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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 0nsr 8901 | The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.) |
Theorem | 0r 8902 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Theorem | 1sr 8903 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Theorem | m1r 8904 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Theorem | addclsr 8905 | Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.) |
Theorem | mulclsr 8906 | Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
Theorem | dmaddsr 8907 | Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
Theorem | dmmulsr 8908 | Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
Theorem | addcomsr 8909 | Addition of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.) |
Theorem | addasssr 8910 | Addition of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.) |
Theorem | mulcomsr 8911 | Multiplication of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.) |
Theorem | mulasssr 8912 | Multiplication of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.) |
Theorem | distrsr 8913 | Multiplication of signed reals is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.) |
Theorem | m1p1sr 8914 | Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
Theorem | m1m1sr 8915 | Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | ltsosr 8916 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.) |
Theorem | 0lt1sr 8917 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
Theorem | 1ne0sr 8918 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
Theorem | 0idsr 8919 | The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.) |
Theorem | 1idsr 8920 | 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.) |
Theorem | 00sr 8921 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.) |
Theorem | ltasr 8922 | Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | pn0sr 8923 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | negexsr 8924* | Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) (New usage is discouraged.) |
Theorem | recexsrlem 8925* | The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Theorem | addgt0sr 8926 | The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | mulgt0sr 8927 | The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | sqgt0sr 8928 | The square of a nonzero signed real is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | recexsr 8929* | The reciprocal of a nonzero signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Theorem | mappsrpr 8930 | Mapping from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | ltpsrpr 8931 | Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | map2psrpr 8932* | Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | supsrlem 8933* | Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | supsr 8934* | A non-empty, bounded set of signed reals has a supremum. (Cotributed by Mario Carneiro, 15-Jun-2013.) (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Syntax | cc 8935 | Class of complex numbers. |
Syntax | cr 8936 | Class of real numbers. |
Syntax | cc0 8937 | Extend class notation to include the complex number 0. |
Syntax | c1 8938 | Extend class notation to include the complex number 1. |
Syntax | ci 8939 | Extend class notation to include the complex number i. |
Syntax | caddc 8940 | Addition on complex numbers. |
Syntax | cltrr 8941 | 'Less than' predicate (defined over real subset of complex numbers). |
Syntax | cmul 8942 | Multiplication on complex numbers. The token is a center dot. |
Definition | df-c 8943 | Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 8970. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-0 8944 | Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-1 8945 | Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-i 8946 | Define the complex number (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-r 8947 | Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-add 8948* | Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
Definition | df-mul 8949* | Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Definition | df-lt 8950* | Define 'less than' on the real subset of complex numbers. Proofs should typically use instead; see df-ltxr 9072. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | opelcn 8951 | Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | opelreal 8952 | Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | elreal 8953* | Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
Theorem | elreal2 8954 | Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | 0ncn 8955 | 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.) |
Theorem | ltrelre 8956 | 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | addcnsr 8957 | Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
Theorem | mulcnsr 8958 | Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Theorem | eqresr 8959 | Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | addresr 8960 | Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | mulresr 8961 | Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | ltresr 8962 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | ltresr2 8963 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | dfcnqs 8964 | Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in from those in . The trick involves qsid 6920, 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 8943), 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.) |
Theorem | addcnsrec 8965 | Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 8964 and mulcnsrec 8966. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
Theorem | mulcnsrec 8966 |
Technical trick to permit re-use of some equivalence class lemmas for
operation laws. The trick involves ecid 6919,
which shows that the coset of
the converse epsilon relation (which is not an equivalence relation)
leaves a set unchanged. See also dfcnqs 8964.
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 8666. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
Theorem | axaddf 8967 | 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 8973. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 9016. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
Theorem | axmulf 8968 | 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 8975. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 9017. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
Theorem | axcnex 8969 | The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 10554), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 4275 in later theorems by invoking the axiom ax-cnex 8993 instead of cnexALT 10554. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
Theorem | axresscn 8970 | 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 8994. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
Theorem | ax1cn 8971 | 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 8995. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.) |
Theorem | axicn 8972 | 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 8996. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.) |
Theorem | axaddcl 8973 | 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 8997 be used later. Instead, in most cases use addcl 9019. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.) |
Theorem | axaddrcl 8974 | 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 8998 be used later. Instead, in most cases use readdcl 9020. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
Theorem | axmulcl 8975 | 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 8999 be used later. Instead, in most cases use mulcl 9021. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
Theorem | axmulrcl 8976 | 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 9000 be used later. Instead, in most cases use remulcl 9022. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.) |
Theorem | axmulcom 8977 | 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 9001 be used later. Instead, use mulcom 9023. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.) |
Theorem | axaddass 8978 | 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 9002 be used later. Instead, use addass 9024. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Theorem | axmulass 8979 | 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 9003. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
Theorem | axdistr 8980 | 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 9004 be used later. Instead, use adddi 9026. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Theorem | axi2m1 8981 | 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 9005. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
Theorem | ax1ne0 8982 | 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 9006. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.) |
Theorem | ax1rid 8983 | 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 9035, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 9007. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) |
Theorem | axrnegex 8984* | 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 9008. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Theorem | axrrecex 8985* | 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 9009. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Theorem | axcnre 8986* | 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 9010. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | axpre-lttri 8987 | 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 9094. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 9011. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | axpre-lttrn 8988 | 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 9095. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 9012. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | axpre-ltadd 8989 | 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 9096. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 9013. (Contributed by NM, 11-May-1996.) (New usage is discouraged.) |
Theorem | axpre-mulgt0 8990 | 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 9097. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 9014. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | axpre-sup 8991* | 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 9098. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 9015. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | wuncn 8992 | 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 | ||
Axiom | ax-cnex 8993 | The complex numbers form a set. This axiom is redundant - see cnexALT 10554- but we provide this axiom because the justification theorem axcnex 8969 does not use ax-rep 4275 even though the redundancy proof does. Proofs should normally use cnex 9018 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-resscn 8994 | The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 8970. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-1cn 8995 | 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 8971. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-icn 8996 | is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 8972. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-addcl 8997 | Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 8973. Proofs should normally use addcl 9019 instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-addrcl 8998 | Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 8974. Proofs should normally use readdcl 9020 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulcl 8999 | Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 8975. Proofs should normally use mulcl 9021 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulrcl 9000 | Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 8976. Proofs should normally use remulcl 9022 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
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