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Theorem List for Intuitionistic Logic Explorer - 6701-6800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltexprlemopu 6701* The upper cut of our constructed difference is open. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 21-Dec-2019.)

Theoremltexprlemupu 6702* The upper cut of our constructed difference is upper. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 21-Dec-2019.)

Theoremltexprlemrnd 6703* Our constructed difference is rounded. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 17-Dec-2019.)

Theoremltexprlemdisj 6704* Our constructed difference is disjoint. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 17-Dec-2019.)

Theoremltexprlemloc 6705* Our constructed difference is located. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 17-Dec-2019.)

Theoremltexprlempr 6706* Our constructed difference is a positive real. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 17-Dec-2019.)

Theoremltexprlemfl 6707* Lemma for ltexpri 6711. One directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)

Theoremltexprlemrl 6708* Lemma for ltexpri 6711. Reverse directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)

Theoremltexprlemfu 6709* Lemma for ltexpri 6711. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)

Theoremltexprlemru 6710* Lemma for ltexpri 6711. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)

Theoremltexpri 6711* Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)

Theoremaddcanprleml 6712 Lemma for addcanprg 6714. (Contributed by Jim Kingdon, 25-Dec-2019.)

Theoremaddcanprlemu 6713 Lemma for addcanprg 6714. (Contributed by Jim Kingdon, 25-Dec-2019.)

Theoremaddcanprg 6714 Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by Jim Kingdon, 24-Dec-2019.)

Theoremlteupri 6715* The difference from ltexpri 6711 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)

Theoremltaprlem 6716 Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.)

Theoremltaprg 6717 Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.)

Theoremprplnqu 6718* Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.)

Theoremaddextpr 6719 Strong extensionality of addition (ordering version). This is similar to addext 7601 but for positive reals and based on less-than rather than apartness. (Contributed by Jim Kingdon, 17-Feb-2020.)

Theoremrecexprlemell 6720* Membership in the lower cut of . Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)

Theoremrecexprlemelu 6721* Membership in the upper cut of . Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)

Theoremrecexprlemm 6722* is inhabited. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)

Theoremrecexprlemopl 6723* The lower cut of is open. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 28-Dec-2019.)

Theoremrecexprlemlol 6724* The lower cut of is lower. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 28-Dec-2019.)

Theoremrecexprlemopu 6725* The upper cut of is open. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 28-Dec-2019.)

Theoremrecexprlemupu 6726* The upper cut of is upper. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 28-Dec-2019.)

Theoremrecexprlemrnd 6727* is rounded. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)

Theoremrecexprlemdisj 6728* is disjoint. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)

Theoremrecexprlemloc 6729* is located. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)

Theoremrecexprlempr 6730* is a positive real. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)

Theoremrecexprlem1ssl 6731* The lower cut of one is a subset of the lower cut of . Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)

Theoremrecexprlem1ssu 6732* The upper cut of one is a subset of the upper cut of . Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)

Theoremrecexprlemss1l 6733* The lower cut of is a subset of the lower cut of one. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)

Theoremrecexprlemss1u 6734* The upper cut of is a subset of the upper cut of one. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)

Theoremrecexprlemex 6735* is the reciprocal of . Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)

Theoremrecexpr 6736* The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)

Theoremaptiprleml 6737 Lemma for aptipr 6739. (Contributed by Jim Kingdon, 28-Jan-2020.)

Theoremaptiprlemu 6738 Lemma for aptipr 6739. (Contributed by Jim Kingdon, 28-Jan-2020.)

Theoremaptipr 6739 Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.)

Theoremltmprr 6740 Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.)

Theoremarchpr 6741* For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer is embedded into the reals as described at nnprlu 6651. (Contributed by Jim Kingdon, 22-Apr-2020.)

Theoremcaucvgprlemcanl 6742* Lemma for cauappcvgprlemladdrl 6755. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.)

Theoremcauappcvgprlemm 6743* Lemma for cauappcvgpr 6760. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.)

Theoremcauappcvgprlemopl 6744* Lemma for cauappcvgpr 6760. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)

Theoremcauappcvgprlemlol 6745* Lemma for cauappcvgpr 6760. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 4-Aug-2020.)

Theoremcauappcvgprlemopu 6746* Lemma for cauappcvgpr 6760. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)

Theoremcauappcvgprlemupu 6747* Lemma for cauappcvgpr 6760. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 4-Aug-2020.)

Theoremcauappcvgprlemrnd 6748* Lemma for cauappcvgpr 6760. The putative limit is rounded. (Contributed by Jim Kingdon, 18-Jul-2020.)

Theoremcauappcvgprlemdisj 6749* Lemma for cauappcvgpr 6760. The putative limit is disjoint. (Contributed by Jim Kingdon, 18-Jul-2020.)

Theoremcauappcvgprlemloc 6750* Lemma for cauappcvgpr 6760. The putative limit is located. (Contributed by Jim Kingdon, 18-Jul-2020.)

Theoremcauappcvgprlemcl 6751* Lemma for cauappcvgpr 6760. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.)

Theoremcauappcvgprlemladdfu 6752* Lemma for cauappcvgprlemladd 6756. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)

Theoremcauappcvgprlemladdfl 6753* Lemma for cauappcvgprlemladd 6756. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)

Theoremcauappcvgprlemladdru 6754* Lemma for cauappcvgprlemladd 6756. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)

Theoremcauappcvgprlemladdrl 6755* Lemma for cauappcvgprlemladd 6756. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)

Theoremcauappcvgprlemladd 6756* Lemma for cauappcvgpr 6760. This takes and offsets it by the positive fraction . (Contributed by Jim Kingdon, 23-Jun-2020.)

Theoremcauappcvgprlem1 6757* Lemma for cauappcvgpr 6760. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)

Theoremcauappcvgprlem2 6758* Lemma for cauappcvgpr 6760. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)

Theoremcauappcvgprlemlim 6759* Lemma for cauappcvgpr 6760. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-2020.)

Theoremcauappcvgpr 6760* A Cauchy approximation has a limit. A Cauchy approximation, here , is similar to a Cauchy sequence but is indexed by the desired tolerance (that is, how close together terms needs to be) rather than by natural numbers. This is basically Theorem 11.2.12 of [HoTT], p. (varies) with a few differences such as that we are proving the existence of a limit without anything about how fast it converges (that is, mere existence instead of existence, in HoTT terms), and that the codomain of is rather than . We also specify that every term needs to be larger than a fraction , to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This proof (including its lemmas) is similar to the proofs of caucvgpr 6780 and caucvgprpr 6810 but is somewhat simpler, so reading this one first may help understanding the other two.

(Contributed by Jim Kingdon, 19-Jun-2020.)

Theoremarchrecnq 6761* Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.)

Theoremarchrecpr 6762* Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.)

Theoremcaucvgprlemk 6763 Lemma for caucvgpr 6780. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.)

Theoremcaucvgprlemnkj 6764* Lemma for caucvgpr 6780. Part of disjointness. (Contributed by Jim Kingdon, 23-Oct-2020.)

Theoremcaucvgprlemnbj 6765* Lemma for caucvgpr 6780. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-Oct-2020.)

Theoremcaucvgprlemm 6766* Lemma for caucvgpr 6780. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.)

Theoremcaucvgprlemopl 6767* Lemma for caucvgpr 6780. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)

Theoremcaucvgprlemlol 6768* Lemma for caucvgpr 6780. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-2020.)

Theoremcaucvgprlemopu 6769* Lemma for caucvgpr 6780. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)

Theoremcaucvgprlemupu 6770* Lemma for caucvgpr 6780. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 20-Oct-2020.)

Theoremcaucvgprlemrnd 6771* Lemma for caucvgpr 6780. The putative limit is rounded. (Contributed by Jim Kingdon, 27-Sep-2020.)

Theoremcaucvgprlemdisj 6772* Lemma for caucvgpr 6780. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.)

Theoremcaucvgprlemloc 6773* Lemma for caucvgpr 6780. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.)

Theoremcaucvgprlemcl 6774* Lemma for caucvgpr 6780. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.)

Theoremcaucvgprlemladdfu 6775* Lemma for caucvgpr 6780. Adding after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)

Theoremcaucvgprlemladdrl 6776* Lemma for caucvgpr 6780. Adding after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 8-Oct-2020.)

Theoremcaucvgprlem1 6777* Lemma for caucvgpr 6780. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)

Theoremcaucvgprlem2 6778* Lemma for caucvgpr 6780. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)

Theoremcaucvgprlemlim 6779* Lemma for caucvgpr 6780. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.)

Theoremcaucvgpr 6780* A Cauchy sequence of positive fractions 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 fraction , to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This proof (including its lemmas) is similar to the proofs of cauappcvgpr 6760 and caucvgprpr 6810. Reading cauappcvgpr 6760 first (the simplest of the three) might help understanding the other two.

(Contributed by Jim Kingdon, 18-Jun-2020.)

Theoremcaucvgprprlemk 6781* Lemma for caucvgprpr 6810. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.)

Theoremcaucvgprprlemloccalc 6782* Lemma for caucvgprpr 6810. Rearranging some expressions for caucvgprprlemloc 6801. (Contributed by Jim Kingdon, 8-Feb-2021.)

Theoremcaucvgprprlemell 6783* Lemma for caucvgprpr 6810. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.)

Theoremcaucvgprprlemelu 6784* Lemma for caucvgprpr 6810. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-Jan-2021.)

Theoremcaucvgprprlemcbv 6785* Lemma for caucvgprpr 6810. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.)

Theoremcaucvgprprlemval 6786* Lemma for caucvgprpr 6810. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.)

Theoremcaucvgprprlemnkltj 6787* Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)

Theoremcaucvgprprlemnkeqj 6788* Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)

Theoremcaucvgprprlemnjltk 6789* Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)

Theoremcaucvgprprlemnkj 6790* Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 20-Jan-2021.)

Theoremcaucvgprprlemnbj 6791* Lemma for caucvgprpr 6810. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-2021.)

Theoremcaucvgprprlemml 6792* Lemma for caucvgprpr 6810. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)

Theoremcaucvgprprlemmu 6793* Lemma for caucvgprpr 6810. The upper cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)

Theoremcaucvgprprlemm 6794* Lemma for caucvgprpr 6810. The putative limit is inhabited. (Contributed by Jim Kingdon, 21-Dec-2020.)

Theoremcaucvgprprlemopl 6795* Lemma for caucvgprpr 6810. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)

Theoremcaucvgprprlemlol 6796* Lemma for caucvgprpr 6810. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.)

Theoremcaucvgprprlemopu 6797* Lemma for caucvgprpr 6810. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)

Theoremcaucvgprprlemupu 6798* Lemma for caucvgprpr 6810. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 21-Dec-2020.)

Theoremcaucvgprprlemrnd 6799* Lemma for caucvgprpr 6810. The putative limit is rounded. (Contributed by Jim Kingdon, 21-Dec-2020.)

Theoremcaucvgprprlemdisj 6800* Lemma for caucvgprpr 6810. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.)

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