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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | archrecnq 7201* | Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.) |
Theorem | archrecpr 7202* | Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.) |
Theorem | caucvgprlemk 7203 | Lemma for caucvgpr 7220. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.) |
Theorem | caucvgprlemnkj 7204* | Lemma for caucvgpr 7220. Part of disjointness. (Contributed by Jim Kingdon, 23-Oct-2020.) |
Theorem | caucvgprlemnbj 7205* | Lemma for caucvgpr 7220. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-Oct-2020.) |
Theorem | caucvgprlemm 7206* | Lemma for caucvgpr 7220. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.) |
Theorem | caucvgprlemopl 7207* | Lemma for caucvgpr 7220. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.) |
Theorem | caucvgprlemlol 7208* | Lemma for caucvgpr 7220. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-2020.) |
Theorem | caucvgprlemopu 7209* | Lemma for caucvgpr 7220. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.) |
Theorem | caucvgprlemupu 7210* | Lemma for caucvgpr 7220. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 20-Oct-2020.) |
Theorem | caucvgprlemrnd 7211* | Lemma for caucvgpr 7220. The putative limit is rounded. (Contributed by Jim Kingdon, 27-Sep-2020.) |
Theorem | caucvgprlemdisj 7212* | Lemma for caucvgpr 7220. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.) |
Theorem | caucvgprlemloc 7213* | Lemma for caucvgpr 7220. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.) |
Theorem | caucvgprlemcl 7214* | Lemma for caucvgpr 7220. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.) |
Theorem | caucvgprlemladdfu 7215* | Lemma for caucvgpr 7220. Adding after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.) |
Theorem | caucvgprlemladdrl 7216* | Lemma for caucvgpr 7220. Adding after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 8-Oct-2020.) |
Theorem | caucvgprlem1 7217* | Lemma for caucvgpr 7220. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.) |
Theorem | caucvgprlem2 7218* | Lemma for caucvgpr 7220. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.) |
Theorem | caucvgprlemlim 7219* | Lemma for caucvgpr 7220. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.) |
Theorem | caucvgpr 7220* |
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 7200 and caucvgprpr 7250. Reading cauappcvgpr 7200 first (the simplest of the three) might help understanding the other two. (Contributed by Jim Kingdon, 18-Jun-2020.) |
Theorem | caucvgprprlemk 7221* | Lemma for caucvgprpr 7250. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.) |
Theorem | caucvgprprlemloccalc 7222* | Lemma for caucvgprpr 7250. Rearranging some expressions for caucvgprprlemloc 7241. (Contributed by Jim Kingdon, 8-Feb-2021.) |
Theorem | caucvgprprlemell 7223* | Lemma for caucvgprpr 7250. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.) |
Theorem | caucvgprprlemelu 7224* | Lemma for caucvgprpr 7250. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-Jan-2021.) |
Theorem | caucvgprprlemcbv 7225* | Lemma for caucvgprpr 7250. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.) |
Theorem | caucvgprprlemval 7226* | Lemma for caucvgprpr 7250. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.) |
Theorem | caucvgprprlemnkltj 7227* | Lemma for caucvgprpr 7250. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.) |
Theorem | caucvgprprlemnkeqj 7228* | Lemma for caucvgprpr 7250. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.) |
Theorem | caucvgprprlemnjltk 7229* | Lemma for caucvgprpr 7250. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.) |
Theorem | caucvgprprlemnkj 7230* | Lemma for caucvgprpr 7250. Part of disjointness. (Contributed by Jim Kingdon, 20-Jan-2021.) |
Theorem | caucvgprprlemnbj 7231* | Lemma for caucvgprpr 7250. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-2021.) |
Theorem | caucvgprprlemml 7232* | Lemma for caucvgprpr 7250. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.) |
Theorem | caucvgprprlemmu 7233* | Lemma for caucvgprpr 7250. The upper cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.) |
Theorem | caucvgprprlemm 7234* | Lemma for caucvgprpr 7250. The putative limit is inhabited. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemopl 7235* | Lemma for caucvgprpr 7250. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemlol 7236* | Lemma for caucvgprpr 7250. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemopu 7237* | Lemma for caucvgprpr 7250. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemupu 7238* | Lemma for caucvgprpr 7250. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemrnd 7239* | Lemma for caucvgprpr 7250. The putative limit is rounded. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemdisj 7240* | Lemma for caucvgprpr 7250. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemloc 7241* | Lemma for caucvgprpr 7250. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemcl 7242* | Lemma for caucvgprpr 7250. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-2020.) |
Theorem | caucvgprprlemclphr 7243* | Lemma for caucvgprpr 7250. The putative limit is a positive real. Like caucvgprprlemcl 7242 but without a distinct variable constraint between and . (Contributed by Jim Kingdon, 19-Jun-2021.) |
Theorem | caucvgprprlemexbt 7244* | Lemma for caucvgprpr 7250. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.) |
Theorem | caucvgprprlemexb 7245* | Lemma for caucvgprpr 7250. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.) |
Theorem | caucvgprprlemaddq 7246* | Lemma for caucvgprpr 7250. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.) |
Theorem | caucvgprprlem1 7247* | Lemma for caucvgprpr 7250. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.) |
Theorem | caucvgprprlem2 7248* | Lemma for caucvgprpr 7250. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.) |
Theorem | caucvgprprlemlim 7249* | Lemma for caucvgprpr 7250. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.) |
Theorem | caucvgprpr 7250* |
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 7220 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 7200) might help in understanding this one, as they are slightly simpler but similarly structured. (Contributed by Jim Kingdon, 14-Nov-2020.) |
Definition | df-enr 7251* | 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.) |
Definition | df-nr 7252 | 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.) |
Definition | df-plr 7253* | 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.) |
Definition | df-mr 7254* | 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.) |
Definition | df-ltr 7255* | 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.) |
Definition | df-0r 7256 | 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.) |
Definition | df-1r 7257 | 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.) |
Definition | df-m1r 7258 | 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.) |
Theorem | enrbreq 7259 | Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) |
Theorem | enrer 7260 | 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.) |
Theorem | enreceq 7261 | Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) |
Theorem | enrex 7262 | The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) |
Theorem | ltrelsr 7263 | Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) |
Theorem | addcmpblnr 7264 | Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) |
Theorem | mulcmpblnrlemg 7265 | Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.) |
Theorem | mulcmpblnr 7266 | Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) |
Theorem | prsrlem1 7267* | Decomposing signed reals into positive reals. Lemma for addsrpr 7270 and mulsrpr 7271. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | addsrmo 7268* | There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | mulsrmo 7269* | There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | addsrpr 7270 | Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Theorem | mulsrpr 7271 | Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Theorem | ltsrprg 7272 | Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.) |
Theorem | gt0srpr 7273 | Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) |
Theorem | 0nsr 7274 | The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) |
Theorem | 0r 7275 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | 1sr 7276 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | m1r 7277 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | addclsr 7278 | Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) |
Theorem | mulclsr 7279 | Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) |
Theorem | addcomsrg 7280 | Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | addasssrg 7281 | Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | mulcomsrg 7282 | Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | mulasssrg 7283 | Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | distrsrg 7284 | Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.) |
Theorem | m1p1sr 7285 | Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) |
Theorem | m1m1sr 7286 | Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) |
Theorem | lttrsr 7287* | Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Theorem | ltposr 7288 | Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Theorem | ltsosr 7289 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) |
Theorem | 0lt1sr 7290 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) |
Theorem | 1ne0sr 7291 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) |
Theorem | 0idsr 7292 | The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) |
Theorem | 1idsr 7293 | 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.) |
Theorem | 00sr 7294 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) |
Theorem | ltasrg 7295 | Ordering property of addition. (Contributed by NM, 10-May-1996.) |
Theorem | pn0sr 7296 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) |
Theorem | negexsr 7297* | Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) |
Theorem | recexgt0sr 7298* | The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) |
Theorem | recexsrlem 7299* | The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) |
Theorem | addgt0sr 7300 | The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) |
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