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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | caucvgprprlemloc 6801* | Lemma for caucvgprpr 6810. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemcl 6802* | Lemma for caucvgprpr 6810. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-2020.) |
Theorem | caucvgprprlemclphr 6803* | Lemma for caucvgprpr 6810. The putative limit is a positive real. Like caucvgprprlemcl 6802 but without a distinct variable constraint between and . (Contributed by Jim Kingdon, 19-Jun-2021.) |
Theorem | caucvgprprlemexbt 6804* | Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.) |
Theorem | caucvgprprlemexb 6805* | Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.) |
Theorem | caucvgprprlemaddq 6806* | Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.) |
Theorem | caucvgprprlem1 6807* | Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.) |
Theorem | caucvgprprlem2 6808* | Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.) |
Theorem | caucvgprprlemlim 6809* | Lemma for caucvgprpr 6810. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.) |
Theorem | caucvgprpr 6810* |
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 6780 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 6760) might help in understanding this one, as they are slightly simpler but similarly structured. (Contributed by Jim Kingdon, 14-Nov-2020.) |
Definition | df-enr 6811* | 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 6812 | 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 6813* | 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 6814* | 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 6815* | 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 6816 | 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 6817 | 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 6818 | 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 6819 | Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) |
Theorem | enrer 6820 | 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 6821 | Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) |
Theorem | enrex 6822 | The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) |
Theorem | ltrelsr 6823 | Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) |
Theorem | addcmpblnr 6824 | Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) |
Theorem | mulcmpblnrlemg 6825 | Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.) |
Theorem | mulcmpblnr 6826 | Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) |
Theorem | prsrlem1 6827* | Decomposing signed reals into positive reals. Lemma for addsrpr 6830 and mulsrpr 6831. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | addsrmo 6828* | There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | mulsrmo 6829* | There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | addsrpr 6830 | Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Theorem | mulsrpr 6831 | Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Theorem | ltsrprg 6832 | Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.) |
Theorem | gt0srpr 6833 | Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) |
Theorem | 0nsr 6834 | The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) |
Theorem | 0r 6835 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | 1sr 6836 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | m1r 6837 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | addclsr 6838 | Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) |
Theorem | mulclsr 6839 | Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) |
Theorem | addcomsrg 6840 | Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | addasssrg 6841 | Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | mulcomsrg 6842 | Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | mulasssrg 6843 | Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | distrsrg 6844 | Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.) |
Theorem | m1p1sr 6845 | Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) |
Theorem | m1m1sr 6846 | Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) |
Theorem | lttrsr 6847* | Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Theorem | ltposr 6848 | Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Theorem | ltsosr 6849 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) |
Theorem | 0lt1sr 6850 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) |
Theorem | 1ne0sr 6851 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) |
Theorem | 0idsr 6852 | The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) |
Theorem | 1idsr 6853 | 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.) |
Theorem | 00sr 6854 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) |
Theorem | ltasrg 6855 | Ordering property of addition. (Contributed by NM, 10-May-1996.) |
Theorem | pn0sr 6856 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) |
Theorem | negexsr 6857* | Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) |
Theorem | recexgt0sr 6858* | The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) |
Theorem | recexsrlem 6859* | 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 6860 | The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) |
Theorem | ltadd1sr 6861 | Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.) |
Theorem | mulgt0sr 6862 | The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) |
Theorem | aptisr 6863 | Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.) |
Theorem | mulextsr1lem 6864 | Lemma for mulextsr1 6865. (Contributed by Jim Kingdon, 17-Feb-2020.) |
Theorem | mulextsr1 6865 | Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.) |
Theorem | archsr 6866* | 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 6867* | Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | prsrcl 6868 | Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | prsrpos 6869 | Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | prsradd 6870 | Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | prsrlt 6871 | Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | prsrriota 6872* | Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | caucvgsrlemcl 6873* | Lemma for caucvgsr 6886. Terms of the sequence from caucvgsrlemgt1 6879 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.) |
Theorem | caucvgsrlemasr 6874* | Lemma for caucvgsr 6886. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.) |
Theorem | caucvgsrlemfv 6875* | Lemma for caucvgsr 6886. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | caucvgsrlemf 6876* | Lemma for caucvgsr 6886. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.) |
Theorem | caucvgsrlemcau 6877* | Lemma for caucvgsr 6886. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.) |
Theorem | caucvgsrlembound 6878* | Lemma for caucvgsr 6886. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | caucvgsrlemgt1 6879* | Lemma for caucvgsr 6886. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.) |
Theorem | caucvgsrlemoffval 6880* | Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlemofff 6881* | Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlemoffcau 6882* | Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlemoffgt1 6883* | Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlemoffres 6884* | Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlembnd 6885* | Lemma for caucvgsr 6886. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.) |
Theorem | caucvgsr 6886* |
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 6810 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 6885). 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 6881). 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 6810 to get a limit (see caucvgsrlemgt1 6879). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 6879). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 6884). (Contributed by Jim Kingdon, 20-Jun-2021.) |
Syntax | cc 6887 | Class of complex numbers. |
Syntax | cr 6888 | Class of real numbers. |
Syntax | cc0 6889 | Extend class notation to include the complex number 0. |
Syntax | c1 6890 | Extend class notation to include the complex number 1. |
Syntax | ci 6891 | Extend class notation to include the complex number i. |
Syntax | caddc 6892 | Addition on complex numbers. |
Syntax | cltrr 6893 | 'Less than' predicate (defined over real subset of complex numbers). |
Syntax | cmul 6894 | Multiplication on complex numbers. The token is a center dot. |
Definition | df-c 6895 | Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.) |
Definition | df-0 6896 | Define the complex number 0. (Contributed by NM, 22-Feb-1996.) |
Definition | df-1 6897 | Define the complex number 1. (Contributed by NM, 22-Feb-1996.) |
Definition | df-i 6898 | Define the complex number (the imaginary unit). (Contributed by NM, 22-Feb-1996.) |
Definition | df-r 6899 | Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) |
Definition | df-add 6900* | Define addition over complex numbers. (Contributed by NM, 28-May-1995.) |
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