _V
 
Theoremelimne0 8825 Hypothesis for weak deduction theorem to eliminate  A  =/=  0. (Contributed by NM, 15-May-1999.)
 |- 
 if ( A  =/=  0 ,  A , 
 1 )  =/=  0
 
Theoremadddir 8826 Distributive law for complex numbers. (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 ) ) )
 
Theorem0cn 8827 0 is a complex number. (Contributed by NM, 19-Feb-2005.)
 |-  0  e.  CC
 
Theoremc0ex 8828 0 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  0  e.  _V
 
Theorem1ex 8829 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  1  e.  _V
 
Theoremcnre 8830* Alias for ax-cnre 8806, for naming consistency. (Contributed by Mario Carneiro, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
 
Theoremmulid1 8831  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 )
 
Theoremmulid2 8832 Identity law for multiplication. Note: see mulid1 8831 for commuted version. (Contributed by NM, 8-Oct-1999.)
 |-  ( A  e.  CC  ->  ( 1  x.  A )  =  A )
 
Theorem1re 8833  1 is a real number. This used to be one of our postulates for complex numbers, but Eric Schmidt discovered that it could be derived from a weaker postulate, ax-1cn 8791, by exploiting properties of the imaginary unit  _i. (Contributed by Eric Schmidt, 11-Apr-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  1  e.  RR
 
Theorem0re 8834  0 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  0  e.  RR
 
Theoremmulid1i 8835 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
 |-  A  e.  CC   =>    |-  ( A  x.  1 )  =  A
 
Theoremmulid2i 8836 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
 |-  A  e.  CC   =>    |-  ( 1  x.  A )  =  A
 
Theoremaddcli 8837 Closure law for addition. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  +  B )  e.  CC
 
Theoremmulcli 8838 Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  x.  B )  e.  CC
 
Theoremmulcomi 8839 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  x.  B )  =  ( B  x.  A )
 
Theoremmulcomli 8840 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  ( A  x.  B )  =  C   =>    |-  ( B  x.  A )  =  C
 
Theoremaddassi 8841 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 )
 )
 
Theoremmulassi 8842 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 ) )
 
Theoremadddii 8843 Distributive law. (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 ) )
 
Theoremadddiri 8844 Distributive law. (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 ) )
 
Theoremrecni 8845 A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
 |-  A  e.  RR   =>    |-  A  e.  CC
 
Theoremreaddcli 8846 Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  +  B )  e.  RR
 
Theoremremulcli 8847 Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  x.  B )  e.  RR
 
Theoremmulid1d 8848 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  1 )  =  A )
 
Theoremmulid2d 8849 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 1  x.  A )  =  A )
 
Theoremaddcld 8850 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 8851 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 8852 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 8853 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 8854 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 8855 Distributive law. (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 8856 Distributive law. (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 ) ) )
 
Theoremrecnd 8857 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremreaddcld 8858 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 8859 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 )
 
5.2.2  Infinity and the extended real number system
 
Syntaxcpnf 8860 Plus infinity.
 class  +oo
 
Syntaxcmnf 8861 Minus infinity.
 class  -oo
 
Syntaxcxr 8862 The set of extended reals (includes plus and minus infinity).
 class  RR*
 
Syntaxclt 8863 'Less than' predicate (extended to include the extended reals).
 class  <
 
Syntaxcle 8864 Extend wff notation to include the 'less than or equal to' relation.
 class  <_
 
Definitiondf-pnf 8865 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 8866). 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 8870, mnfnre 8871, and pnfnemnf 10455.

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 8866 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 8871 and pnfnemnf 10455). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
 |- 
 -oo  =  ~P  +oo
 
Definitiondf-xr 8867 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 8868* 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 8869 Define 'less than or equal to' on the extended real subset of complex numbers. Theorem leloe 8904 relates it to 'less than' for reals. (Contributed by NM, 13-Oct-2005.)
 |- 
 <_  =  ( ( RR*  X.  RR* )  \  `'  <  )
 
Theorempnfnre 8870 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
 |- 
 +oo  e/  RR
 
Theoremmnfnre 8871 Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
 |- 
 -oo  e/  RR
 
Theoremressxr 8872 The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
 |- 
 RR  C_  RR*
 
Theoremrexr 8873 A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR  ->  A  e.  RR* )
 
Theorem0xr 8874 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
 |-  0  e.  RR*
 
Theoremrenepnf 8875 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 8876 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 8877 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  e.  RR* )
 
Theoremrenepnfd 8878 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  =/=  +oo )
 
Theoremrenemnfd 8879 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  =/=  -oo )
 
Theoremrexri 8880 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
 |-  A  e.  RR   =>    |-  A  e.  RR*
 
Theoremrenfdisj 8881 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 8882 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 <  C_  ( RR*  X.  RR* )
 
Theoremltrel 8883 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
 |- 
 Rel  <
 
Theoremlerelxr 8884 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 <_  C_  ( RR*  X.  RR* )
 
Theoremlerel 8885 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |- 
 Rel  <_
 
Theoremxrlenlt 8886 '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 ) )
 
Theoremxrltnle 8887 'Less than' expressed in terms of 'less than or equal to', for extended reals. (Contributed by NM, 6-Feb-2007.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -.  B  <_  A )
 )
 
Theoremssxr 8888 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 8889 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 8890 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 8807 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 8891 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. This restates ax-pre-lttrn 8808 with ordering on the extended reals. New proofs should use lttr 8895 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 8892 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 8809 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 8893 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 8810 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 8894* 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 8811 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 8895 Alias for axlttrn 8891, for naming consistency with lttri 8941. New proofs should generally use this instead of ax-pre-lttrn 8808. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <  B 
 /\  B  <  C )  ->  A  <  C ) )
 
Theoremmulgt0 8896 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 8897 '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 8898 '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 8899 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
 |- 
 <  Or  RR
 
Theoremlttri2 8900 Consequence of trichotomy. (Contributed by NM, 9-Oct-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =/=  B  <-> 
 ( A  <  B  \/  B  <  A ) ) )
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