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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 0nsr 8701 | 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 8702 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Theorem | 1sr 8703 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Theorem | m1r 8704 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Theorem | addclsr 8705 | Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.) |
Theorem | mulclsr 8706 | Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
Theorem | dmaddsr 8707 | Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
Theorem | dmmulsr 8708 | Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
Theorem | addcomsr 8709 | 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 8710 | 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 8711 | 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 8712 | 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 8713 | 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 8714 | Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
Theorem | m1m1sr 8715 | Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | ltsosr 8716 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.) |
Theorem | 0lt1sr 8717 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
Theorem | 1ne0sr 8718 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
Theorem | 0idsr 8719 | 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 8720 | 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.) |
Theorem | 00sr 8721 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.) |
Theorem | ltasr 8722 | Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | pn0sr 8723 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | negexsr 8724* | 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 8725* | 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 8726 | The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | mulgt0sr 8727 | The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | sqgt0sr 8728 | The square of a nonzero signed real is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | recexsr 8729* | 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 8730 | 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 8731 | 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 8732* | Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | supsrlem 8733* | Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | supsr 8734* | 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 8735 | Class of complex numbers. |
Syntax | cr 8736 | Class of real numbers. |
Syntax | cc0 8737 | Extend class notation to include the complex number 0. |
Syntax | c1 8738 | Extend class notation to include the complex number 1. |
Syntax | ci 8739 | Extend class notation to include the complex number i. |
Syntax | caddc 8740 | Addition on complex numbers. |
Syntax | cltrr 8741 | 'Less than' predicate (defined over real subset of complex numbers). |
Syntax | cmul 8742 | Multiplication on complex numbers. The token is a center dot. |
Definition | df-c 8743 | Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 8770. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-0 8744 | Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-1 8745 | Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-i 8746 | Define the complex number (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-r 8747 | Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Definition | df-add 8748* | Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
Definition | df-mul 8749* | Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Definition | df-lt 8750* | Define 'less than' on the real subset of complex numbers. Proofs should typically use instead; see df-ltxr 8872. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | opelcn 8751 | Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Theorem | opelreal 8752 | Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | elreal 8753* | Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
Theorem | elreal2 8754 | Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Theorem | 0ncn 8755 | 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 8756 | 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | addcnsr 8757 | Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
Theorem | mulcnsr 8758 | Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Theorem | eqresr 8759 | Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | addresr 8760 | Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | mulresr 8761 | Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
Theorem | ltresr 8762 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | ltresr2 8763 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
Theorem | dfcnqs 8764 | Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in from those in . The trick involves qsid 6725, 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 8743), 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 8765 | Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 8764 and mulcnsrec 8766. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
Theorem | mulcnsrec 8766 |
Technical trick to permit re-use of some equivalence class lemmas for
operation laws. The trick involves ecid 6724,
which shows that the coset of
the converse epsilon relation (which is not an equivalence relation)
leaves a set unchanged. See also dfcnqs 8764.
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 8466. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.) |
Theorem | axaddf 8767 | 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 8773. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 8816. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
Theorem | axmulf 8768 | 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 8775. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 8817. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.) |
Theorem | axcnex 8769 | The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 10350), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus we can avoid ax-rep 4131 in later theorems by invoking the axiom ax-cnex 8793 instead of cnexALT 10350. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
Theorem | axresscn 8770 | 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 8794. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
Theorem | ax1cn 8771 | 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 8795. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.) |
Theorem | axicn 8772 | 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 8796. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.) |
Theorem | axaddcl 8773 | 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 8797 be used later. Instead, in most cases use addcl 8819. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.) |
Theorem | axaddrcl 8774 | 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 8798 be used later. Instead, in most cases use readdcl 8820. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
Theorem | axmulcl 8775 | 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 8799 be used later. Instead, in most cases use mulcl 8821. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
Theorem | axmulrcl 8776 | 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 8800 be used later. Instead, in most cases use remulcl 8822. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.) |
Theorem | axmulcom 8777 | 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 8801 be used later. Instead, use mulcom 8823. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.) |
Theorem | axaddass 8778 | 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 8802 be used later. Instead, use addass 8824. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Theorem | axmulass 8779 | 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 8803. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
Theorem | axdistr 8780 | 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 8804 be used later. Instead, use adddi 8826. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Theorem | axi2m1 8781 | 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 8805. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
Theorem | ax1ne0 8782 | 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 8806. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.) |
Theorem | ax1rid 8783 | 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 8835, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 8807. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) |
Theorem | axrnegex 8784* | 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 8808. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Theorem | axrrecex 8785* | 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 8809. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Theorem | axcnre 8786* | 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 8810. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | axpre-lttri 8787 | 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 8894. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 8811. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | axpre-lttrn 8788 | 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 8895. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 8812. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | axpre-ltadd 8789 | 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 8896. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 8813. (Contributed by NM, 11-May-1996.) (New usage is discouraged.) |
Theorem | axpre-mulgt0 8790 | 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 8897. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 8814. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | axpre-sup 8791* | 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 8898. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 8815. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | wuncn 8792 | 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 8793 | The complex numbers form a set. This axiom is redundant - see cnexALT 10350- but we provide this axiom because the justification theorem axcnex 8769 does not use ax-rep 4131 even though the redundancy proof does. Proofs should normally use cnex 8818 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-resscn 8794 | The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 8770. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-1cn 8795 | 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 8771. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-icn 8796 | is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 8772. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-addcl 8797 | Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 8773. Proofs should normally use addcl 8819 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 8798 | Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 8774. Proofs should normally use readdcl 8820 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulcl 8799 | Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 8775. Proofs should normally use mulcl 8821 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulrcl 8800 | Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 8776. Proofs should normally use remulcl 8822 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
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