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Type | Label | Description |
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Statement | ||
Theorem | ltexprlemlol 7601* | The lower cut of our constructed difference is lower. Lemma for ltexpri 7612. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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Theorem | ltexprlemopu 7602* | The upper cut of our constructed difference is open. Lemma for ltexpri 7612. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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Theorem | ltexprlemupu 7603* | The upper cut of our constructed difference is upper. Lemma for ltexpri 7612. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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Theorem | ltexprlemrnd 7604* | Our constructed difference is rounded. Lemma for ltexpri 7612. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemdisj 7605* | Our constructed difference is disjoint. Lemma for ltexpri 7612. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemloc 7606* | Our constructed difference is located. Lemma for ltexpri 7612. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlempr 7607* | Our constructed difference is a positive real. Lemma for ltexpri 7612. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemfl 7608* | Lemma for ltexpri 7612. One direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemrl 7609* | Lemma for ltexpri 7612. Reverse direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemfu 7610* | Lemma for ltexpri 7612. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemru 7611* | Lemma for ltexpri 7612. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexpri 7612* | Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) |
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Theorem | addcanprleml 7613 | Lemma for addcanprg 7615. (Contributed by Jim Kingdon, 25-Dec-2019.) |
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Theorem | addcanprlemu 7614 | Lemma for addcanprg 7615. (Contributed by Jim Kingdon, 25-Dec-2019.) |
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Theorem | addcanprg 7615 | Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by Jim Kingdon, 24-Dec-2019.) |
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Theorem | lteupri 7616* | The difference from ltexpri 7612 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.) |
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Theorem | ltaprlem 7617 | Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) |
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Theorem | ltaprg 7618 | Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.) |
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Theorem | prplnqu 7619* | Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.) |
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Theorem | addextpr 7620 | Strong extensionality of addition (ordering version). This is similar to addext 8567 but for positive reals and based on less-than rather than apartness. (Contributed by Jim Kingdon, 17-Feb-2020.) |
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Theorem | recexprlemell 7621* |
Membership in the lower cut of ![]() |
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Theorem | recexprlemelu 7622* |
Membership in the upper cut of ![]() |
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Theorem | recexprlemm 7623* |
![]() |
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Theorem | recexprlemopl 7624* |
The lower cut of ![]() |
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Theorem | recexprlemlol 7625* |
The lower cut of ![]() |
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Theorem | recexprlemopu 7626* |
The upper cut of ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | recexprlemupu 7627* |
The upper cut of ![]() |
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Theorem | recexprlemrnd 7628* |
![]() |
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Theorem | recexprlemdisj 7629* |
![]() |
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Theorem | recexprlemloc 7630* |
![]() |
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Theorem | recexprlempr 7631* |
![]() |
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Theorem | recexprlem1ssl 7632* |
The lower cut of one is a subset of the lower cut of ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | recexprlem1ssu 7633* |
The upper cut of one is a subset of the upper cut of ![]() ![]() ![]() |
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Theorem | recexprlemss1l 7634* |
The lower cut of ![]() ![]() ![]() |
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Theorem | recexprlemss1u 7635* |
The upper cut of ![]() ![]() ![]() |
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Theorem | recexprlemex 7636* |
![]() ![]() |
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Theorem | recexpr 7637* | 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.) |
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Theorem | aptiprleml 7638 | Lemma for aptipr 7640. (Contributed by Jim Kingdon, 28-Jan-2020.) |
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Theorem | aptiprlemu 7639 | Lemma for aptipr 7640. (Contributed by Jim Kingdon, 28-Jan-2020.) |
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Theorem | aptipr 7640 | Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.) |
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Theorem | ltmprr 7641 | Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.) |
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Theorem | archpr 7642* |
For any positive real, there is an integer that is greater than it.
This is also known as the "archimedean property". The integer
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Theorem | caucvgprlemcanl 7643* | Lemma for cauappcvgprlemladdrl 7656. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.) |
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Theorem | cauappcvgprlemm 7644* | Lemma for cauappcvgpr 7661. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.) |
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Theorem | cauappcvgprlemopl 7645* | Lemma for cauappcvgpr 7661. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.) |
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Theorem | cauappcvgprlemlol 7646* | Lemma for cauappcvgpr 7661. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 4-Aug-2020.) |
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Theorem | cauappcvgprlemopu 7647* | Lemma for cauappcvgpr 7661. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.) |
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Theorem | cauappcvgprlemupu 7648* | Lemma for cauappcvgpr 7661. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 4-Aug-2020.) |
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Theorem | cauappcvgprlemrnd 7649* | Lemma for cauappcvgpr 7661. The putative limit is rounded. (Contributed by Jim Kingdon, 18-Jul-2020.) |
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Theorem | cauappcvgprlemdisj 7650* | Lemma for cauappcvgpr 7661. The putative limit is disjoint. (Contributed by Jim Kingdon, 18-Jul-2020.) |
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Theorem | cauappcvgprlemloc 7651* | Lemma for cauappcvgpr 7661. The putative limit is located. (Contributed by Jim Kingdon, 18-Jul-2020.) |
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Theorem | cauappcvgprlemcl 7652* | Lemma for cauappcvgpr 7661. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.) |
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Theorem | cauappcvgprlemladdfu 7653* | Lemma for cauappcvgprlemladd 7657. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
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Theorem | cauappcvgprlemladdfl 7654* | Lemma for cauappcvgprlemladd 7657. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
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Theorem | cauappcvgprlemladdru 7655* | Lemma for cauappcvgprlemladd 7657. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
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Theorem | cauappcvgprlemladdrl 7656* | Lemma for cauappcvgprlemladd 7657. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
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Theorem | cauappcvgprlemladd 7657* |
Lemma for cauappcvgpr 7661. This takes ![]() ![]() |
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Theorem | cauappcvgprlem1 7658* | Lemma for cauappcvgpr 7661. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.) |
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Theorem | cauappcvgprlem2 7659* | Lemma for cauappcvgpr 7661. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.) |
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Theorem | cauappcvgprlemlim 7660* | Lemma for cauappcvgpr 7661. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-2020.) |
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Theorem | cauappcvgpr 7661* |
A Cauchy approximation has a limit. A Cauchy approximation, here
![]() ![]() ![]() ![]() ![]() This proof (including its lemmas) is similar to the proofs of caucvgpr 7681 and caucvgprpr 7711 but is somewhat simpler, so reading this one first may help understanding the other two. (Contributed by Jim Kingdon, 19-Jun-2020.) |
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Theorem | archrecnq 7662* | Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.) |
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Theorem | archrecpr 7663* | Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.) |
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Theorem | caucvgprlemk 7664 | Lemma for caucvgpr 7681. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.) |
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Theorem | caucvgprlemnkj 7665* | Lemma for caucvgpr 7681. Part of disjointness. (Contributed by Jim Kingdon, 23-Oct-2020.) |
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Theorem | caucvgprlemnbj 7666* | Lemma for caucvgpr 7681. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-Oct-2020.) |
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Theorem | caucvgprlemm 7667* | Lemma for caucvgpr 7681. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.) |
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Theorem | caucvgprlemopl 7668* | Lemma for caucvgpr 7681. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.) |
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Theorem | caucvgprlemlol 7669* | Lemma for caucvgpr 7681. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-2020.) |
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Theorem | caucvgprlemopu 7670* | Lemma for caucvgpr 7681. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.) |
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Theorem | caucvgprlemupu 7671* | Lemma for caucvgpr 7681. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 20-Oct-2020.) |
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Theorem | caucvgprlemrnd 7672* | Lemma for caucvgpr 7681. The putative limit is rounded. (Contributed by Jim Kingdon, 27-Sep-2020.) |
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Theorem | caucvgprlemdisj 7673* | Lemma for caucvgpr 7681. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.) |
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Theorem | caucvgprlemloc 7674* | Lemma for caucvgpr 7681. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.) |
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Theorem | caucvgprlemcl 7675* | Lemma for caucvgpr 7681. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.) |
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Theorem | caucvgprlemladdfu 7676* |
Lemma for caucvgpr 7681. Adding ![]() |
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Theorem | caucvgprlemladdrl 7677* |
Lemma for caucvgpr 7681. Adding ![]() |
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Theorem | caucvgprlem1 7678* | Lemma for caucvgpr 7681. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.) |
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Theorem | caucvgprlem2 7679* | Lemma for caucvgpr 7681. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.) |
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Theorem | caucvgprlemlim 7680* | Lemma for caucvgpr 7681. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.) |
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Theorem | caucvgpr 7681* |
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
![]() ![]() ![]() ![]() This proof (including its lemmas) is similar to the proofs of cauappcvgpr 7661 and caucvgprpr 7711. Reading cauappcvgpr 7661 first (the simplest of the three) might help understanding the other two. (Contributed by Jim Kingdon, 18-Jun-2020.) |
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Theorem | caucvgprprlemk 7682* | Lemma for caucvgprpr 7711. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.) |
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Theorem | caucvgprprlemloccalc 7683* | Lemma for caucvgprpr 7711. Rearranging some expressions for caucvgprprlemloc 7702. (Contributed by Jim Kingdon, 8-Feb-2021.) |
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Theorem | caucvgprprlemell 7684* | Lemma for caucvgprpr 7711. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.) |
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Theorem | caucvgprprlemelu 7685* | Lemma for caucvgprpr 7711. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-Jan-2021.) |
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Theorem | caucvgprprlemcbv 7686* | Lemma for caucvgprpr 7711. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.) |
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Theorem | caucvgprprlemval 7687* | Lemma for caucvgprpr 7711. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.) |
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Theorem | caucvgprprlemnkltj 7688* | Lemma for caucvgprpr 7711. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.) |
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Theorem | caucvgprprlemnkeqj 7689* | Lemma for caucvgprpr 7711. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.) |
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Theorem | caucvgprprlemnjltk 7690* | Lemma for caucvgprpr 7711. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.) |
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Theorem | caucvgprprlemnkj 7691* | Lemma for caucvgprpr 7711. Part of disjointness. (Contributed by Jim Kingdon, 20-Jan-2021.) |
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Theorem | caucvgprprlemnbj 7692* | Lemma for caucvgprpr 7711. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-2021.) |
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Theorem | caucvgprprlemml 7693* | Lemma for caucvgprpr 7711. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.) |
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Theorem | caucvgprprlemmu 7694* | Lemma for caucvgprpr 7711. The upper cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.) |
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Theorem | caucvgprprlemm 7695* | Lemma for caucvgprpr 7711. The putative limit is inhabited. (Contributed by Jim Kingdon, 21-Dec-2020.) |
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Theorem | caucvgprprlemopl 7696* | Lemma for caucvgprpr 7711. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.) |
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Theorem | caucvgprprlemlol 7697* | Lemma for caucvgprpr 7711. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.) |
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Theorem | caucvgprprlemopu 7698* | Lemma for caucvgprpr 7711. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.) |
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Theorem | caucvgprprlemupu 7699* | Lemma for caucvgprpr 7711. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 21-Dec-2020.) |
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Theorem | caucvgprprlemrnd 7700* | Lemma for caucvgprpr 7711. The putative limit is rounded. (Contributed by Jim Kingdon, 21-Dec-2020.) |
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