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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | enrex 8901 | The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.) |
Theorem | ltrelsr 8902 | Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.) |
Theorem | addcmpblnr 8903 | Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
Theorem | mulcmpblnrlem 8904 | Lemma used in lemma showing compatibility of multiplication. (Contributed by NM, 4-Sep-1995.) (New usage is discouraged.) |
Theorem | mulcmpblnr 8905 | Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) (New usage is discouraged.) |
Theorem | addsrpr 8906 | Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.) |
Theorem | mulsrpr 8907 | Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.) |
Theorem | ltsrpr 8908 | Ordering of signed reals in terms of positive reals. (Contributed by NM, 20-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.) |
Theorem | gt0srpr 8909 | Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | 0nsr 8910 | 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 8911 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Theorem | 1sr 8912 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Theorem | m1r 8913 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Theorem | addclsr 8914 | Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.) |
Theorem | mulclsr 8915 | Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
Theorem | dmaddsr 8916 | Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
Theorem | dmmulsr 8917 | Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
Theorem | addcomsr 8918 | 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 8919 | 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 8920 | 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 8921 | 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 8922 | 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 8923 | Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
Theorem | m1m1sr 8924 | Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | ltsosr 8925 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.) |
Theorem | 0lt1sr 8926 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
Theorem | 1ne0sr 8927 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
Theorem | 0idsr 8928 | 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 8929 | 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.) |
Theorem | 00sr 8930 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.) |
Theorem | ltasr 8931 | Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | pn0sr 8932 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | negexsr 8933* | 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 8934* | 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 8935 | The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | mulgt0sr 8936 | The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | sqgt0sr 8937 | The square of a nonzero signed real is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | recexsr 8938* | 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 8939 | 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 8940 | 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 8941* | Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | supsrlem 8942* | Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | supsr 8943* | 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 8944 | Class of complex numbers. |
Syntax | cr 8945 | Class of real numbers. |
Syntax | cc0 8946 | Extend class notation to include the complex number 0. |
Syntax | c1 8947 | Extend class notation to include the complex number 1. |
Syntax | ci 8948 | Extend class notation to include the complex number i. |
Syntax | caddc 8949 | Addition on complex numbers. |
Syntax | cltrr 8950 | 'Less than' predicate (defined over real subset of complex numbers). |
Syntax | cmul 8951 | Multiplication on complex numbers. The token is a center dot. |
Definition | df-c 8952 | Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 8979. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-0 8953 | Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-1 8954 | Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-i 8955 | Define the complex number (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-r 8956 | Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-add 8957* | Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
Definition | df-mul 8958* | Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Definition | df-lt 8959* | Define 'less than' on the real subset of complex numbers. Proofs should typically use instead; see df-ltxr 9081. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | opelcn 8960 | Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | opelreal 8961 | Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | elreal 8962* | Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
Theorem | elreal2 8963 | Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | 0ncn 8964 | 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 8965 | 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | addcnsr 8966 | Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
Theorem | mulcnsr 8967 | Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Theorem | eqresr 8968 | Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | addresr 8969 | Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | mulresr 8970 | Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | ltresr 8971 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | ltresr2 8972 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | dfcnqs 8973 | Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in from those in . The trick involves qsid 6929, 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 8952), 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 8974 | Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 8973 and mulcnsrec 8975. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
Theorem | mulcnsrec 8975 |
Technical trick to permit re-use of some equivalence class lemmas for
operation laws. The trick involves ecid 6928,
which shows that the coset of
the converse epsilon relation (which is not an equivalence relation)
leaves a set unchanged. See also dfcnqs 8973.
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 8675. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
Theorem | axaddf 8976 | 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 8982. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 9025. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
Theorem | axmulf 8977 | 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 8984. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 9026. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
Theorem | axcnex 8978 | The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 10564), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 4280 in later theorems by invoking the axiom ax-cnex 9002 instead of cnexALT 10564. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
Theorem | axresscn 8979 | 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 9003. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
Theorem | ax1cn 8980 | 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 9004. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.) |
Theorem | axicn 8981 | 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 9005. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.) |
Theorem | axaddcl 8982 | 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 9006 be used later. Instead, in most cases use addcl 9028. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.) |
Theorem | axaddrcl 8983 | 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 9007 be used later. Instead, in most cases use readdcl 9029. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
Theorem | axmulcl 8984 | 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 9008 be used later. Instead, in most cases use mulcl 9030. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
Theorem | axmulrcl 8985 | 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 9009 be used later. Instead, in most cases use remulcl 9031. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.) |
Theorem | axmulcom 8986 | 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 9010 be used later. Instead, use mulcom 9032. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.) |
Theorem | axaddass 8987 | 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 9011 be used later. Instead, use addass 9033. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Theorem | axmulass 8988 | 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 9012. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
Theorem | axdistr 8989 | 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 9013 be used later. Instead, use adddi 9035. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Theorem | axi2m1 8990 | 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 9014. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
Theorem | ax1ne0 8991 | 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 9015. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.) |
Theorem | ax1rid 8992 | 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 9044, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 9016. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) |
Theorem | axrnegex 8993* | 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 9017. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Theorem | axrrecex 8994* | 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 9018. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Theorem | axcnre 8995* | 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 9019. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | axpre-lttri 8996 | 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 9103. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 9020. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | axpre-lttrn 8997 | 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 9104. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 9021. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | axpre-ltadd 8998 | 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 9105. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 9022. (Contributed by NM, 11-May-1996.) (New usage is discouraged.) |
Theorem | axpre-mulgt0 8999 | 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 9106. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 9023. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | axpre-sup 9000* | 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 9107. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 9024. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
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