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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | caucvgprprlemupu 7501* | Lemma for caucvgprpr 7513. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemrnd 7502* | Lemma for caucvgprpr 7513. The putative limit is rounded. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemdisj 7503* | Lemma for caucvgprpr 7513. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemloc 7504* | Lemma for caucvgprpr 7513. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Theorem | caucvgprprlemcl 7505* | Lemma for caucvgprpr 7513. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-2020.) |
Theorem | caucvgprprlemclphr 7506* | Lemma for caucvgprpr 7513. The putative limit is a positive real. Like caucvgprprlemcl 7505 but without a distinct variable constraint between and . (Contributed by Jim Kingdon, 19-Jun-2021.) |
Theorem | caucvgprprlemexbt 7507* | Lemma for caucvgprpr 7513. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.) |
Theorem | caucvgprprlemexb 7508* | Lemma for caucvgprpr 7513. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.) |
Theorem | caucvgprprlemaddq 7509* | Lemma for caucvgprpr 7513. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.) |
Theorem | caucvgprprlem1 7510* | Lemma for caucvgprpr 7513. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.) |
Theorem | caucvgprprlem2 7511* | Lemma for caucvgprpr 7513. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.) |
Theorem | caucvgprprlemlim 7512* | Lemma for caucvgprpr 7513. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.) |
Theorem | caucvgprpr 7513* |
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 7483 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 7463) might help in understanding this one, as they are slightly simpler but similarly structured. (Contributed by Jim Kingdon, 14-Nov-2020.) |
Theorem | suplocexprlemell 7514* | Lemma for suplocexpr 7526. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlem2b 7515 | Lemma for suplocexpr 7526. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlemss 7516* | Lemma for suplocexpr 7526. is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Theorem | suplocexprlemml 7517* | Lemma for suplocexpr 7526. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Theorem | suplocexprlemrl 7518* | Lemma for suplocexpr 7526. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlemmu 7519* | Lemma for suplocexpr 7526. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Theorem | suplocexprlemru 7520* | Lemma for suplocexpr 7526. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlemdisj 7521* | Lemma for suplocexpr 7526. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlemloc 7522* | Lemma for suplocexpr 7526. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.) |
Theorem | suplocexprlemex 7523* | Lemma for suplocexpr 7526. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Theorem | suplocexprlemub 7524* | Lemma for suplocexpr 7526. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.) |
Theorem | suplocexprlemlub 7525* | Lemma for suplocexpr 7526. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.) |
Theorem | suplocexpr 7526* | An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.) |
Definition | df-enr 7527* | 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 7528 | 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 7529* | 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 7530* | 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 7531* | 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 7532 | 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 7533 | 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 7534 | 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 7535 | Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) |
Theorem | enrer 7536 | 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 7537 | Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) |
Theorem | enrex 7538 | The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) |
Theorem | ltrelsr 7539 | Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) |
Theorem | addcmpblnr 7540 | Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) |
Theorem | mulcmpblnrlemg 7541 | Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.) |
Theorem | mulcmpblnr 7542 | Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) |
Theorem | prsrlem1 7543* | Decomposing signed reals into positive reals. Lemma for addsrpr 7546 and mulsrpr 7547. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | addsrmo 7544* | There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | mulsrmo 7545* | There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
Theorem | addsrpr 7546 | Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Theorem | mulsrpr 7547 | Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Theorem | ltsrprg 7548 | Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.) |
Theorem | gt0srpr 7549 | Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) |
Theorem | 0nsr 7550 | The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) |
Theorem | 0r 7551 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | 1sr 7552 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | m1r 7553 | The constant is a signed real. (Contributed by NM, 9-Aug-1995.) |
Theorem | addclsr 7554 | Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) |
Theorem | mulclsr 7555 | Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) |
Theorem | addcomsrg 7556 | Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | addasssrg 7557 | Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | mulcomsrg 7558 | Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | mulasssrg 7559 | Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Theorem | distrsrg 7560 | Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.) |
Theorem | m1p1sr 7561 | Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) |
Theorem | m1m1sr 7562 | Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) |
Theorem | lttrsr 7563* | Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Theorem | ltposr 7564 | Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Theorem | ltsosr 7565 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) |
Theorem | 0lt1sr 7566 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) |
Theorem | 1ne0sr 7567 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) |
Theorem | 0idsr 7568 | The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) |
Theorem | 1idsr 7569 | 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.) |
Theorem | 00sr 7570 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) |
Theorem | ltasrg 7571 | Ordering property of addition. (Contributed by NM, 10-May-1996.) |
Theorem | pn0sr 7572 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) |
Theorem | negexsr 7573* | Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) |
Theorem | recexgt0sr 7574* | The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) |
Theorem | recexsrlem 7575* | 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 7576 | The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) |
Theorem | ltadd1sr 7577 | Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.) |
Theorem | ltm1sr 7578 | Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.) |
Theorem | mulgt0sr 7579 | The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) |
Theorem | aptisr 7580 | Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.) |
Theorem | mulextsr1lem 7581 | Lemma for mulextsr1 7582. (Contributed by Jim Kingdon, 17-Feb-2020.) |
Theorem | mulextsr1 7582 | Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.) |
Theorem | archsr 7583* | 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 7584* | Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | prsrcl 7585 | Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | prsrpos 7586 | Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | prsradd 7587 | Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | prsrlt 7588 | Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | prsrriota 7589* | Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | caucvgsrlemcl 7590* | Lemma for caucvgsr 7603. Terms of the sequence from caucvgsrlemgt1 7596 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.) |
Theorem | caucvgsrlemasr 7591* | Lemma for caucvgsr 7603. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.) |
Theorem | caucvgsrlemfv 7592* | Lemma for caucvgsr 7603. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Theorem | caucvgsrlemf 7593* | Lemma for caucvgsr 7603. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.) |
Theorem | caucvgsrlemcau 7594* | Lemma for caucvgsr 7603. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.) |
Theorem | caucvgsrlembound 7595* | Lemma for caucvgsr 7603. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.) |
Theorem | caucvgsrlemgt1 7596* | Lemma for caucvgsr 7603. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.) |
Theorem | caucvgsrlemoffval 7597* | Lemma for caucvgsr 7603. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlemofff 7598* | Lemma for caucvgsr 7603. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlemoffcau 7599* | Lemma for caucvgsr 7603. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
Theorem | caucvgsrlemoffgt1 7600* | Lemma for caucvgsr 7603. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.) |
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