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Type | Label | Description |
---|---|---|
Statement | ||
Axiom | ax-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.) |
Axiom | ax-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.) |
Axiom | ax-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.) |
Axiom | ax-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.) |
Theorem | cnex 7005 | Alias for ax-cnex 6975. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Theorem | addcl 7006 | Alias for ax-addcl 6980, for naming consistency with addcli 7031. Use this theorem instead of ax-addcl 6980 or axaddcl 6940. (Contributed by NM, 10-Mar-2008.) |
Theorem | readdcl 7007 | Alias for ax-addrcl 6981, for naming consistency with readdcli 7040. (Contributed by NM, 10-Mar-2008.) |
Theorem | mulcl 7008 | Alias for ax-mulcl 6982, for naming consistency with mulcli 7032. (Contributed by NM, 10-Mar-2008.) |
Theorem | remulcl 7009 | Alias for ax-mulrcl 6983, for naming consistency with remulcli 7041. (Contributed by NM, 10-Mar-2008.) |
Theorem | mulcom 7010 | Alias for ax-mulcom 6985, for naming consistency with mulcomi 7033. (Contributed by NM, 10-Mar-2008.) |
Theorem | addass 7011 | Alias for ax-addass 6986, for naming consistency with addassi 7035. (Contributed by NM, 10-Mar-2008.) |
Theorem | mulass 7012 | Alias for ax-mulass 6987, for naming consistency with mulassi 7036. (Contributed by NM, 10-Mar-2008.) |
Theorem | adddi 7013 | Alias for ax-distr 6988, for naming consistency with adddii 7037. (Contributed by NM, 10-Mar-2008.) |
Theorem | recn 7014 | A real number is a complex number. (Contributed by NM, 10-Aug-1999.) |
Theorem | reex 7015 | The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Theorem | reelprrecn 7016 | Reals are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | cnelprrecn 7017 | Complex numbers are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | adddir 7018 | Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.) |
Theorem | 0cn 7019 | 0 is a complex number. (Contributed by NM, 19-Feb-2005.) |
Theorem | 0cnd 7020 | 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | c0ex 7021 | 0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.) |
Theorem | 1ex 7022 | 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.) |
Theorem | cnre 7023* | Alias for ax-cnre 6995, for naming consistency. (Contributed by NM, 3-Jan-2013.) |
Theorem | mulid1 7024 | is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) |
Theorem | mulid2 7025 | Identity law for multiplication. Note: see mulid1 7024 for commuted version. (Contributed by NM, 8-Oct-1999.) |
Theorem | 1re 7026 | is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.) |
Theorem | 0re 7027 | is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) |
Theorem | 0red 7028 | is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.) |
Theorem | mulid1i 7029 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
Theorem | mulid2i 7030 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
Theorem | addcli 7031 | Closure law for addition. (Contributed by NM, 23-Nov-1994.) |
Theorem | mulcli 7032 | Closure law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Theorem | mulcomi 7033 | Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Theorem | mulcomli 7034 | Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Theorem | addassi 7035 | Associative law for addition. (Contributed by NM, 23-Nov-1994.) |
Theorem | mulassi 7036 | Associative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Theorem | adddii 7037 | Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.) |
Theorem | adddiri 7038 | Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.) |
Theorem | recni 7039 | A real number is a complex number. (Contributed by NM, 1-Mar-1995.) |
Theorem | readdcli 7040 | Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.) |
Theorem | remulcli 7041 | Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.) |
Theorem | 1red 7042 | 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
Theorem | 1cnd 7043 | 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
Theorem | mulid1d 7044 | Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | mulid2d 7045 | Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | addcld 7046 | Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | mulcld 7047 | Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | mulcomd 7048 | Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | addassd 7049 | Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | mulassd 7050 | Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | adddid 7051 | Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | adddird 7052 | Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | joinlmuladdmuld 7053 | Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.) |
Theorem | recnd 7054 | Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.) |
Theorem | readdcld 7055 | Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | remulcld 7056 | Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
Syntax | cpnf 7057 | Plus infinity. |
Syntax | cmnf 7058 | Minus infinity. |
Syntax | cxr 7059 | The set of extended reals (includes plus and minus infinity). |
Syntax | clt 7060 | 'Less than' predicate (extended to include the extended reals). |
Syntax | cle 7061 | Extend wff notation to include the 'less than or equal to' relation. |
Definition | df-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.) |
Definition | df-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.) |
Definition | df-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.) |
Definition | df-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.) |
Definition | df-le 7066 | Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.) |
Theorem | pnfnre 7067 | Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
Theorem | mnfnre 7068 | Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
Theorem | ressxr 7069 | The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.) |
Theorem | rexpssxrxp 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.) |
Theorem | rexr 7071 | A standard real is an extended real. (Contributed by NM, 14-Oct-2005.) |
Theorem | 0xr 7072 | Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.) |
Theorem | renepnf 7073 | No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Theorem | renemnf 7074 | No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Theorem | rexrd 7075 | A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | renepnfd 7076 | No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | renemnfd 7077 | No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | rexri 7078 | A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.) |
Theorem | renfdisj 7079 | The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Theorem | ltrelxr 7080 | 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Theorem | ltrel 7081 | 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.) |
Theorem | lerelxr 7082 | 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Theorem | lerel 7083 | 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | xrlenlt 7084 | 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.) |
Theorem | ltxrlt 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.) |
Theorem | axltirr 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.) |
Theorem | axltwlin 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.) |
Theorem | axlttrn 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.) |
Theorem | axltadd 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.) |
Theorem | axapti 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.) |
Theorem | axmulgt0 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.) |
Theorem | lttr 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.) |
Theorem | mulgt0 7093 | The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.) |
Theorem | lenlt 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.) |
Theorem | ltnr 7095 | 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
Theorem | ltso 7096 | 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.) |
Theorem | gtso 7097 | 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.) |
Theorem | lttri3 7098 | Tightness of real apartness. (Contributed by NM, 5-May-1999.) |
Theorem | letri3 7099 | Tightness of real apartness. (Contributed by NM, 14-May-1999.) |
Theorem | ltleletr 7100 | Transitive law, weaker form of . (Contributed by AV, 14-Oct-2018.) |
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