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Type | Label | Description |
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Statement | ||
Theorem | pitonnlem1 6701* | Lemma for pitonn 6704. Two ways to write the number one. (Contributed by Jim Kingdon, 24-Apr-2020.) |
Theorem | pitonnlem1p1 6702 | Lemma for pitonn 6704. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.) |
Theorem | pitonnlem2 6703* | Lemma for pitonn 6704. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.) |
Theorem | pitonn 6704* | Mapping from to . (Contributed by Jim Kingdon, 22-Apr-2020.) |
Theorem | axcnex 6705 | The complex numbers form a set. Use cnex 6763 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
Theorem | axresscn 6706 | The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-resscn 6735. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
Theorem | ax1cn 6707 | 1 is a complex number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1cn 6736. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.) |
Theorem | ax1re 6708 |
1 is a real number. Axiom for real and complex numbers, derived from set
theory. This construction-dependent theorem should not be referenced
directly; instead, use ax-1re 6737.
In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 6736 and the other axioms. It is not known whether we can do so here, but the Metamath Proof Explorer proof (accessed 13-Jan-2020) uses excluded middle. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.) |
Theorem | axicn 6709 | is a complex number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-icn 6738. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.) |
Theorem | axaddcl 6710 | Closure law for addition of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcl 6739 be used later. Instead, in most cases use addcl 6764. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.) |
Theorem | axaddrcl 6711 | Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addrcl 6740 be used later. Instead, in most cases use readdcl 6765. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
Theorem | axmulcl 6712 | Closure law for multiplication of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcl 6741 be used later. Instead, in most cases use mulcl 6766. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
Theorem | axmulrcl 6713 | Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulrcl 6742 be used later. Instead, in most cases use remulcl 6767. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.) |
Theorem | axaddcom 6714 |
Addition commutes. Axiom for real and complex numbers, derived from set
theory. This construction-dependent theorem should not be referenced
directly, nor should the proven axiom ax-addcom 6743 be used later.
Instead, use addcom 6907.
In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on real number trichotomy and it is not known whether it is possible to prove this from the other axioms without it. (Contributed by Jim Kingdon, 17-Jan-2020.) (New usage is discouraged.) |
Theorem | axmulcom 6715 | Multiplication of complex numbers is commutative. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcom 6744 be used later. Instead, use mulcom 6768. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.) |
Theorem | axaddass 6716 | 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 for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addass 6745 be used later. Instead, use addass 6769. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Theorem | axmulass 6717 | Multiplication of complex numbers is associative. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulass 6746. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
Theorem | axdistr 6718 | Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-distr 6747 be used later. Instead, use adddi 6771. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.) |
Theorem | axi2m1 6719 | i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-i2m1 6748. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
Theorem | ax0lt1 6720 |
0 is less than 1. Axiom for real and complex numbers, derived from set
theory. This construction-dependent theorem should not be referenced
directly; instead, use ax-0lt1 6749.
The version of this axiom in the Metamath Proof Explorer reads ; here we change it to . The proof of from in the Metamath Proof Explorer (accessed 12-Jan-2020) relies on real number trichotomy. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.) |
Theorem | ax1rid 6721 | is an identity element for real multiplication. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 6750. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) |
Theorem | ax0id 6722 |
is an identity element
for real addition. Axiom for real and
complex numbers, derived from set theory. This construction-dependent
theorem should not be referenced directly; instead, use ax-0id 6751.
In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on excluded middle and it is not known whether it is possible to prove this from the other axioms without excluded middle. (Contributed by Jim Kingdon, 16-Jan-2020.) (New usage is discouraged.) |
Theorem | axrnegex 6723* | Existence of negative of real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 6752. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Theorem | axprecex 6724* |
Existence of positive reciprocal of positive real number. Axiom for
real and complex numbers, derived from set theory. This
construction-dependent theorem should not be referenced directly;
instead, use ax-precex 6753.
In treatments which assume excluded middle, the condition is generally replaced by , and it may not be necessary to state that the reciproacal is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) (New usage is discouraged.) |
Theorem | axcnre 6725* | A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 6754. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | axpre-ltirr 6726 | Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltirr 6755. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.) |
Theorem | axpre-ltwlin 6727 | Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltwlin 6756. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.) |
Theorem | axpre-lttrn 6728 | Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 6757. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Theorem | axpre-apti 6729 |
Apartness of reals is tight. Axiom for real and complex numbers,
derived from set theory. This construction-dependent theorem should not
be referenced directly; instead, use ax-pre-apti 6758.
(Contributed by Jim Kingdon, 29-Jan-2020.) (New usage is discouraged.) |
Theorem | axpre-ltadd 6730 | Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 6759. (Contributed by NM, 11-May-1996.) (New usage is discouraged.) |
Theorem | axpre-mulgt0 6731 | The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 6760. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Theorem | axpre-mulext 6732 |
Strong extensionality of multiplication (expressed in terms of
).
Axiom for real and complex numbers, derived from set
theory. This construction-dependent theorem should not be referenced
directly; instead, use ax-pre-mulext 6761.
(Contributed by Jim Kingdon, 18-Feb-2020.) (New usage is discouraged.) |
Theorem | axarch 6733* |
Archimedean axiom. The Archimedean property is more naturally stated
once we have defined . Unless we find another way to state it,
we'll just use the right hand side of dfnn2 7657 in stating what we mean by
"natural number" in the context of this axiom.
This construction-dependent theorem should not be referenced directly; instead, use ax-arch 6762. (Contributed by Jim Kingdon, 22-Apr-2020.) (New usage is discouraged.) |
Axiom | ax-cnex 6734 | The complex numbers form a set. Proofs should normally use cnex 6763 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-resscn 6735 | The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, justified by theorem axresscn 6706. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-1cn 6736 | 1 is a complex number. Axiom for real and complex numbers, justified by theorem ax1cn 6707. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-1re 6737 | 1 is a real number. Axiom for real and complex numbers, justified by theorem ax1re 6708. Proofs should use 1re 6784 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.) |
Axiom | ax-icn 6738 | is a complex number. Axiom for real and complex numbers, justified by theorem axicn 6709. (Contributed by NM, 1-Mar-1995.) |
Axiom | ax-addcl 6739 | Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by theorem axaddcl 6710. Proofs should normally use addcl 6764 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 6740 | Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axaddrcl 6711. Proofs should normally use readdcl 6765 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulcl 6741 | Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by theorem axmulcl 6712. Proofs should normally use mulcl 6766 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulrcl 6742 | Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 6713. Proofs should normally use remulcl 6767 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-addcom 6743 | Addition commutes. Axiom for real and complex numbers, justified by theorem axaddcom 6714. Proofs should normally use addcom 6907 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.) |
Axiom | ax-mulcom 6744 | Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by theorem axmulcom 6715. Proofs should normally use mulcom 6768 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-addass 6745 | Addition of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axaddass 6716. Proofs should normally use addass 6769 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-mulass 6746 | Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 6717. Proofs should normally use mulass 6770 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-distr 6747 | Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, justified by theorem axdistr 6718. Proofs should normally use adddi 6771 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Axiom | ax-i2m1 6748 | i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, justified by theorem axi2m1 6719. (Contributed by NM, 29-Jan-1995.) |
Theorem | ax-0lt1 6749 | 0 is less than 1. Axiom for real and complex numbers, justified by theorem ax0lt1 6720. Proofs should normally use 0lt1 6898 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.) |
Axiom | ax-1rid 6750 | is an identity element for real multiplication. Axiom for real and complex numbers, justified by theorem ax1rid 6721. (Contributed by NM, 29-Jan-1995.) |
Axiom | ax-0id 6751 |
is an identity element
for real addition. Axiom for real and
complex numbers, justified by theorem ax0id 6722.
Proofs should normally use addid1 6908 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.) |
Axiom | ax-rnegex 6752* | Existence of negative of real number. Axiom for real and complex numbers, justified by theorem axrnegex 6723. (Contributed by Eric Schmidt, 21-May-2007.) |
Axiom | ax-precex 6753* | Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by theorem axprecex 6724. (Contributed by Jim Kingdon, 6-Feb-2020.) |
Axiom | ax-cnre 6754* | A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom for real and complex numbers, justified by theorem axcnre 6725. For naming consistency, use cnre 6781 for new proofs. (New usage is discouraged.) (Contributed by NM, 9-May-1999.) |
Axiom | ax-pre-ltirr 6755 | Real number less-than is irreflexive. Axiom for real and complex numbers, justified by theorem ax-pre-ltirr 6755. (Contributed by Jim Kingdon, 12-Jan-2020.) |
Axiom | ax-pre-ltwlin 6756 | Real number less-than is weakly linear. Axiom for real and complex numbers, justified by theorem axpre-ltwlin 6727. (Contributed by Jim Kingdon, 12-Jan-2020.) |
Axiom | ax-pre-lttrn 6757 | Ordering on reals is transitive. Axiom for real and complex numbers, justified by theorem axpre-lttrn 6728. (Contributed by NM, 13-Oct-2005.) |
Axiom | ax-pre-apti 6758 | Apartness of reals is tight. Axiom for real and complex numbers, justified by theorem axpre-apti 6729. (Contributed by Jim Kingdon, 29-Jan-2020.) |
Axiom | ax-pre-ltadd 6759 | Ordering property of addition on reals. Axiom for real and complex numbers, justified by theorem axpre-ltadd 6730. (Contributed by NM, 13-Oct-2005.) |
Axiom | ax-pre-mulgt0 6760 | The product of two positive reals is positive. Axiom for real and complex numbers, justified by theorem axpre-mulgt0 6731. (Contributed by NM, 13-Oct-2005.) |
Axiom | ax-pre-mulext 6761 |
Strong extensionality of multiplication (expressed in terms of ).
Axiom for real and complex numbers, justified by theorem axpre-mulext 6732
(Contributed by Jim Kingdon, 18-Feb-2020.) |
Axiom | ax-arch 6762* |
Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for
real and complex numbers, justified by theorem axarch 6733.
This axiom should not be used directly; instead use arch 7914 (which is the same, but stated in terms of and ). (Contributed by Jim Kingdon, 2-May-2020.) (New usage is discouraged.) |
Theorem | cnex 6763 | Alias for ax-cnex 6734. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Theorem | addcl 6764 | Alias for ax-addcl 6739, for naming consistency with addcli 6789. Use this theorem instead of ax-addcl 6739 or axaddcl 6710. (Contributed by NM, 10-Mar-2008.) |
Theorem | readdcl 6765 | Alias for ax-addrcl 6740, for naming consistency with readdcli 6798. (Contributed by NM, 10-Mar-2008.) |
Theorem | mulcl 6766 | Alias for ax-mulcl 6741, for naming consistency with mulcli 6790. (Contributed by NM, 10-Mar-2008.) |
Theorem | remulcl 6767 | Alias for ax-mulrcl 6742, for naming consistency with remulcli 6799. (Contributed by NM, 10-Mar-2008.) |
Theorem | mulcom 6768 | Alias for ax-mulcom 6744, for naming consistency with mulcomi 6791. (Contributed by NM, 10-Mar-2008.) |
Theorem | addass 6769 | Alias for ax-addass 6745, for naming consistency with addassi 6793. (Contributed by NM, 10-Mar-2008.) |
Theorem | mulass 6770 | Alias for ax-mulass 6746, for naming consistency with mulassi 6794. (Contributed by NM, 10-Mar-2008.) |
Theorem | adddi 6771 | Alias for ax-distr 6747, for naming consistency with adddii 6795. (Contributed by NM, 10-Mar-2008.) |
Theorem | recn 6772 | A real number is a complex number. (Contributed by NM, 10-Aug-1999.) |
Theorem | reex 6773 | The real numbers form a set. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Theorem | reelprrecn 6774 | Reals are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | cnelprrecn 6775 | Complex numbers are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | adddir 6776 | Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.) |
Theorem | 0cn 6777 | 0 is a complex number. (Contributed by NM, 19-Feb-2005.) |
Theorem | 0cnd 6778 | 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | c0ex 6779 | 0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.) |
Theorem | 1ex 6780 | 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.) |
Theorem | cnre 6781* | Alias for ax-cnre 6754, for naming consistency. (Contributed by NM, 3-Jan-2013.) |
Theorem | mulid1 6782 | is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) |
Theorem | mulid2 6783 | Identity law for multiplication. Note: see mulid1 6782 for commuted version. (Contributed by NM, 8-Oct-1999.) |
Theorem | 1re 6784 | is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.) |
Theorem | 0re 6785 | is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) |
Theorem | 0red 6786 | is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.) |
Theorem | mulid1i 6787 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
Theorem | mulid2i 6788 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
Theorem | addcli 6789 | Closure law for addition. (Contributed by NM, 23-Nov-1994.) |
Theorem | mulcli 6790 | Closure law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Theorem | mulcomi 6791 | Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Theorem | mulcomli 6792 | Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Theorem | addassi 6793 | Associative law for addition. (Contributed by NM, 23-Nov-1994.) |
Theorem | mulassi 6794 | Associative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Theorem | adddii 6795 | Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.) |
Theorem | adddiri 6796 | Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.) |
Theorem | recni 6797 | A real number is a complex number. (Contributed by NM, 1-Mar-1995.) |
Theorem | readdcli 6798 | Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.) |
Theorem | remulcli 6799 | Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.) |
Theorem | 1red 6800 | 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
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