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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | caucvgprprlemlim 6901* | Lemma for caucvgprpr 6902. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.) |
Theorem | caucvgprpr 6902* |
A Cauchy sequence of positive reals with a modulus of convergence
converges to a positive real. This is basically Corollary 11.2.13 of
[HoTT], p. (varies) (one key difference
being that this is for
positive reals rather than signed reals). Also, the HoTT book theorem
has a modulus of convergence (that is, a rate of convergence)
specified by (11.2.9) in HoTT whereas this theorem fixes the rate of
convergence to say that all terms after the nth term must be within
of the nth term (it should later be able
to prove versions
of this theorem with a different fixed rate or a modulus of
convergence supplied as a hypothesis). We also specify that every
term needs to be larger than a given value , to avoid the case
where we have positive terms which "converge" to zero (which
is not a
positive real).
This is similar to caucvgpr 6872 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 6852) might help in understanding this one, as they are slightly simpler but similarly structured. (Contributed by Jim Kingdon, 14-Nov-2020.) |
Definition | df-enr 6903* | Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) |
Definition | df-nr 6904 | Define class of signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) |
Definition | df-plr 6905* | Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) |
Definition | df-mr 6906* | Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) |
Definition | df-ltr 6907* | Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.4 of [Gleason] p. 127. (Contributed by NM, 14-Feb-1996.) |
Definition | df-0r 6908 | Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.) |
Definition | df-1r 6909 | Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.) |
Definition | df-m1r 6910 | Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 9-Aug-1995.) |
Theorem | enrbreq 6911 | Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) |
Theorem | enrer 6912 | The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) |
Theorem | enreceq 6913 | Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) |
Theorem | enrex 6914 | The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) |
Theorem | ltrelsr 6915 | Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) |
Theorem | addcmpblnr 6916 | Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) |
Theorem | mulcmpblnrlemg 6917 | Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.) |
Theorem | mulcmpblnr 6918 | Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) |
Theorem | prsrlem1 6919* | Decomposing signed reals into positive reals. Lemma for addsrpr 6922 and mulsrpr 6923. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | addsrmo 6920* | There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | mulsrmo 6921* | There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | addsrpr 6922 | Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Theorem | mulsrpr 6923 | Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Theorem | ltsrprg 6924 | Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.) |
Theorem | gt0srpr 6925 | Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) |
Theorem | 0nsr 6926 | The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) |
Theorem | 0r 6927 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | 1sr 6928 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | m1r 6929 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | addclsr 6930 | Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) |
Theorem | mulclsr 6931 | Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) |
Theorem | addcomsrg 6932 | Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | addasssrg 6933 | Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | mulcomsrg 6934 | Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | mulasssrg 6935 | Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | distrsrg 6936 | Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.) |
Theorem | m1p1sr 6937 | Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) |
Theorem | m1m1sr 6938 | Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) |
Theorem | lttrsr 6939* | Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Theorem | ltposr 6940 | Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Theorem | ltsosr 6941 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) |
Theorem | 0lt1sr 6942 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) |
Theorem | 1ne0sr 6943 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) |
Theorem | 0idsr 6944 | The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) |
Theorem | 1idsr 6945 | 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.) |
Theorem | 00sr 6946 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) |
Theorem | ltasrg 6947 | Ordering property of addition. (Contributed by NM, 10-May-1996.) |
Theorem | pn0sr 6948 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) |
Theorem | negexsr 6949* | Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) |
Theorem | recexgt0sr 6950* | The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) |
Theorem | recexsrlem 6951* | The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) |
Theorem | addgt0sr 6952 | The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) |
Theorem | ltadd1sr 6953 | Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.) |
Theorem | mulgt0sr 6954 | The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) |
Theorem | aptisr 6955 | Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.) |
Theorem | mulextsr1lem 6956 | Lemma for mulextsr1 6957. (Contributed by Jim Kingdon, 17-Feb-2020.) |
Theorem | mulextsr1 6957 | Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.) |
Theorem | archsr 6958* | For any signed real, there is an integer that is greater than it. This is also known as the "archimedean property". The expression , is the embedding of the positive integer into the signed reals. (Contributed by Jim Kingdon, 23-Apr-2020.) |
Theorem | srpospr 6959* | Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | prsrcl 6960 | Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | prsrpos 6961 | Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | prsradd 6962 | Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | prsrlt 6963 | Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | prsrriota 6964* | Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | caucvgsrlemcl 6965* | Lemma for caucvgsr 6978. Terms of the sequence from caucvgsrlemgt1 6971 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.) |
Theorem | caucvgsrlemasr 6966* | Lemma for caucvgsr 6978. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.) |
Theorem | caucvgsrlemfv 6967* | Lemma for caucvgsr 6978. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | caucvgsrlemf 6968* | Lemma for caucvgsr 6978. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.) |
Theorem | caucvgsrlemcau 6969* | Lemma for caucvgsr 6978. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.) |
Theorem | caucvgsrlembound 6970* | Lemma for caucvgsr 6978. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | caucvgsrlemgt1 6971* | Lemma for caucvgsr 6978. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.) |
Theorem | caucvgsrlemoffval 6972* | Lemma for caucvgsr 6978. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlemofff 6973* | Lemma for caucvgsr 6978. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlemoffcau 6974* | Lemma for caucvgsr 6978. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlemoffgt1 6975* | Lemma for caucvgsr 6978. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlemoffres 6976* | Lemma for caucvgsr 6978. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlembnd 6977* | Lemma for caucvgsr 6978. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.) |
Theorem | caucvgsr 6978* |
A Cauchy sequence of signed reals with a modulus of convergence
converges to a signed real. This is basically Corollary 11.2.13 of
[HoTT], p. (varies). The HoTT book
theorem has a modulus of
convergence (that is, a rate of convergence) specified by (11.2.9) in
HoTT whereas this theorem fixes the rate of convergence to say that
all terms after the nth term must be within of the nth
term
(it should later be able to prove versions of this theorem with a
different fixed rate or a modulus of convergence supplied as a
hypothesis).
This is similar to caucvgprpr 6902 but is for signed reals rather than positive reals. Here is an outline of how we prove it: 1. Choose a lower bound for the sequence (see caucvgsrlembnd 6977). 2. Offset each element of the sequence so that each element of the resulting sequence is greater than one (greater than zero would not suffice, because the limit as well as the elements of the sequence need to be positive) (see caucvgsrlemofff 6973). 3. Since a signed real (element of ) which is greater than zero can be mapped to a positive real (element of ), perform that mapping on each element of the sequence and invoke caucvgprpr 6902 to get a limit (see caucvgsrlemgt1 6971). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 6971). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 6976). (Contributed by Jim Kingdon, 20-Jun-2021.) |
Syntax | cc 6979 | Class of complex numbers. |
Syntax | cr 6980 | Class of real numbers. |
Syntax | cc0 6981 | Extend class notation to include the complex number 0. |
Syntax | c1 6982 | Extend class notation to include the complex number 1. |
Syntax | ci 6983 | Extend class notation to include the complex number i. |
Syntax | caddc 6984 | Addition on complex numbers. |
Syntax | cltrr 6985 | 'Less than' predicate (defined over real subset of complex numbers). |
Syntax | cmul 6986 | Multiplication on complex numbers. The token is a center dot. |
Definition | df-c 6987 | Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.) |
Definition | df-0 6988 | Define the complex number 0. (Contributed by NM, 22-Feb-1996.) |
Definition | df-1 6989 | Define the complex number 1. (Contributed by NM, 22-Feb-1996.) |
Definition | df-i 6990 | Define the complex number (the imaginary unit). (Contributed by NM, 22-Feb-1996.) |
Definition | df-r 6991 | Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) |
Definition | df-add 6992* | Define addition over complex numbers. (Contributed by NM, 28-May-1995.) |
Definition | df-mul 6993* | Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) |
Definition | df-lt 6994* | Define 'less than' on the real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) |
Theorem | opelcn 6995 | Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) |
Theorem | opelreal 6996 | Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) |
Theorem | elreal 6997* | Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) |
Theorem | elrealeu 6998* | The real number mapping in elreal 6997 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.) |
Theorem | elreal2 6999 | Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) |
Theorem | 0ncn 7000 | 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.) |
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