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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | caucvgprprlemdisj 7601* | Lemma for caucvgprpr 7611. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemloc 7602* | Lemma for caucvgprpr 7611. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemcl 7603* | Lemma for caucvgprpr 7611. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-2020.) |
Theorem | caucvgprprlemclphr 7604* | Lemma for caucvgprpr 7611. The putative limit is a positive real. Like caucvgprprlemcl 7603 but without a distinct variable constraint between and . (Contributed by Jim Kingdon, 19-Jun-2021.) |
Theorem | caucvgprprlemexbt 7605* | Lemma for caucvgprpr 7611. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.) |
Theorem | caucvgprprlemexb 7606* | Lemma for caucvgprpr 7611. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.) |
Theorem | caucvgprprlemaddq 7607* | Lemma for caucvgprpr 7611. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.) |
Theorem | caucvgprprlem1 7608* | Lemma for caucvgprpr 7611. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.) |
Theorem | caucvgprprlem2 7609* | Lemma for caucvgprpr 7611. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.) |
Theorem | caucvgprprlemlim 7610* | Lemma for caucvgprpr 7611. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.) |
Theorem | caucvgprpr 7611* |
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 7581 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 7561) might help in understanding this one, as they are slightly simpler but similarly structured. (Contributed by Jim Kingdon, 14-Nov-2020.) |
Theorem | suplocexprlemell 7612* | Lemma for suplocexpr 7624. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlem2b 7613 | Lemma for suplocexpr 7624. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlemss 7614* | Lemma for suplocexpr 7624. is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Theorem | suplocexprlemml 7615* | Lemma for suplocexpr 7624. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Theorem | suplocexprlemrl 7616* | Lemma for suplocexpr 7624. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlemmu 7617* | Lemma for suplocexpr 7624. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Theorem | suplocexprlemru 7618* | Lemma for suplocexpr 7624. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlemdisj 7619* | Lemma for suplocexpr 7624. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlemloc 7620* | Lemma for suplocexpr 7624. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlemex 7621* | Lemma for suplocexpr 7624. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Theorem | suplocexprlemub 7622* | Lemma for suplocexpr 7624. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.) |
Theorem | suplocexprlemlub 7623* | Lemma for suplocexpr 7624. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.) |
Theorem | suplocexpr 7624* | An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Definition | df-enr 7625* | 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 7626 | 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 7627* | 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 7628* | 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 7629* | 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 7630 | 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 7631 | 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 7632 | 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 7633 | Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) |
Theorem | enrer 7634 | 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 7635 | Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) |
Theorem | enrex 7636 | The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) |
Theorem | ltrelsr 7637 | Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) |
Theorem | addcmpblnr 7638 | Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) |
Theorem | mulcmpblnrlemg 7639 | Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.) |
Theorem | mulcmpblnr 7640 | Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) |
Theorem | prsrlem1 7641* | Decomposing signed reals into positive reals. Lemma for addsrpr 7644 and mulsrpr 7645. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | addsrmo 7642* | There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | mulsrmo 7643* | There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | addsrpr 7644 | Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Theorem | mulsrpr 7645 | Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Theorem | ltsrprg 7646 | Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.) |
Theorem | gt0srpr 7647 | Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) |
Theorem | 0nsr 7648 | The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) |
Theorem | 0r 7649 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | 1sr 7650 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | m1r 7651 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | addclsr 7652 | Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) |
Theorem | mulclsr 7653 | Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) |
Theorem | addcomsrg 7654 | Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | addasssrg 7655 | Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | mulcomsrg 7656 | Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | mulasssrg 7657 | Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | distrsrg 7658 | Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.) |
Theorem | m1p1sr 7659 | Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) |
Theorem | m1m1sr 7660 | Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) |
Theorem | lttrsr 7661* | Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Theorem | ltposr 7662 | Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Theorem | ltsosr 7663 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) |
Theorem | 0lt1sr 7664 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) |
Theorem | 1ne0sr 7665 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) |
Theorem | 0idsr 7666 | The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) |
Theorem | 1idsr 7667 | 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.) |
Theorem | 00sr 7668 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) |
Theorem | ltasrg 7669 | Ordering property of addition. (Contributed by NM, 10-May-1996.) |
Theorem | pn0sr 7670 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) |
Theorem | negexsr 7671* | Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) |
Theorem | recexgt0sr 7672* | The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) |
Theorem | recexsrlem 7673* | 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 7674 | The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) |
Theorem | ltadd1sr 7675 | Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.) |
Theorem | ltm1sr 7676 | Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.) |
Theorem | mulgt0sr 7677 | The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) |
Theorem | aptisr 7678 | Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.) |
Theorem | mulextsr1lem 7679 | Lemma for mulextsr1 7680. (Contributed by Jim Kingdon, 17-Feb-2020.) |
Theorem | mulextsr1 7680 | Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.) |
Theorem | archsr 7681* | For any signed real, there is an integer that is greater than it. This is also known as the "archimedean property". The expression , is the embedding of the positive integer into the signed reals. (Contributed by Jim Kingdon, 23-Apr-2020.) |
Theorem | srpospr 7682* | Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | prsrcl 7683 | Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | prsrpos 7684 | Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | prsradd 7685 | Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | prsrlt 7686 | Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | prsrriota 7687* | Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | caucvgsrlemcl 7688* | Lemma for caucvgsr 7701. Terms of the sequence from caucvgsrlemgt1 7694 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.) |
Theorem | caucvgsrlemasr 7689* | Lemma for caucvgsr 7701. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.) |
Theorem | caucvgsrlemfv 7690* | Lemma for caucvgsr 7701. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | caucvgsrlemf 7691* | Lemma for caucvgsr 7701. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.) |
Theorem | caucvgsrlemcau 7692* | Lemma for caucvgsr 7701. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.) |
Theorem | caucvgsrlembound 7693* | Lemma for caucvgsr 7701. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | caucvgsrlemgt1 7694* | Lemma for caucvgsr 7701. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.) |
Theorem | caucvgsrlemoffval 7695* | Lemma for caucvgsr 7701. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlemofff 7696* | Lemma for caucvgsr 7701. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlemoffcau 7697* | Lemma for caucvgsr 7701. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlemoffgt1 7698* | Lemma for caucvgsr 7701. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlemoffres 7699* | Lemma for caucvgsr 7701. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlembnd 7700* | Lemma for caucvgsr 7701. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.) |
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