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Theorem List for Metamath Proof Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-r 9001 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)

Definitiondf-add 9002* Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)

Definitiondf-mul 9003* Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)

Definitiondf-lt 9004* Define 'less than' on the real subset of complex numbers. Proofs should typically use instead; see df-ltxr 9126. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)

Theoremopelcn 9005 Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)

Theoremopelreal 9006 Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)

Theoremelreal 9007* Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)

Theoremelreal2 9008 Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)

Theorem0ncn 9009 The empty set is not a complex number. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)

Theoremltrelre 9010 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)

Theoremaddcnsr 9011 Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)

Theoremmulcnsr 9012 Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)

Theoremeqresr 9013 Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)

Theoremaddresr 9014 Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)

Theoremmulresr 9015 Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)

Theoremltresr 9016 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)

Theoremltresr2 9017 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)

Theoremdfcnqs 9018 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in from those in . The trick involves qsid 6971, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) acts as an identity divisor for the quotient set operation. This lets us "pretend" that is a quotient set, even though it is not (compare df-c 8997), and allows us to reuse some of the equivalence class lemmas we developed for the transition from positive reals to signed reals, etc. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)

Theoremaddcnsrec 9019 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 9018 and mulcnsrec 9020. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)

Theoremmulcnsrec 9020 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 6970, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 9018.

Note: This is the last lemma (from which the axioms will be derived) in the construction of real and complex numbers. The construction starts at cnpi 8720. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)

5.1.2  Final derivation of real and complex number postulates

Theoremaxaddf 9021 Addition is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axaddcl 9027. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 9070. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)

Theoremaxmulf 9022 Multiplication is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axmulcl 9029. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 9071. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)

Theoremaxcnex 9023 The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 10609), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 4321 in later theorems by invoking the axiom ax-cnex 9047 instead of cnexALT 10609. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)

Theoremaxresscn 9024 The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-resscn 9048. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)

Theoremax1cn 9025 1 is a complex number. Axiom 2 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1cn 9049. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)

Theoremaxicn 9026 is a complex number. Axiom 3 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-icn 9050. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)

Theoremaxaddcl 9027 Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcl 9051 be used later. Instead, in most cases use addcl 9073. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)

Theoremaxaddrcl 9028 Closure law for addition in the real subfield of complex numbers. Axiom 5 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addrcl 9052 be used later. Instead, in most cases use readdcl 9074. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)

Theoremaxmulcl 9029 Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcl 9053 be used later. Instead, in most cases use mulcl 9075. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)

Theoremaxmulrcl 9030 Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulrcl 9054 be used later. Instead, in most cases use remulcl 9076. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)

Theoremaxmulcom 9031 Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcom 9055 be used later. Instead, use mulcom 9077. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)

Theoremaxaddass 9032 Addition of complex numbers is associative. This theorem transfers the associative laws for the real and imaginary signed real components of complex number pairs, to complex number addition itself. Axiom 9 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addass 9056 be used later. Instead, use addass 9078. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)

Theoremaxmulass 9033 Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulass 9057. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)

Theoremaxdistr 9034 Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-distr 9058 be used later. Instead, use adddi 9080. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)

Theoremaxi2m1 9035 i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 12 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-i2m1 9059. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)

Theoremax1ne0 9036 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1ne0 9060. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)

Theoremax1rid 9037 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, derived from ZF set theory. Weakened from the original axiom in the form of statement in mulid1 9089, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 9061. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)

Theoremaxrnegex 9038* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 9062. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)

Theoremaxrrecex 9039* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rrecex 9063. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)

Theoremaxcnre 9040* 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, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 9064. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)

Theoremaxpre-lttri 9041 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttri 9148. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 9065. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)

Theoremaxpre-lttrn 9042 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttrn 9149. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 9066. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)

Theoremaxpre-ltadd 9043 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axltadd 9150. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 9067. (Contributed by NM, 11-May-1996.) (New usage is discouraged.)

Theoremaxpre-mulgt0 9044 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axmulgt0 9151. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 9068. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)

Theoremaxpre-sup 9045* 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. Note: The more general version with ordering on extended reals is axsup 9152. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 9069. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)

Theoremwuncn 9046 A weak universe containing contains the complex number construction. This theorem is construction-dependent in the literal sense, but will also be satisfied by any other reasonable implementation of the complex numbers. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

5.1.3  Real and complex number postulates restated as axioms

Axiomax-cnex 9047 The complex numbers form a set. This axiom is redundant - see cnexALT 10609- but we provide this axiom because the justification theorem axcnex 9023 does not use ax-rep 4321 even though the redundancy proof does. Proofs should normally use cnex 9072 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)

Axiomax-resscn 9048 The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 9024. (Contributed by NM, 1-Mar-1995.)

Axiomax-1cn 9049 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 9025. (Contributed by NM, 1-Mar-1995.)

Axiomax-icn 9050 is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 9026. (Contributed by NM, 1-Mar-1995.)

Axiomax-addcl 9051 Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 9027. Proofs should normally use addcl 9073 instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-addrcl 9052 Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 9028. Proofs should normally use readdcl 9074 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-mulcl 9053 Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 9029. Proofs should normally use mulcl 9075 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-mulrcl 9054 Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 9030. Proofs should normally use remulcl 9076 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-mulcom 9055 Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 9031. Proofs should normally use mulcom 9077 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-addass 9056 Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 9032. Proofs should normally use addass 9078 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-mulass 9057 Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 9033. Proofs should normally use mulass 9079 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-distr 9058 Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 9034. Proofs should normally use adddi 9080 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Axiomax-i2m1 9059 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 9035. (Contributed by NM, 29-Jan-1995.)

Axiomax-1ne0 9060 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by theorem ax1ne0 9036. (Contributed by NM, 29-Jan-1995.)

Axiomax-1rid 9061 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by theorem ax1rid 9037. Weakened from the original axiom in the form of statement in mulid1 9089, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.)

Axiomax-rnegex 9062* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by theorem axrnegex 9038. (Contributed by Eric Schmidt, 21-May-2007.)

Axiomax-rrecex 9063* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 9039. (Contributed by Eric Schmidt, 11-Apr-2007.)

Axiomax-cnre 9064* 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 9040. For naming consistency, use cnre 9088 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.)

Axiomax-pre-lttri 9065 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri 9041. Note: The more general version for extended reals is axlttri 9148. Normally new proofs would use xrlttri 10733. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Axiomax-pre-lttrn 9066 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by theorem axpre-lttrn 9042. Note: The more general version for extended reals is axlttrn 9149. Normally new proofs would use lttr 9153. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Axiomax-pre-ltadd 9067 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by theorem axpre-ltadd 9043. Normally new proofs would use axltadd 9150. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Axiomax-pre-mulgt0 9068 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by theorem axpre-mulgt0 9044. Normally new proofs would use axmulgt0 9151. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Axiomax-pre-sup 9069* 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 9045. Note: Normally new proofs would use axsup 9152. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Axiomax-addf 9070 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 9073 should be used. Note that uses of ax-addf 9070 can be eliminated by using the defined operation 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 9021. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

Axiomax-mulf 9071 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 9053 should be used. Note that uses of ax-mulf 9071 can be eliminated by using the defined operation in place of , from which this axiom (with the defined operation in place of ) follows as a theorem.

This axiom is justified by theorem axmulf 9022. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

5.2  Derive the basic properties from the field axioms

5.2.1  Some deductions from the field axioms for complex numbers

Theoremcnex 9072 Alias for ax-cnex 9047. See also cnexALT 10609. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremmulcl 9075 Alias for ax-mulcl 9053, for naming consistency with mulcli 9096. (Contributed by NM, 10-Mar-2008.)

Theoremremulcl 9076 Alias for ax-mulrcl 9054, for naming consistency with remulcli 9105. (Contributed by NM, 10-Mar-2008.)

Theoremmulcom 9077 Alias for ax-mulcom 9055, for naming consistency with mulcomi 9097. (Contributed by NM, 10-Mar-2008.)

Theoremmulass 9079 Alias for ax-mulass 9057, for naming consistency with mulassi 9100. (Contributed by NM, 10-Mar-2008.)

Theoremadddi 9080 Alias for ax-distr 9058, for naming consistency with adddii 9101. (Contributed by NM, 10-Mar-2008.)

Theoremrecn 9081 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)

Theoremreex 9082 The real numbers form a set. See also reexALT 10607. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremelimne0 9083 Hypothesis for weak deduction theorem to eliminate . (Contributed by NM, 15-May-1999.)

Theoremadddir 9084 Distributive law for complex numbers. (Contributed by NM, 10-Oct-2004.)

Theorem0cn 9085 0 is a complex number. See also 0cnALT 9296. (Contributed by NM, 19-Feb-2005.)

Theoremc0ex 9086 0 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem1ex 9087 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theoremcnre 9088* Alias for ax-cnre 9064, for naming consistency. (Contributed by Mario Carneiro, 3-Jan-2013.)

Theoremmulid1 9089 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmulid2 9090 Identity law for multiplication. Note: see mulid1 9089 for commuted version. (Contributed by NM, 8-Oct-1999.)

Theorem1re 9091 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 9049, by exploiting properties of the imaginary unit . (Contributed by Eric Schmidt, 11-Apr-2007.) (Revised by Scott Fenton, 3-Jan-2013.)

Theorem0re 9092 is a real number. See also 0reALT 9398. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremmulid1i 9093 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)

Theoremmulid2i 9094 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)

Theoremmulcli 9096 Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)

Theoremmulcomi 9097 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)

Theoremmulcomli 9098 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)