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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | addclsr 8701 | Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.) |
Theorem | mulclsr 8702 | Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
Theorem | dmaddsr 8703 | Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
Theorem | dmmulsr 8704 | Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
Theorem | addcomsr 8705 | 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 8706 | 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 8707 | 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 8708 | 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 8709 | 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 8710 | Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
Theorem | m1m1sr 8711 | Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | ltsosr 8712 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.) |
Theorem | 0lt1sr 8713 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
Theorem | 1ne0sr 8714 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
Theorem | 0idsr 8715 | 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 8716 | 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.) |
Theorem | 00sr 8717 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.) |
Theorem | ltasr 8718 | Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | pn0sr 8719 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | negexsr 8720* | 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 8721* | 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 8722 | The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | mulgt0sr 8723 | The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | sqgt0sr 8724 | The square of a nonzero signed real is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | recexsr 8725* | 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 8726 | 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 8727 | 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 8728* | Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | supsrlem 8729* | Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | supsr 8730* | 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 8731 | Class of complex numbers. |
Syntax | cr 8732 | Class of real numbers. |
Syntax | cc0 8733 | Extend class notation to include the complex number 0. |
Syntax | c1 8734 | Extend class notation to include the complex number 1. |
Syntax | ci 8735 | Extend class notation to include the complex number i. |
Syntax | caddc 8736 | Addition on complex numbers. |
Syntax | cltrr 8737 | 'Less than' predicate (defined over real subset of complex numbers). |
Syntax | cmul 8738 | Multiplication on complex numbers. The token is a center dot. |
Definition | df-c 8739 | Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 8766. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-0 8740 | Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-1 8741 | Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-i 8742 | Define the complex number (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-r 8743 | Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-add 8744* | Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
Definition | df-mul 8745* | Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Definition | df-lt 8746* | Define 'less than' on the real subset of complex numbers. Proofs should typically use instead; see df-ltxr 8868. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | opelcn 8747 | Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | opelreal 8748 | Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | elreal 8749* | Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
Theorem | elreal2 8750 | Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | 0ncn 8751 | 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 8752 | 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | addcnsr 8753 | Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
Theorem | mulcnsr 8754 | Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Theorem | eqresr 8755 | Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | addresr 8756 | Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | mulresr 8757 | Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | ltresr 8758 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | ltresr2 8759 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | dfcnqs 8760 | Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in from those in . The trick involves qsid 6721, 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 8739), 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 8761 | Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 8760 and mulcnsrec 8762. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
Theorem | mulcnsrec 8762 |
Technical trick to permit re-use of some equivalence class lemmas for
operation laws. The trick involves ecid 6720,
which shows that the coset of
the converse epsilon relation (which is not an equivalence relation)
leaves a set unchanged. See also dfcnqs 8760.
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 8462. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
Theorem | axaddf 8763 | 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 8769. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 8812. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
Theorem | axmulf 8764 | 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 8771. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 8813. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
Theorem | axcnex 8765 | The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 10346), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus we can avoid ax-rep 4132 in later theorems by invoking the axiom ax-cnex 8789 instead of cnexALT 10346. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
Theorem | axresscn 8766 | 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 8790. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
Theorem | ax1cn 8767 | 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 8791. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.) |
Theorem | axicn 8768 | 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 8792. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.) |
Theorem | axaddcl 8769 | 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 8793 be used later. Instead, in most cases use addcl 8815. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.) |
Theorem | axaddrcl 8770 | 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 8794 be used later. Instead, in most cases use readdcl 8816. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
Theorem | axmulcl 8771 | 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 8795 be used later. Instead, in most cases use mulcl 8817. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
Theorem | axmulrcl 8772 | 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 8796 be used later. Instead, in most cases use remulcl 8818. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.) |
Theorem | axmulcom 8773 | 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 8797 be used later. Instead, use mulcom 8819. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.) |
Theorem | axaddass 8774 | 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 8798 be used later. Instead, use addass 8820. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Theorem | axmulass 8775 | 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 8799. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
Theorem | axdistr 8776 | 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 8800 be used later. Instead, use adddi 8822. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Theorem | axi2m1 8777 | 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 8801. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
Theorem | ax1ne0 8778 | 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 8802. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.) |
Theorem | ax1rid 8779 | 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 8831, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 8803. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) |
Theorem | axrnegex 8780* | 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 8804. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Theorem | axrrecex 8781* | 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 8805. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Theorem | axcnre 8782* | 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 8806. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | axpre-lttri 8783 | 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 8890. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 8807. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | axpre-lttrn 8784 | 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 8891. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 8808. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | axpre-ltadd 8785 | 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 8892. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 8809. (Contributed by NM, 11-May-1996.) (New usage is discouraged.) |
Theorem | axpre-mulgt0 8786 | 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 8893. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 8810. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | axpre-sup 8787* | 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 8894. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 8811. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | wuncn 8788 | 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 8789 | The complex numbers form a set. This axiom is redundant - see cnexALT 10346- but we provide this axiom because the justification theorem axcnex 8765 does not use ax-rep 4132 even though the redundancy proof does. Proofs should normally use cnex 8814 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-resscn 8790 | The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 8766. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-1cn 8791 | 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 8767. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-icn 8792 | is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 8768. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-addcl 8793 | Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 8769. Proofs should normally use addcl 8815 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 8794 | Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 8770. Proofs should normally use readdcl 8816 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulcl 8795 | Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 8771. Proofs should normally use mulcl 8817 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulrcl 8796 | Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 8772. Proofs should normally use remulcl 8818 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulcom 8797 | Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 8773. Proofs should normally use mulcom 8819 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-addass 8798 | Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 8774. Proofs should normally use addass 8820 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulass 8799 | Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 8775. Proofs should normally use mulass 8821 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-distr 8800 | Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 8776. Proofs should normally use adddi 8822 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
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