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Theorem List for Intuitionistic Logic Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Axiomax-pre-mulgt0 7001 The product of two positive reals is positive. Axiom for real and complex numbers, justified by theorem axpre-mulgt0 6961. (Contributed by NM, 13-Oct-2005.)

Axiomax-pre-mulext 7002 Strong extensionality of multiplication (expressed in terms of ). Axiom for real and complex numbers, justified by theorem axpre-mulext 6962

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

Axiomax-arch 7003* Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for real and complex numbers, justified by theorem axarch 6965.

This axiom should not be used directly; instead use arch 8178 (which is the same, but stated in terms of and ). (Contributed by Jim Kingdon, 2-May-2020.) (New usage is discouraged.)

Axiomax-caucvg 7004* Completeness. Axiom for real and complex numbers, justified by theorem axcaucvg 6974.

A Cauchy sequence (as defined here, which has a rate convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within of the nth term.

This axiom should not be used directly; instead use caucvgre 9580 (which is the same, but stated in terms of the and notations). (Contributed by Jim Kingdon, 19-Jul-2021.) (New usage is discouraged.)

3.2  Derive the basic properties from the field axioms

3.2.1  Some deductions from the field axioms for complex numbers

Theoremcnex 7005 Alias for ax-cnex 6975. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremmulcl 7008 Alias for ax-mulcl 6982, for naming consistency with mulcli 7032. (Contributed by NM, 10-Mar-2008.)

Theoremremulcl 7009 Alias for ax-mulrcl 6983, for naming consistency with remulcli 7041. (Contributed by NM, 10-Mar-2008.)

Theoremmulcom 7010 Alias for ax-mulcom 6985, for naming consistency with mulcomi 7033. (Contributed by NM, 10-Mar-2008.)

Theoremmulass 7012 Alias for ax-mulass 6987, for naming consistency with mulassi 7036. (Contributed by NM, 10-Mar-2008.)

Theoremadddi 7013 Alias for ax-distr 6988, for naming consistency with adddii 7037. (Contributed by NM, 10-Mar-2008.)

Theoremrecn 7014 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)

Theoremreex 7015 The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremreelprrecn 7016 Reals are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremcnelprrecn 7017 Complex numbers are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremadddir 7018 Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.)

Theorem0cn 7019 0 is a complex number. (Contributed by NM, 19-Feb-2005.)

Theorem0cnd 7020 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremc0ex 7021 0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem1ex 7022 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theoremcnre 7023* Alias for ax-cnre 6995, for naming consistency. (Contributed by NM, 3-Jan-2013.)

Theoremmulid1 7024 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmulid2 7025 Identity law for multiplication. Note: see mulid1 7024 for commuted version. (Contributed by NM, 8-Oct-1999.)

Theorem1re 7026 is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.)

Theorem0re 7027 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)

Theorem0red 7028 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)

Theoremmulid1i 7029 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)

Theoremmulid2i 7030 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)

Theoremmulcli 7032 Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)

Theoremmulcomi 7033 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)

Theoremmulcomli 7034 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)

Theoremmulassi 7036 Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)

Theoremadddii 7037 Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)

Theoremadddiri 7038 Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)

Theoremrecni 7039 A real number is a complex number. (Contributed by NM, 1-Mar-1995.)

Theoremreaddcli 7040 Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)

Theoremremulcli 7041 Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)

Theorem1red 7042 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)

Theorem1cnd 7043 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)

Theoremmulid1d 7044 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulid2d 7045 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulcld 7047 Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulcomd 7048 Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulassd 7050 Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadddid 7051 Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadddird 7052 Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)

Theoremjoinlmuladdmuld 7053 Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)

Theoremrecnd 7054 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)

Theoremreaddcld 7055 Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremremulcld 7056 Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)

3.2.2  Infinity and the extended real number system

Syntaxcpnf 7057 Plus infinity.

Syntaxcmnf 7058 Minus infinity.

Syntaxcxr 7059 The set of extended reals (includes plus and minus infinity).

Syntaxclt 7060 'Less than' predicate (extended to include the extended reals).

Syntaxcle 7061 Extend wff notation to include the 'less than or equal to' relation.

Definitiondf-pnf 7062 Define plus infinity. Note that the definition is arbitrary, requiring only that be a set not in and different from (df-mnf 7063). We use to make it independent of the construction of , and Cantor's Theorem will show that it is different from any member of and therefore . See pnfnre 7067 and mnfnre 7068, and we'll also be able to prove .

A simpler possibility is to define as and as , but that approach requires the Axiom of Regularity to show that and are different from each other and from all members of . (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)

Definitiondf-mnf 7063 Define minus infinity as the power set of plus infinity. Note that the definition is arbitrary, requiring only that be a set not in and different from (see mnfnre 7068). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)

Definitiondf-xr 7064 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.)

Definitiondf-ltxr 7065* 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, is primitive and not necessarily a relation on . (Contributed by NM, 13-Oct-2005.)

Definitiondf-le 7066 Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.)

Theorempnfnre 7067 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)

Theoremmnfnre 7068 Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)

Theoremressxr 7069 The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)

Theoremrexpssxrxp 7070 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.)

Theoremrexr 7071 A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)

Theorem0xr 7072 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)

Theoremrenepnf 7073 No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremrenemnf 7074 No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremrexrd 7075 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrenepnfd 7076 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrenemnfd 7077 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrexri 7078 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)

Theoremrenfdisj 7079 The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltrelxr 7080 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremltrel 7081 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)

Theoremlerelxr 7082 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremlerel 7083 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremxrlenlt 7084 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.)

Theoremltxrlt 7085 The standard less-than and the extended real less-than are identical when restricted to the non-extended reals . (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

3.2.3  Restate the ordering postulates with extended real "less than"

Theoremaxltirr 7086 Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 6996 with ordering on the extended reals. New proofs should use ltnr 7095 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)

Theoremaxltwlin 7087 Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 6997 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.)

Theoremaxlttrn 7088 Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 6998 with ordering on the extended reals. New proofs should use lttr 7092 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Theoremaxltadd 7089 Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7000 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)

Theoremaxapti 7090 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 6999 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)

Theoremaxmulgt0 7091 The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7001 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)

3.2.4  Ordering on reals

Theoremlttr 7092 Alias for axlttrn 7088, for naming consistency with lttri 7122. New proofs should generally use this instead of ax-pre-lttrn 6998. (Contributed by NM, 10-Mar-2008.)

Theoremmulgt0 7093 The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.)

Theoremlenlt 7094 '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.)

Theoremltnr 7095 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)

Theoremltso 7096 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)

Theoremgtso 7097 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)

Theoremlttri3 7098 Tightness of real apartness. (Contributed by NM, 5-May-1999.)

Theoremletri3 7099 Tightness of real apartness. (Contributed by NM, 14-May-1999.)

Theoremltleletr 7100 Transitive law, weaker form of . (Contributed by AV, 14-Oct-2018.)

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