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Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Axiomax-pre-mulext 6801 Strong extensionality of multiplication (expressed in terms of  <RR). Axiom for real and complex numbers, justified by theorem axpre-mulext 6772

(Contributed by Jim Kingdon, 18-Feb-2020.)

 RR  RR  C  RR  x.  C  <RR  x.  C  <RR  <RR
 
Axiomax-arch 6802* Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for real and complex numbers, justified by theorem axarch 6773.

This axiom should not be used directly; instead use arch 7954 (which is the same, but stated in terms of 
NN and  <). (Contributed by Jim Kingdon, 2-May-2020.) (New usage is discouraged.)

 RR  n  |^| {  |  1  +  1  }  <RR  n
 
3.2  Derive the basic properties from the field axioms
 
3.2.1  Some deductions from the field axioms for complex numbers
 
Theoremcnex 6803 Alias for ax-cnex 6774. (Contributed by Mario Carneiro, 17-Nov-2014.)

 CC  _V
 
Theoremaddcl 6804 Alias for ax-addcl 6779, for naming consistency with addcli 6829. Use this theorem instead of ax-addcl 6779 or axaddcl 6750. (Contributed by NM, 10-Mar-2008.)
 CC  CC  +  CC
 
Theoremreaddcl 6805 Alias for ax-addrcl 6780, for naming consistency with readdcli 6838. (Contributed by NM, 10-Mar-2008.)
 RR  RR  +  RR
 
Theoremmulcl 6806 Alias for ax-mulcl 6781, for naming consistency with mulcli 6830. (Contributed by NM, 10-Mar-2008.)
 CC  CC  x.  CC
 
Theoremremulcl 6807 Alias for ax-mulrcl 6782, for naming consistency with remulcli 6839. (Contributed by NM, 10-Mar-2008.)
 RR  RR  x.  RR
 
Theoremmulcom 6808 Alias for ax-mulcom 6784, for naming consistency with mulcomi 6831. (Contributed by NM, 10-Mar-2008.)
 CC  CC  x.  x.
 
Theoremaddass 6809 Alias for ax-addass 6785, for naming consistency with addassi 6833. (Contributed by NM, 10-Mar-2008.)
 CC  CC  C  CC  +  +  C  +  +  C
 
Theoremmulass 6810 Alias for ax-mulass 6786, for naming consistency with mulassi 6834. (Contributed by NM, 10-Mar-2008.)
 CC  CC  C  CC  x.  x.  C  x.  x.  C
 
Theoremadddi 6811 Alias for ax-distr 6787, for naming consistency with adddii 6835. (Contributed by NM, 10-Mar-2008.)
 CC  CC  C  CC  x.  +  C  x.  +  x.  C
 
Theoremrecn 6812 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
 RR  CC
 
Theoremreex 6813 The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.)

 RR  _V
 
Theoremreelprrecn 6814 Reals are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

 RR  { RR ,  CC }
 
Theoremcnelprrecn 6815 Complex numbers are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

 CC  { RR ,  CC }
 
Theoremadddir 6816 Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.)
 CC  CC  C  CC  +  x.  C  x.  C  +  x.  C
 
Theorem0cn 6817 0 is a complex number. (Contributed by NM, 19-Feb-2005.)
 0  CC
 
Theorem0cnd 6818 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
 0  CC
 
Theoremc0ex 6819 0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
 0  _V
 
Theorem1ex 6820 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
 1  _V
 
Theoremcnre 6821* Alias for ax-cnre 6794, for naming consistency. (Contributed by NM, 3-Jan-2013.)
 CC  RR  RR  +  _i  x.
 
Theoremmulid1 6822  1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
 CC  x.  1
 
Theoremmulid2 6823 Identity law for multiplication. Note: see mulid1 6822 for commuted version. (Contributed by NM, 8-Oct-1999.)
 CC  1  x.
 
Theorem1re 6824  1 is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.)
 1  RR
 
Theorem0re 6825  0 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
 0  RR
 
Theorem0red 6826  0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
 0  RR
 
Theoremmulid1i 6827 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
 CC   =>     x.  1
 
Theoremmulid2i 6828 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
 CC   =>     1  x.
 
Theoremaddcli 6829 Closure law for addition. (Contributed by NM, 23-Nov-1994.)
 CC   &     CC   =>     +  CC
 
Theoremmulcli 6830 Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
 CC   &     CC   =>     x.  CC
 
Theoremmulcomi 6831 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
 CC   &     CC   =>     x.  x.
 
Theoremmulcomli 6832 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
 CC   &     CC   &     x.  C   =>     x.  C
 
Theoremaddassi 6833 Associative law for addition. (Contributed by NM, 23-Nov-1994.)
 CC   &     CC   &     C  CC   =>     +  +  C  +  +  C
 
Theoremmulassi 6834 Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
 CC   &     CC   &     C  CC   =>     x.  x.  C  x.  x.  C
 
Theoremadddii 6835 Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)
 CC   &     CC   &     C  CC   =>     x.  +  C  x.  +  x.  C
 
Theoremadddiri 6836 Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)
 CC   &     CC   &     C  CC   =>     +  x.  C  x.  C  +  x.  C
 
Theoremrecni 6837 A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
 RR   =>     CC
 
Theoremreaddcli 6838 Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
 RR   &     RR   =>     +  RR
 
Theoremremulcli 6839 Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
 RR   &     RR   =>     x.  RR
 
Theorem1red 6840 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
 1  RR
 
Theorem1cnd 6841 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
 1  CC
 
Theoremmulid1d 6842 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     x.  1
 
Theoremmulid2d 6843 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>    
 1  x.
 
Theoremaddcld 6844 Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     + 
 CC
 
Theoremmulcld 6845 Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     x. 
 CC
 
Theoremmulcomd 6846 Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     x.  x.
 
Theoremaddassd 6847 Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  +  C  +  +  C
 
Theoremmulassd 6848 Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     x.  x.  C  x.  x.  C
 
Theoremadddid 6849 Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     x.  +  C  x.  +  x.  C
 
Theoremadddird 6850 Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  x.  C  x.  C  +  x.  C
 
Theoremrecnd 6851 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
 RR   =>     CC
 
Theoremreaddcld 6852 Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     + 
 RR
 
Theoremremulcld 6853 Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     x. 
 RR
 
3.2.2  Infinity and the extended real number system
 
Syntaxcpnf 6854 Plus infinity.
+oo
 
Syntaxcmnf 6855 Minus infinity.
-oo
 
Syntaxcxr 6856 The set of extended reals (includes plus and minus infinity).
 RR*
 
Syntaxclt 6857 'Less than' predicate (extended to include the extended reals).

 <
 
Syntaxcle 6858 Extend wff notation to include the 'less than or equal to' relation.

 <_
 
Definitiondf-pnf 6859 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 6860). 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 6864 and mnfnre 6865, 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 6860 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 6865). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
-oo  ~P +oo
 
Definitiondf-xr 6861 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 6862* 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.)

 <  { <. ,  >.  |  RR  RR  <RR  }  u.  RR 
 u.  { -oo }  X.  { +oo }  u.  { -oo }  X.  RR
 
Definitiondf-le 6863 Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.)

 <_  RR*  X.  RR*  \  `'  <
 
Theorempnfnre 6864 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
+oo  e/  RR
 
Theoremmnfnre 6865 Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
-oo  e/  RR
 
Theoremressxr 6866 The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)

 RR  C_  RR*
 
Theoremrexpssxrxp 6867 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 6868 A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
 RR  RR*
 
Theorem0xr 6869 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
 0  RR*
 
Theoremrenepnf 6870 No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR  =/= +oo
 
Theoremrenemnf 6871 No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR  =/= -oo
 
Theoremrexrd 6872 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     RR*
 
Theoremrenepnfd 6873 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     =/= +oo
 
Theoremrenemnfd 6874 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     =/= -oo
 
Theoremrexri 6875 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
 RR   =>     RR*
 
Theoremrenfdisj 6876 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 6877 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

 <  C_  RR*  X.  RR*
 
Theoremltrel 6878 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)

 Rel  <
 
Theoremlerelxr 6879 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

 <_  C_  RR*  X.  RR*
 
Theoremlerel 6880 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)

 Rel  <_
 
Theoremxrlenlt 6881 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.)
 RR*  RR*  <_  <
 
Theoremltxrlt 6882 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.)
 RR  RR  <  <RR
 
3.2.3  Restate the ordering postulates with extended real "less than"
 
Theoremaxltirr 6883 Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 6795 with ordering on the extended reals. New proofs should use ltnr 6892 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
 RR  <
 
Theoremaxltwlin 6884 Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 6796 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.)
 RR  RR  C  RR  <  <  C  C  <
 
Theoremaxlttrn 6885 Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 6797 with ordering on the extended reals. New proofs should use lttr 6889 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
 RR  RR  C  RR  <  <  C  <  C
 
Theoremaxltadd 6886 Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 6799 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
 RR  RR  C  RR  <  C  +  <  C  +
 
Theoremaxapti 6887 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 6798 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)
 RR  RR  <  <
 
Theoremaxmulgt0 6888 The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 6800 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
 RR  RR  0  <  0  <  0  <  x.
 
3.2.4  Ordering on reals
 
Theoremlttr 6889 Alias for axlttrn 6885, for naming consistency with lttri 6919. New proofs should generally use this instead of ax-pre-lttrn 6797. (Contributed by NM, 10-Mar-2008.)
 RR  RR  C  RR  <  <  C  <  C
 
Theoremmulgt0 6890 The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.)
 RR  0  <  RR  0  <  0  <  x.
 
Theoremlenlt 6891 '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.)
 RR  RR  <_  <
 
Theoremltnr 6892 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
 RR  <
 
Theoremltso 6893 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)

 <  Or  RR
 
Theoremgtso 6894 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
 `'  <  Or  RR
 
Theoremlttri3 6895 Tightness of real apartness. (Contributed by NM, 5-May-1999.)
 RR  RR  <  <
 
Theoremletri3 6896 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
 RR  RR  <_  <_
 
Theoremltleletr 6897 Transitive law, weaker form of  <  <_  C  <  C. (Contributed by AV, 14-Oct-2018.)
 RR  RR  C  RR  <  <_  C  <_  C
 
Theoremletr 6898 Transitive law. (Contributed by NM, 12-Nov-1999.)
 RR  RR  C  RR  <_  <_  C  <_  C
 
Theoremleid 6899 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
 RR  <_
 
Theoremltne 6900 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)
 RR  <  =/=
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