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Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcaucvgprprlemdisj 7601* Lemma for caucvgprpr 7611. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.)

Theoremcaucvgprprlemloc 7602* Lemma for caucvgprpr 7611. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.)

Theoremcaucvgprprlemcl 7603* Lemma for caucvgprpr 7611. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-2020.)

Theoremcaucvgprprlemclphr 7604* Lemma for caucvgprpr 7611. The putative limit is a positive real. Like caucvgprprlemcl 7603 but without a distinct variable constraint between and . (Contributed by Jim Kingdon, 19-Jun-2021.)

Theoremcaucvgprprlemexbt 7605* Lemma for caucvgprpr 7611. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.)

Theoremcaucvgprprlemexb 7606* Lemma for caucvgprpr 7611. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.)

Theoremcaucvgprprlemaddq 7607* Lemma for caucvgprpr 7611. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.)

Theoremcaucvgprprlem1 7608* Lemma for caucvgprpr 7611. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)

Theoremcaucvgprprlem2 7609* Lemma for caucvgprpr 7611. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)

Theoremcaucvgprprlemlim 7610* Lemma for caucvgprpr 7611. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.)

Theoremcaucvgprpr 7611* 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 7581 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 7561) might help in understanding this one, as they are slightly simpler but similarly structured. (Contributed by Jim Kingdon, 14-Nov-2020.)

Theoremsuplocexprlemell 7612* Lemma for suplocexpr 7624. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)

Theoremsuplocexprlem2b 7613 Lemma for suplocexpr 7624. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)

Theoremsuplocexprlemss 7614* Lemma for suplocexpr 7624. is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.)

Theoremsuplocexprlemml 7615* Lemma for suplocexpr 7624. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)

Theoremsuplocexprlemrl 7616* Lemma for suplocexpr 7624. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)

Theoremsuplocexprlemmu 7617* Lemma for suplocexpr 7624. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)

Theoremsuplocexprlemru 7618* Lemma for suplocexpr 7624. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)

Theoremsuplocexprlemdisj 7619* Lemma for suplocexpr 7624. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.)

Theoremsuplocexprlemloc 7620* Lemma for suplocexpr 7624. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.)

Theoremsuplocexprlemex 7621* Lemma for suplocexpr 7624. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.)

Theoremsuplocexprlemub 7622* Lemma for suplocexpr 7624. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)

Theoremsuplocexprlemlub 7623* Lemma for suplocexpr 7624. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)

Theoremsuplocexpr 7624* An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.)

Definitiondf-enr 7625* 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 7626 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 7627* 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 7628* 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 7629* 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 7630 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 7631 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 7632 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 7633 Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.)

Theoremenrer 7634 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 7635 Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.)

Theoremenrex 7636 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.)

Theoremltrelsr 7637 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)

Theoremmulcmpblnrlemg 7639 Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.)

Theoremmulcmpblnr 7640 Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.)

Theoremprsrlem1 7641* Decomposing signed reals into positive reals. Lemma for addsrpr 7644 and mulsrpr 7645. (Contributed by Jim Kingdon, 30-Dec-2019.)

Theoremaddsrmo 7642* There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)

Theoremmulsrmo 7643* There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)

Theoremaddsrpr 7644 Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremmulsrpr 7645 Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremltsrprg 7646 Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.)

Theoremgt0srpr 7647 Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.)

Theorem0nsr 7648 The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.)

Theorem0r 7649 The constant is a signed real. (Contributed by NM, 9-Aug-1995.)

Theorem1sr 7650 The constant is a signed real. (Contributed by NM, 9-Aug-1995.)

Theoremm1r 7651 The constant is a signed real. (Contributed by NM, 9-Aug-1995.)

Theoremaddclsr 7652 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.)

Theoremmulclsr 7653 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.)

Theoremaddcomsrg 7654 Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)

Theoremaddasssrg 7655 Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)

Theoremmulcomsrg 7656 Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)

Theoremmulasssrg 7657 Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)

Theoremdistrsrg 7658 Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.)

Theoremm1p1sr 7659 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.)

Theoremm1m1sr 7660 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.)

Theoremlttrsr 7661* Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.)

Theoremltposr 7662 Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.)

Theoremltsosr 7663 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)

Theorem0lt1sr 7664 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.)

Theorem1ne0sr 7665 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.)

Theorem0idsr 7666 The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.)

Theorem1idsr 7667 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)

Theorem00sr 7668 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)

Theoremltasrg 7669 Ordering property of addition. (Contributed by NM, 10-May-1996.)

Theorempn0sr 7670 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.)

Theoremnegexsr 7671* Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.)

Theoremrecexgt0sr 7672* The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.)

Theoremrecexsrlem 7673* 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 7674 The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.)

Theoremltadd1sr 7675 Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.)

Theoremltm1sr 7676 Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.)

Theoremmulgt0sr 7677 The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.)

Theoremaptisr 7678 Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)

Theoremmulextsr1lem 7679 Lemma for mulextsr1 7680. (Contributed by Jim Kingdon, 17-Feb-2020.)

Theoremmulextsr1 7680 Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.)

Theoremarchsr 7681* 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 7682* Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.)

Theoremprsrcl 7683 Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.)

Theoremprsrpos 7684 Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.)

Theoremprsrlt 7686 Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.)

Theoremprsrriota 7687* Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)

Theoremcaucvgsrlemcl 7688* Lemma for caucvgsr 7701. Terms of the sequence from caucvgsrlemgt1 7694 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.)

Theoremcaucvgsrlemasr 7689* Lemma for caucvgsr 7701. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.)

Theoremcaucvgsrlemfv 7690* Lemma for caucvgsr 7701. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)

Theoremcaucvgsrlemf 7691* Lemma for caucvgsr 7701. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)

Theoremcaucvgsrlemcau 7692* Lemma for caucvgsr 7701. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)

Theoremcaucvgsrlembound 7693* Lemma for caucvgsr 7701. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.)

Theoremcaucvgsrlemgt1 7694* Lemma for caucvgsr 7701. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.)

Theoremcaucvgsrlemoffval 7695* Lemma for caucvgsr 7701. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)

Theoremcaucvgsrlemofff 7696* Lemma for caucvgsr 7701. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)

Theoremcaucvgsrlemoffcau 7697* Lemma for caucvgsr 7701. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)

Theoremcaucvgsrlemoffgt1 7698* Lemma for caucvgsr 7701. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)

Theoremcaucvgsrlemoffres 7699* Lemma for caucvgsr 7701. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)

Theoremcaucvgsrlembnd 7700* Lemma for caucvgsr 7701. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.)

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