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Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1cnd 6801 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
 1  CC
 
Theoremmulid1d 6802 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     x.  1
 
Theoremmulid2d 6803 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>    
 1  x.
 
Theoremaddcld 6804 Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     + 
 CC
 
Theoremmulcld 6805 Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     x. 
 CC
 
Theoremmulcomd 6806 Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     x.  x.
 
Theoremaddassd 6807 Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  +  C  +  +  C
 
Theoremmulassd 6808 Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     x.  x.  C  x.  x.  C
 
Theoremadddid 6809 Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     x.  +  C  x.  +  x.  C
 
Theoremadddird 6810 Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  x.  C  x.  C  +  x.  C
 
Theoremrecnd 6811 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
 RR   =>     CC
 
Theoremreaddcld 6812 Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     + 
 RR
 
Theoremremulcld 6813 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 6814 Plus infinity.
+oo
 
Syntaxcmnf 6815 Minus infinity.
-oo
 
Syntaxcxr 6816 The set of extended reals (includes plus and minus infinity).
 RR*
 
Syntaxclt 6817 'Less than' predicate (extended to include the extended reals).

 <
 
Syntaxcle 6818 Extend wff notation to include the 'less than or equal to' relation.

 <_
 
Definitiondf-pnf 6819 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 6820). 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 6824 and mnfnre 6825, 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 6820 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 6825). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
-oo  ~P +oo
 
Definitiondf-xr 6821 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 6822* 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 6823 Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.)

 <_  RR*  X.  RR*  \  `'  <
 
Theorempnfnre 6824 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
+oo  e/  RR
 
Theoremmnfnre 6825 Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
-oo  e/  RR
 
Theoremressxr 6826 The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)

 RR  C_  RR*
 
Theoremrexpssxrxp 6827 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 6828 A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
 RR  RR*
 
Theorem0xr 6829 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
 0  RR*
 
Theoremrenepnf 6830 No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR  =/= +oo
 
Theoremrenemnf 6831 No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR  =/= -oo
 
Theoremrexrd 6832 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     RR*
 
Theoremrenepnfd 6833 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     =/= +oo
 
Theoremrenemnfd 6834 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   =>     =/= -oo
 
Theoremrexri 6835 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
 RR   =>     RR*
 
Theoremrenfdisj 6836 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 6837 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

 <  C_  RR*  X.  RR*
 
Theoremltrel 6838 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)

 Rel  <
 
Theoremlerelxr 6839 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

 <_  C_  RR*  X.  RR*
 
Theoremlerel 6840 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)

 Rel  <_
 
Theoremxrlenlt 6841 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.)
 RR*  RR*  <_  <
 
Theoremltxrlt 6842 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 6843 Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 6755 with ordering on the extended reals. New proofs should use ltnr 6852 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
 RR  <
 
Theoremaxltwlin 6844 Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 6756 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.)
 RR  RR  C  RR  <  <  C  C  <
 
Theoremaxlttrn 6845 Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 6757 with ordering on the extended reals. New proofs should use lttr 6849 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
 RR  RR  C  RR  <  <  C  <  C
 
Theoremaxltadd 6846 Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 6759 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
 RR  RR  C  RR  <  C  +  <  C  +
 
Theoremaxapti 6847 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 6758 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)
 RR  RR  <  <
 
Theoremaxmulgt0 6848 The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 6760 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 6849 Alias for axlttrn 6845, for naming consistency with lttri 6879. New proofs should generally use this instead of ax-pre-lttrn 6757. (Contributed by NM, 10-Mar-2008.)
 RR  RR  C  RR  <  <  C  <  C
 
Theoremmulgt0 6850 The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.)
 RR  0  <  RR  0  <  0  <  x.
 
Theoremlenlt 6851 '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 6852 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
 RR  <
 
Theoremltso 6853 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)

 <  Or  RR
 
Theoremgtso 6854 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
 `'  <  Or  RR
 
Theoremlttri3 6855 Tightness of real apartness. (Contributed by NM, 5-May-1999.)
 RR  RR  <  <
 
Theoremletri3 6856 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
 RR  RR  <_  <_
 
Theoremltleletr 6857 Transitive law, weaker form of  <  <_  C  <  C. (Contributed by AV, 14-Oct-2018.)
 RR  RR  C  RR  <  <_  C  <_  C
 
Theoremletr 6858 Transitive law. (Contributed by NM, 12-Nov-1999.)
 RR  RR  C  RR  <_  <_  C  <_  C
 
Theoremleid 6859 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
 RR  <_
 
Theoremltne 6860 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)
 RR  <  =/=
 
Theoremltnsym 6861 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
 RR  RR  <  <
 
Theoremltle 6862 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)
 RR  RR  <  <_
 
Theoremlelttr 6863 Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 23-May-1999.)
 RR  RR  C  RR  <_  <  C  <  C
 
Theoremltletr 6864 Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.)
 RR  RR  C  RR  <  <_  C  <  C
 
Theoremltnsym2 6865 'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR  RR 
 <  <
 
Theoremeqle 6866 Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
 RR  <_
 
Theoremltnri 6867 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
 RR   =>     <
 
Theoremeqlei 6868 Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.) (Revised by Alexander van der Vekens, 20-Mar-2018.)
 RR   =>     <_
 
Theoremeqlei2 6869 Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
 RR   =>     <_
 
Theoremgtneii 6870 'Less than' implies not equal. (Contributed by Mario Carneiro, 30-Sep-2013.)
 RR   &     <    =>     =/=
 
Theoremltneii 6871 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
 RR   &     <    =>     =/=
 
Theoremlttri3i 6872 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
 RR   &     RR   =>     <  <
 
Theoremletri3i 6873 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
 RR   &     RR   =>     <_  <_
 
Theoremltnsymi 6874 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
 RR   &     RR   =>     <  <
 
Theoremlenlti 6875 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)
 RR   &     RR   =>     <_  <
 
Theoremltlei 6876 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)
 RR   &     RR   =>     <  <_
 
Theoremltleii 6877 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.)
 RR   &     RR   &     <    =>     <_
 
Theoremltnei 6878 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
 RR   &     RR   =>     <  =/=
 
Theoremlttri 6879 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
 RR   &     RR   &     C  RR   =>     <  <  C  <  C
 
Theoremlelttri 6880 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.)
 RR   &     RR   &     C  RR   =>     <_  <  C  <  C
 
Theoremltletri 6881 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.)
 RR   &     RR   &     C  RR   =>     <  <_  C  <  C
 
Theoremletri 6882 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)
 RR   &     RR   &     C  RR   =>     <_  <_  C  <_  C
 
Theoremle2tri3i 6883 Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)
 RR   &     RR   &     C  RR   =>     <_  <_  C  C  <_  C  C
 
Theoremmulgt0i 6884 The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
 RR   &     RR   =>     0  <  0  <  0  <  x.
 
Theoremmulgt0ii 6885 The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
 RR   &     RR   &     0  <    &     0  <    =>     0  <  x.
 
Theoremltnrd 6886 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   =>     <
 
Theoremgtned 6887 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     <    =>     =/=
 
Theoremltned 6888 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     <    =>     =/=
 
Theoremlttri3d 6889 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     <  <
 
Theoremletri3d 6890 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>    
 <_  <_
 
Theoremlenltd 6891 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     <_  <
 
Theoremltled 6892 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     <    =>     <_
 
Theoremltnsymd 6893 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     <    =>     <
 
Theoremmulgt0d 6894 The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     0  <    &     0  <    =>     0  <  x.
 
Theoremletrd 6895 Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)
 RR   &     RR   &     C  RR   &     <_    &     <_  C   =>     <_  C
 
Theoremlelttrd 6896 Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)
 RR   &     RR   &     C  RR   &     <_    &     <  C   =>     <  C
 
Theoremlttrd 6897 Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)
 RR   &     RR   &     C  RR   &     <    &     <  C   =>     <  C
 
Theorem0lt1 6898 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
 
3.2.5  Initial properties of the complex numbers
 
Theoremmul12 6899 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
 CC  CC  C  CC  x.  x.  C  x.  x.  C
 
Theoremmul32 6900 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
 CC  CC  C  CC  x.  x.  C  x.  C  x.
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