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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | caucvgprlemlim 7501* | Lemma for caucvgpr 7502. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.) |
Theorem | caucvgpr 7502* |
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 7482 and caucvgprpr 7532. Reading cauappcvgpr 7482 first (the simplest of the three) might help understanding the other two. (Contributed by Jim Kingdon, 18-Jun-2020.) |
Theorem | caucvgprprlemk 7503* | Lemma for caucvgprpr 7532. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.) |
Theorem | caucvgprprlemloccalc 7504* | Lemma for caucvgprpr 7532. Rearranging some expressions for caucvgprprlemloc 7523. (Contributed by Jim Kingdon, 8-Feb-2021.) |
Theorem | caucvgprprlemell 7505* | Lemma for caucvgprpr 7532. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.) |
Theorem | caucvgprprlemelu 7506* | Lemma for caucvgprpr 7532. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-Jan-2021.) |
Theorem | caucvgprprlemcbv 7507* | Lemma for caucvgprpr 7532. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.) |
Theorem | caucvgprprlemval 7508* | Lemma for caucvgprpr 7532. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.) |
Theorem | caucvgprprlemnkltj 7509* | Lemma for caucvgprpr 7532. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.) |
Theorem | caucvgprprlemnkeqj 7510* | Lemma for caucvgprpr 7532. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.) |
Theorem | caucvgprprlemnjltk 7511* | Lemma for caucvgprpr 7532. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.) |
Theorem | caucvgprprlemnkj 7512* | Lemma for caucvgprpr 7532. Part of disjointness. (Contributed by Jim Kingdon, 20-Jan-2021.) |
Theorem | caucvgprprlemnbj 7513* | Lemma for caucvgprpr 7532. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-2021.) |
Theorem | caucvgprprlemml 7514* | Lemma for caucvgprpr 7532. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.) |
Theorem | caucvgprprlemmu 7515* | Lemma for caucvgprpr 7532. The upper cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.) |
Theorem | caucvgprprlemm 7516* | Lemma for caucvgprpr 7532. The putative limit is inhabited. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemopl 7517* | Lemma for caucvgprpr 7532. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemlol 7518* | Lemma for caucvgprpr 7532. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemopu 7519* | Lemma for caucvgprpr 7532. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemupu 7520* | Lemma for caucvgprpr 7532. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemrnd 7521* | Lemma for caucvgprpr 7532. The putative limit is rounded. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemdisj 7522* | Lemma for caucvgprpr 7532. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemloc 7523* | Lemma for caucvgprpr 7532. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemcl 7524* | Lemma for caucvgprpr 7532. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-2020.) |
Theorem | caucvgprprlemclphr 7525* | Lemma for caucvgprpr 7532. The putative limit is a positive real. Like caucvgprprlemcl 7524 but without a distinct variable constraint between and . (Contributed by Jim Kingdon, 19-Jun-2021.) |
Theorem | caucvgprprlemexbt 7526* | Lemma for caucvgprpr 7532. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.) |
Theorem | caucvgprprlemexb 7527* | Lemma for caucvgprpr 7532. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.) |
Theorem | caucvgprprlemaddq 7528* | Lemma for caucvgprpr 7532. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.) |
Theorem | caucvgprprlem1 7529* | Lemma for caucvgprpr 7532. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.) |
Theorem | caucvgprprlem2 7530* | Lemma for caucvgprpr 7532. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.) |
Theorem | caucvgprprlemlim 7531* | Lemma for caucvgprpr 7532. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.) |
Theorem | caucvgprpr 7532* |
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 7502 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 7482) might help in understanding this one, as they are slightly simpler but similarly structured. (Contributed by Jim Kingdon, 14-Nov-2020.) |
Theorem | suplocexprlemell 7533* | Lemma for suplocexpr 7545. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlem2b 7534 | Lemma for suplocexpr 7545. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlemss 7535* | Lemma for suplocexpr 7545. is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Theorem | suplocexprlemml 7536* | Lemma for suplocexpr 7545. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Theorem | suplocexprlemrl 7537* | Lemma for suplocexpr 7545. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlemmu 7538* | Lemma for suplocexpr 7545. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Theorem | suplocexprlemru 7539* | Lemma for suplocexpr 7545. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlemdisj 7540* | Lemma for suplocexpr 7545. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlemloc 7541* | Lemma for suplocexpr 7545. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlemex 7542* | Lemma for suplocexpr 7545. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Theorem | suplocexprlemub 7543* | Lemma for suplocexpr 7545. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.) |
Theorem | suplocexprlemlub 7544* | Lemma for suplocexpr 7545. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.) |
Theorem | suplocexpr 7545* | An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Definition | df-enr 7546* | 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 7547 | 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 7548* | 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 7549* | 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 7550* | 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 7551 | 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 7552 | 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 7553 | 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 7554 | Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) |
Theorem | enrer 7555 | 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 7556 | Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) |
Theorem | enrex 7557 | The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) |
Theorem | ltrelsr 7558 | Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) |
Theorem | addcmpblnr 7559 | Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) |
Theorem | mulcmpblnrlemg 7560 | Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.) |
Theorem | mulcmpblnr 7561 | Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) |
Theorem | prsrlem1 7562* | Decomposing signed reals into positive reals. Lemma for addsrpr 7565 and mulsrpr 7566. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | addsrmo 7563* | There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | mulsrmo 7564* | There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | addsrpr 7565 | Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Theorem | mulsrpr 7566 | Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Theorem | ltsrprg 7567 | Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.) |
Theorem | gt0srpr 7568 | Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) |
Theorem | 0nsr 7569 | The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) |
Theorem | 0r 7570 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | 1sr 7571 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | m1r 7572 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | addclsr 7573 | Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) |
Theorem | mulclsr 7574 | Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) |
Theorem | addcomsrg 7575 | Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | addasssrg 7576 | Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | mulcomsrg 7577 | Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | mulasssrg 7578 | Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | distrsrg 7579 | Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.) |
Theorem | m1p1sr 7580 | Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) |
Theorem | m1m1sr 7581 | Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) |
Theorem | lttrsr 7582* | Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Theorem | ltposr 7583 | Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Theorem | ltsosr 7584 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) |
Theorem | 0lt1sr 7585 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) |
Theorem | 1ne0sr 7586 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) |
Theorem | 0idsr 7587 | The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) |
Theorem | 1idsr 7588 | 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.) |
Theorem | 00sr 7589 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) |
Theorem | ltasrg 7590 | Ordering property of addition. (Contributed by NM, 10-May-1996.) |
Theorem | pn0sr 7591 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) |
Theorem | negexsr 7592* | Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) |
Theorem | recexgt0sr 7593* | The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) |
Theorem | recexsrlem 7594* | 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 7595 | The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) |
Theorem | ltadd1sr 7596 | Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.) |
Theorem | ltm1sr 7597 | Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.) |
Theorem | mulgt0sr 7598 | The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) |
Theorem | aptisr 7599 | Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.) |
Theorem | mulextsr1lem 7600 | Lemma for mulextsr1 7601. (Contributed by Jim Kingdon, 17-Feb-2020.) |
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