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Theorem | mulnqpru 7401 | Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.) |
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Theorem | mullocprlem 7402 | Calculations for mullocpr 7403. (Contributed by Jim Kingdon, 10-Dec-2019.) |
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Theorem | mullocpr 7403* |
Locatedness of multiplication on positive reals. Lemma 12.9 in
[BauerTaylor], p. 56 (but where both
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Theorem | mulclpr 7404 | Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.) |
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Theorem | mulnqprlemrl 7405* | Lemma for mulnqpr 7409. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.) |
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Theorem | mulnqprlemru 7406* | Lemma for mulnqpr 7409. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.) |
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Theorem | mulnqprlemfl 7407* | Lemma for mulnqpr 7409. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.) |
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Theorem | mulnqprlemfu 7408* | Lemma for mulnqpr 7409. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.) |
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Theorem | mulnqpr 7409* | Multiplication of fractions embedded into positive reals. One can either multiply the fractions as fractions, or embed them into positive reals and multiply them as positive reals, and get the same result. (Contributed by Jim Kingdon, 18-Jul-2021.) |
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Theorem | addcomprg 7410 | Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.) |
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Theorem | addassprg 7411 | Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.) |
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Theorem | mulcomprg 7412 | Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.) |
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Theorem | mulassprg 7413 | Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.) |
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Theorem | distrlem1prl 7414 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
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Theorem | distrlem1pru 7415 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
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Theorem | distrlem4prl 7416* | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
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Theorem | distrlem4pru 7417* | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
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Theorem | distrlem5prl 7418 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
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Theorem | distrlem5pru 7419 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
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Theorem | distrprg 7420 | Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.) |
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Theorem | ltprordil 7421 | If a positive real is less than a second positive real, its lower cut is a subset of the second's lower cut. (Contributed by Jim Kingdon, 23-Dec-2019.) |
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Theorem | 1idprl 7422 | Lemma for 1idpr 7424. (Contributed by Jim Kingdon, 13-Dec-2019.) |
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Theorem | 1idpru 7423 | Lemma for 1idpr 7424. (Contributed by Jim Kingdon, 13-Dec-2019.) |
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Theorem | 1idpr 7424 | 1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.) |
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Theorem | ltnqpr 7425* |
We can order fractions via ![]() ![]() |
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Theorem | ltnqpri 7426* |
We can order fractions via ![]() ![]() |
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Theorem | ltpopr 7427 | Positive real 'less than' is a partial ordering. Remark ("< is transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p. (varies). Lemma for ltsopr 7428. (Contributed by Jim Kingdon, 15-Dec-2019.) |
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Theorem | ltsopr 7428 | Positive real 'less than' is a weak linear order (in the sense of df-iso 4227). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.) |
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Theorem | ltaddpr 7429 | The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) |
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Theorem | ltexprlemell 7430* | Element in lower cut of the constructed difference. Lemma for ltexpri 7445. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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Theorem | ltexprlemelu 7431* | Element in upper cut of the constructed difference. Lemma for ltexpri 7445. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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Theorem | ltexprlemm 7432* | Our constructed difference is inhabited. Lemma for ltexpri 7445. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemopl 7433* | The lower cut of our constructed difference is open. Lemma for ltexpri 7445. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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Theorem | ltexprlemlol 7434* | The lower cut of our constructed difference is lower. Lemma for ltexpri 7445. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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Theorem | ltexprlemopu 7435* | The upper cut of our constructed difference is open. Lemma for ltexpri 7445. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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Theorem | ltexprlemupu 7436* | The upper cut of our constructed difference is upper. Lemma for ltexpri 7445. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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Theorem | ltexprlemrnd 7437* | Our constructed difference is rounded. Lemma for ltexpri 7445. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemdisj 7438* | Our constructed difference is disjoint. Lemma for ltexpri 7445. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemloc 7439* | Our constructed difference is located. Lemma for ltexpri 7445. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlempr 7440* | Our constructed difference is a positive real. Lemma for ltexpri 7445. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemfl 7441* | Lemma for ltexpri 7445. One direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemrl 7442* | Lemma for ltexpri 7445. Reverse direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemfu 7443* | Lemma for ltexpri 7445. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemru 7444* | Lemma for ltexpri 7445. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexpri 7445* | 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 7446 | Lemma for addcanprg 7448. (Contributed by Jim Kingdon, 25-Dec-2019.) |
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Theorem | addcanprlemu 7447 | Lemma for addcanprg 7448. (Contributed by Jim Kingdon, 25-Dec-2019.) |
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Theorem | addcanprg 7448 | 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 7449* | The difference from ltexpri 7445 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.) |
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Theorem | ltaprlem 7450 | Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) |
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Theorem | ltaprg 7451 | 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 7452* | 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 7453 | Strong extensionality of addition (ordering version). This is similar to addext 8396 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 7454* |
Membership in the lower cut of ![]() |
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Theorem | recexprlemelu 7455* |
Membership in the upper cut of ![]() |
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Theorem | recexprlemm 7456* |
![]() |
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Theorem | recexprlemopl 7457* |
The lower cut of ![]() |
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Theorem | recexprlemlol 7458* |
The lower cut of ![]() |
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Theorem | recexprlemopu 7459* |
The upper cut of ![]() |
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Theorem | recexprlemupu 7460* |
The upper cut of ![]() |
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Theorem | recexprlemrnd 7461* |
![]() |
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Theorem | recexprlemdisj 7462* |
![]() |
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Theorem | recexprlemloc 7463* |
![]() |
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Theorem | recexprlempr 7464* |
![]() |
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Theorem | recexprlem1ssl 7465* |
The lower cut of one is a subset of the lower cut of ![]() ![]() ![]() |
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Theorem | recexprlem1ssu 7466* |
The upper cut of one is a subset of the upper cut of ![]() ![]() ![]() |
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Theorem | recexprlemss1l 7467* |
The lower cut of ![]() ![]() ![]() |
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Theorem | recexprlemss1u 7468* |
The upper cut of ![]() ![]() ![]() |
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Theorem | recexprlemex 7469* |
![]() ![]() |
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Theorem | recexpr 7470* | 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 7471 | Lemma for aptipr 7473. (Contributed by Jim Kingdon, 28-Jan-2020.) |
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Theorem | aptiprlemu 7472 | Lemma for aptipr 7473. (Contributed by Jim Kingdon, 28-Jan-2020.) |
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Theorem | aptipr 7473 | Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.) |
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Theorem | ltmprr 7474 | Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.) |
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Theorem | archpr 7475* |
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 7476* | Lemma for cauappcvgprlemladdrl 7489. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.) |
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Theorem | cauappcvgprlemm 7477* | Lemma for cauappcvgpr 7494. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.) |
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Theorem | cauappcvgprlemopl 7478* | Lemma for cauappcvgpr 7494. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.) |
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Theorem | cauappcvgprlemlol 7479* | Lemma for cauappcvgpr 7494. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 4-Aug-2020.) |
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Theorem | cauappcvgprlemopu 7480* | Lemma for cauappcvgpr 7494. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.) |
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Theorem | cauappcvgprlemupu 7481* | Lemma for cauappcvgpr 7494. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 4-Aug-2020.) |
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Theorem | cauappcvgprlemrnd 7482* | Lemma for cauappcvgpr 7494. The putative limit is rounded. (Contributed by Jim Kingdon, 18-Jul-2020.) |
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Theorem | cauappcvgprlemdisj 7483* | Lemma for cauappcvgpr 7494. The putative limit is disjoint. (Contributed by Jim Kingdon, 18-Jul-2020.) |
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Theorem | cauappcvgprlemloc 7484* | Lemma for cauappcvgpr 7494. The putative limit is located. (Contributed by Jim Kingdon, 18-Jul-2020.) |
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Theorem | cauappcvgprlemcl 7485* | Lemma for cauappcvgpr 7494. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cauappcvgprlemladdfu 7486* | Lemma for cauappcvgprlemladd 7490. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
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Theorem | cauappcvgprlemladdfl 7487* | Lemma for cauappcvgprlemladd 7490. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cauappcvgprlemladdru 7488* | Lemma for cauappcvgprlemladd 7490. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cauappcvgprlemladdrl 7489* | Lemma for cauappcvgprlemladd 7490. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cauappcvgprlemladd 7490* |
Lemma for cauappcvgpr 7494. This takes ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cauappcvgprlem1 7491* | Lemma for cauappcvgpr 7494. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cauappcvgprlem2 7492* | Lemma for cauappcvgpr 7494. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cauappcvgprlemlim 7493* | Lemma for cauappcvgpr 7494. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cauappcvgpr 7494* |
A Cauchy approximation has a limit. A Cauchy approximation, here
![]() ![]() ![]() ![]() ![]() This proof (including its lemmas) is similar to the proofs of caucvgpr 7514 and caucvgprpr 7544 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 7495* | Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.) |
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Theorem | archrecpr 7496* | Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | caucvgprlemk 7497 | Lemma for caucvgpr 7514. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | caucvgprlemnkj 7498* | Lemma for caucvgpr 7514. Part of disjointness. (Contributed by Jim Kingdon, 23-Oct-2020.) |
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Theorem | caucvgprlemnbj 7499* | Lemma for caucvgpr 7514. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-Oct-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | caucvgprlemm 7500* | Lemma for caucvgpr 7514. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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