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Type | Label | Description |
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Statement | ||
Theorem | mulassprg 7201 | 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 7202 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
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Theorem | distrlem1pru 7203 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
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Theorem | distrlem4prl 7204* | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
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Theorem | distrlem4pru 7205* | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
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Theorem | distrlem5prl 7206 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
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Theorem | distrlem5pru 7207 | Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.) |
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Theorem | distrprg 7208 | 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 7209 | 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 7210 | Lemma for 1idpr 7212. (Contributed by Jim Kingdon, 13-Dec-2019.) |
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Theorem | 1idpru 7211 | Lemma for 1idpr 7212. (Contributed by Jim Kingdon, 13-Dec-2019.) |
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Theorem | 1idpr 7212 | 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 7213* |
We can order fractions via ![]() ![]() |
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Theorem | ltnqpri 7214* |
We can order fractions via ![]() ![]() |
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Theorem | ltpopr 7215 | 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 7216. (Contributed by Jim Kingdon, 15-Dec-2019.) |
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Theorem | ltsopr 7216 | Positive real 'less than' is a weak linear order (in the sense of df-iso 4133). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.) |
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Theorem | ltaddpr 7217 | 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 7218* | Element in lower cut of the constructed difference. Lemma for ltexpri 7233. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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Theorem | ltexprlemelu 7219* | Element in upper cut of the constructed difference. Lemma for ltexpri 7233. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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Theorem | ltexprlemm 7220* | Our constructed difference is inhabited. Lemma for ltexpri 7233. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemopl 7221* | The lower cut of our constructed difference is open. Lemma for ltexpri 7233. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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Theorem | ltexprlemlol 7222* | The lower cut of our constructed difference is lower. Lemma for ltexpri 7233. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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Theorem | ltexprlemopu 7223* | The upper cut of our constructed difference is open. Lemma for ltexpri 7233. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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Theorem | ltexprlemupu 7224* | The upper cut of our constructed difference is upper. Lemma for ltexpri 7233. (Contributed by Jim Kingdon, 21-Dec-2019.) |
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Theorem | ltexprlemrnd 7225* | Our constructed difference is rounded. Lemma for ltexpri 7233. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemdisj 7226* | Our constructed difference is disjoint. Lemma for ltexpri 7233. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemloc 7227* | Our constructed difference is located. Lemma for ltexpri 7233. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlempr 7228* | Our constructed difference is a positive real. Lemma for ltexpri 7233. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemfl 7229* | Lemma for ltexpri 7233. One directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemrl 7230* | Lemma for ltexpri 7233. Reverse directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemfu 7231* | Lemma for ltexpri 7233. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexprlemru 7232* | Lemma for ltexpri 7233. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
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Theorem | ltexpri 7233* | 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 7234 | Lemma for addcanprg 7236. (Contributed by Jim Kingdon, 25-Dec-2019.) |
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Theorem | addcanprlemu 7235 | Lemma for addcanprg 7236. (Contributed by Jim Kingdon, 25-Dec-2019.) |
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Theorem | addcanprg 7236 | 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 7237* | The difference from ltexpri 7233 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.) |
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Theorem | ltaprlem 7238 | Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) |
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Theorem | ltaprg 7239 | 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 7240* | 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 7241 | Strong extensionality of addition (ordering version). This is similar to addext 8148 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 7242* |
Membership in the lower cut of ![]() |
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Theorem | recexprlemelu 7243* |
Membership in the upper cut of ![]() |
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Theorem | recexprlemm 7244* |
![]() |
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Theorem | recexprlemopl 7245* |
The lower cut of ![]() |
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Theorem | recexprlemlol 7246* |
The lower cut of ![]() |
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Theorem | recexprlemopu 7247* |
The upper cut of ![]() |
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Theorem | recexprlemupu 7248* |
The upper cut of ![]() |
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Theorem | recexprlemrnd 7249* |
![]() |
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Theorem | recexprlemdisj 7250* |
![]() |
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Theorem | recexprlemloc 7251* |
![]() |
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Theorem | recexprlempr 7252* |
![]() |
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Theorem | recexprlem1ssl 7253* |
The lower cut of one is a subset of the lower cut of ![]() ![]() ![]() |
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Theorem | recexprlem1ssu 7254* |
The upper cut of one is a subset of the upper cut of ![]() ![]() ![]() |
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Theorem | recexprlemss1l 7255* |
The lower cut of ![]() ![]() ![]() |
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Theorem | recexprlemss1u 7256* |
The upper cut of ![]() ![]() ![]() |
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Theorem | recexprlemex 7257* |
![]() ![]() |
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Theorem | recexpr 7258* | 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 7259 | Lemma for aptipr 7261. (Contributed by Jim Kingdon, 28-Jan-2020.) |
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Theorem | aptiprlemu 7260 | Lemma for aptipr 7261. (Contributed by Jim Kingdon, 28-Jan-2020.) |
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Theorem | aptipr 7261 | Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.) |
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Theorem | ltmprr 7262 | Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.) |
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Theorem | archpr 7263* |
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 7264* | Lemma for cauappcvgprlemladdrl 7277. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.) |
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Theorem | cauappcvgprlemm 7265* | Lemma for cauappcvgpr 7282. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.) |
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Theorem | cauappcvgprlemopl 7266* | Lemma for cauappcvgpr 7282. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.) |
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Theorem | cauappcvgprlemlol 7267* | Lemma for cauappcvgpr 7282. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 4-Aug-2020.) |
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Theorem | cauappcvgprlemopu 7268* | Lemma for cauappcvgpr 7282. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.) |
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Theorem | cauappcvgprlemupu 7269* | Lemma for cauappcvgpr 7282. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 4-Aug-2020.) |
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Theorem | cauappcvgprlemrnd 7270* | Lemma for cauappcvgpr 7282. The putative limit is rounded. (Contributed by Jim Kingdon, 18-Jul-2020.) |
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Theorem | cauappcvgprlemdisj 7271* | Lemma for cauappcvgpr 7282. The putative limit is disjoint. (Contributed by Jim Kingdon, 18-Jul-2020.) |
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Theorem | cauappcvgprlemloc 7272* | Lemma for cauappcvgpr 7282. The putative limit is located. (Contributed by Jim Kingdon, 18-Jul-2020.) |
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Theorem | cauappcvgprlemcl 7273* | Lemma for cauappcvgpr 7282. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.) |
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Theorem | cauappcvgprlemladdfu 7274* | Lemma for cauappcvgprlemladd 7278. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
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Theorem | cauappcvgprlemladdfl 7275* | Lemma for cauappcvgprlemladd 7278. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
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Theorem | cauappcvgprlemladdru 7276* | Lemma for cauappcvgprlemladd 7278. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
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Theorem | cauappcvgprlemladdrl 7277* | Lemma for cauappcvgprlemladd 7278. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
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Theorem | cauappcvgprlemladd 7278* |
Lemma for cauappcvgpr 7282. This takes ![]() ![]() |
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Theorem | cauappcvgprlem1 7279* | Lemma for cauappcvgpr 7282. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.) |
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Theorem | cauappcvgprlem2 7280* | Lemma for cauappcvgpr 7282. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.) |
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Theorem | cauappcvgprlemlim 7281* | Lemma for cauappcvgpr 7282. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-2020.) |
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Theorem | cauappcvgpr 7282* |
A Cauchy approximation has a limit. A Cauchy approximation, here
![]() ![]() ![]() ![]() ![]() This proof (including its lemmas) is similar to the proofs of caucvgpr 7302 and caucvgprpr 7332 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 7283* | Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.) |
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Theorem | archrecpr 7284* | Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.) |
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Theorem | caucvgprlemk 7285 | Lemma for caucvgpr 7302. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.) |
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Theorem | caucvgprlemnkj 7286* | Lemma for caucvgpr 7302. Part of disjointness. (Contributed by Jim Kingdon, 23-Oct-2020.) |
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Theorem | caucvgprlemnbj 7287* | Lemma for caucvgpr 7302. 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 7288* | Lemma for caucvgpr 7302. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.) |
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Theorem | caucvgprlemopl 7289* | Lemma for caucvgpr 7302. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.) |
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Theorem | caucvgprlemlol 7290* | Lemma for caucvgpr 7302. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-2020.) |
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Theorem | caucvgprlemopu 7291* | Lemma for caucvgpr 7302. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.) |
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Theorem | caucvgprlemupu 7292* | Lemma for caucvgpr 7302. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 20-Oct-2020.) |
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Theorem | caucvgprlemrnd 7293* | Lemma for caucvgpr 7302. The putative limit is rounded. (Contributed by Jim Kingdon, 27-Sep-2020.) |
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Theorem | caucvgprlemdisj 7294* | Lemma for caucvgpr 7302. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.) |
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Theorem | caucvgprlemloc 7295* | Lemma for caucvgpr 7302. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.) |
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Theorem | caucvgprlemcl 7296* | Lemma for caucvgpr 7302. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.) |
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Theorem | caucvgprlemladdfu 7297* |
Lemma for caucvgpr 7302. Adding ![]() |
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Theorem | caucvgprlemladdrl 7298* |
Lemma for caucvgpr 7302. Adding ![]() |
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Theorem | caucvgprlem1 7299* | Lemma for caucvgpr 7302. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.) |
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Theorem | caucvgprlem2 7300* | Lemma for caucvgpr 7302. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.) |
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