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Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcaucvgprprlemloc 6801* Lemma for caucvgprpr 6810. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.)

Theoremcaucvgprprlemcl 6802* Lemma for caucvgprpr 6810. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-2020.)

Theoremcaucvgprprlemclphr 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.)

Theoremcaucvgprprlemexbt 6804* Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.)

Theoremcaucvgprprlemexb 6805* Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.)

Theoremcaucvgprprlemaddq 6806* Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.)

Theoremcaucvgprprlem1 6807* Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)

Theoremcaucvgprprlem2 6808* Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)

Theoremcaucvgprprlemlim 6809* Lemma for caucvgprpr 6810. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.)

Theoremcaucvgprpr 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.)

Definitiondf-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.)

Definitiondf-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.)

Definitiondf-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.)

Definitiondf-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.)

Definitiondf-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.)

Definitiondf-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.)

Definitiondf-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.)

Definitiondf-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.)

Theoremenrbreq 6819 Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.)

Theoremenrer 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.)

Theoremenreceq 6821 Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.)

Theoremenrex 6822 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.)

Theoremltrelsr 6823 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)

Theoremmulcmpblnrlemg 6825 Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.)

Theoremmulcmpblnr 6826 Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.)

Theoremprsrlem1 6827* Decomposing signed reals into positive reals. Lemma for addsrpr 6830 and mulsrpr 6831. (Contributed by Jim Kingdon, 30-Dec-2019.)

Theoremaddsrmo 6828* There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)

Theoremmulsrmo 6829* There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)

Theoremaddsrpr 6830 Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremmulsrpr 6831 Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremltsrprg 6832 Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.)

Theoremgt0srpr 6833 Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.)

Theorem0nsr 6834 The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.)

Theorem0r 6835 The constant is a signed real. (Contributed by NM, 9-Aug-1995.)

Theorem1sr 6836 The constant is a signed real. (Contributed by NM, 9-Aug-1995.)

Theoremm1r 6837 The constant is a signed real. (Contributed by NM, 9-Aug-1995.)

Theoremaddclsr 6838 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.)

Theoremmulclsr 6839 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.)

Theoremaddcomsrg 6840 Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)

Theoremaddasssrg 6841 Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)

Theoremmulcomsrg 6842 Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)

Theoremmulasssrg 6843 Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)

Theoremdistrsrg 6844 Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.)

Theoremm1p1sr 6845 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.)

Theoremm1m1sr 6846 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.)

Theoremlttrsr 6847* Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.)

Theoremltposr 6848 Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.)

Theoremltsosr 6849 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)

Theorem0lt1sr 6850 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.)

Theorem1ne0sr 6851 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.)

Theorem0idsr 6852 The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.)

Theorem1idsr 6853 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)

Theorem00sr 6854 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)

Theoremltasrg 6855 Ordering property of addition. (Contributed by NM, 10-May-1996.)

Theorempn0sr 6856 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.)

Theoremnegexsr 6857* Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.)

Theoremrecexgt0sr 6858* The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.)

Theoremrecexsrlem 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.)

Theoremaddgt0sr 6860 The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.)

Theoremltadd1sr 6861 Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.)

Theoremmulgt0sr 6862 The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.)

Theoremaptisr 6863 Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)

Theoremmulextsr1lem 6864 Lemma for mulextsr1 6865. (Contributed by Jim Kingdon, 17-Feb-2020.)

Theoremmulextsr1 6865 Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.)

Theoremarchsr 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.)

Theoremsrpospr 6867* Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.)

Theoremprsrcl 6868 Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.)

Theoremprsrpos 6869 Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.)

Theoremprsrlt 6871 Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.)

Theoremprsrriota 6872* Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)

Theoremcaucvgsrlemcl 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.)

Theoremcaucvgsrlemasr 6874* Lemma for caucvgsr 6886. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.)

Theoremcaucvgsrlemfv 6875* Lemma for caucvgsr 6886. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)

Theoremcaucvgsrlemf 6876* Lemma for caucvgsr 6886. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)

Theoremcaucvgsrlemcau 6877* Lemma for caucvgsr 6886. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)

Theoremcaucvgsrlembound 6878* Lemma for caucvgsr 6886. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.)

Theoremcaucvgsrlemgt1 6879* Lemma for caucvgsr 6886. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.)

Theoremcaucvgsrlemoffval 6880* Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)

Theoremcaucvgsrlemofff 6881* Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)

Theoremcaucvgsrlemoffcau 6882* Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)

Theoremcaucvgsrlemoffgt1 6883* Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)

Theoremcaucvgsrlemoffres 6884* Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)

Theoremcaucvgsrlembnd 6885* Lemma for caucvgsr 6886. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.)

Theoremcaucvgsr 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.)

Syntaxcc 6887 Class of complex numbers.

Syntaxcr 6888 Class of real numbers.

Syntaxcc0 6889 Extend class notation to include the complex number 0.

Syntaxc1 6890 Extend class notation to include the complex number 1.

Syntaxci 6891 Extend class notation to include the complex number i.

Syntaxcltrr 6893 'Less than' predicate (defined over real subset of complex numbers).

Syntaxcmul 6894 Multiplication on complex numbers. The token is a center dot.

Definitiondf-c 6895 Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.)

Definitiondf-0 6896 Define the complex number 0. (Contributed by NM, 22-Feb-1996.)

Definitiondf-1 6897 Define the complex number 1. (Contributed by NM, 22-Feb-1996.)

Definitiondf-i 6898 Define the complex number (the imaginary unit). (Contributed by NM, 22-Feb-1996.)

Definitiondf-r 6899 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.)