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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mulcomsr 8801 | 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 8802 | 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 8803 | 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 8804 | Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
Theorem | m1m1sr 8805 | Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | ltsosr 8806 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.) |
Theorem | 0lt1sr 8807 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
Theorem | 1ne0sr 8808 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
Theorem | 0idsr 8809 | 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 8810 | 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.) |
Theorem | 00sr 8811 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.) |
Theorem | ltasr 8812 | Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | pn0sr 8813 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | negexsr 8814* | 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 8815* | 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 8816 | The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | mulgt0sr 8817 | The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | sqgt0sr 8818 | The square of a nonzero signed real is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | recexsr 8819* | 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 8820 | 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 8821 | 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 8822* | Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | supsrlem 8823* | Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | supsr 8824* | 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 8825 | Class of complex numbers. |
Syntax | cr 8826 | Class of real numbers. |
Syntax | cc0 8827 | Extend class notation to include the complex number 0. |
Syntax | c1 8828 | Extend class notation to include the complex number 1. |
Syntax | ci 8829 | Extend class notation to include the complex number i. |
Syntax | caddc 8830 | Addition on complex numbers. |
Syntax | cltrr 8831 | 'Less than' predicate (defined over real subset of complex numbers). |
Syntax | cmul 8832 | Multiplication on complex numbers. The token is a center dot. |
Definition | df-c 8833 | Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 8860. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-0 8834 | Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-1 8835 | Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-i 8836 | Define the complex number (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-r 8837 | Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-add 8838* | Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
Definition | df-mul 8839* | Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Definition | df-lt 8840* | Define 'less than' on the real subset of complex numbers. Proofs should typically use instead; see df-ltxr 8962. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | opelcn 8841 | Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | opelreal 8842 | Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | elreal 8843* | Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
Theorem | elreal2 8844 | Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | 0ncn 8845 | 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 8846 | 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | addcnsr 8847 | Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
Theorem | mulcnsr 8848 | Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Theorem | eqresr 8849 | Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | addresr 8850 | Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | mulresr 8851 | Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | ltresr 8852 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | ltresr2 8853 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | dfcnqs 8854 | Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in from those in . The trick involves qsid 6812, 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 8833), 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 8855 | Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 8854 and mulcnsrec 8856. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
Theorem | mulcnsrec 8856 |
Technical trick to permit re-use of some equivalence class lemmas for
operation laws. The trick involves ecid 6811,
which shows that the coset of
the converse epsilon relation (which is not an equivalence relation)
leaves a set unchanged. See also dfcnqs 8854.
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 8556. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
Theorem | axaddf 8857 | 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 8863. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 8906. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
Theorem | axmulf 8858 | 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 8865. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 8907. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
Theorem | axcnex 8859 | The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 10442), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 4212 in later theorems by invoking the axiom ax-cnex 8883 instead of cnexALT 10442. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
Theorem | axresscn 8860 | 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 8884. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
Theorem | ax1cn 8861 | 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 8885. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.) |
Theorem | axicn 8862 | 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 8886. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.) |
Theorem | axaddcl 8863 | 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 8887 be used later. Instead, in most cases use addcl 8909. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.) |
Theorem | axaddrcl 8864 | 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 8888 be used later. Instead, in most cases use readdcl 8910. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
Theorem | axmulcl 8865 | 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 8889 be used later. Instead, in most cases use mulcl 8911. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
Theorem | axmulrcl 8866 | 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 8890 be used later. Instead, in most cases use remulcl 8912. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.) |
Theorem | axmulcom 8867 | 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 8891 be used later. Instead, use mulcom 8913. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.) |
Theorem | axaddass 8868 | 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 8892 be used later. Instead, use addass 8914. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Theorem | axmulass 8869 | 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 8893. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
Theorem | axdistr 8870 | 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 8894 be used later. Instead, use adddi 8916. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Theorem | axi2m1 8871 | 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 8895. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
Theorem | ax1ne0 8872 | 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 8896. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.) |
Theorem | ax1rid 8873 | 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 8925, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 8897. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) |
Theorem | axrnegex 8874* | 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 8898. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Theorem | axrrecex 8875* | 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 8899. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Theorem | axcnre 8876* | 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 8900. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | axpre-lttri 8877 | 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 8984. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 8901. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | axpre-lttrn 8878 | 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 8985. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 8902. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | axpre-ltadd 8879 | 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 8986. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 8903. (Contributed by NM, 11-May-1996.) (New usage is discouraged.) |
Theorem | axpre-mulgt0 8880 | 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 8987. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 8904. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | axpre-sup 8881* | 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 8988. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 8905. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | wuncn 8882 | 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 8883 | The complex numbers form a set. This axiom is redundant - see cnexALT 10442- but we provide this axiom because the justification theorem axcnex 8859 does not use ax-rep 4212 even though the redundancy proof does. Proofs should normally use cnex 8908 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-resscn 8884 | The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 8860. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-1cn 8885 | 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 8861. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-icn 8886 | is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 8862. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-addcl 8887 | Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 8863. Proofs should normally use addcl 8909 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 8888 | Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 8864. Proofs should normally use readdcl 8910 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulcl 8889 | Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 8865. Proofs should normally use mulcl 8911 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulrcl 8890 | Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 8866. Proofs should normally use remulcl 8912 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulcom 8891 | Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 8867. Proofs should normally use mulcom 8913 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-addass 8892 | Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 8868. Proofs should normally use addass 8914 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulass 8893 | Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 8869. Proofs should normally use mulass 8915 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-distr 8894 | Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 8870. Proofs should normally use adddi 8916 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-i2m1 8895 | 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 8871. (Contributed by NM, 29-Jan-1995.) |
Axiom | ax-1ne0 8896 | 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by theorem ax1ne0 8872. (Contributed by NM, 29-Jan-1995.) |
Axiom | ax-1rid 8897 | is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by theorem ax1rid 8873. Weakened from the original axiom in the form of statement in mulid1 8925, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.) |
Axiom | ax-rnegex 8898* | Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by theorem axrnegex 8874. (Contributed by Eric Schmidt, 21-May-2007.) |
Axiom | ax-rrecex 8899* | Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 8875. (Contributed by Eric Schmidt, 11-Apr-2007.) |
Axiom | ax-cnre 8900* | 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 8876. For naming consistency, use cnre 8924 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.) |
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