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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | modqsubdir 10301 | Distribute the modulo operation over a subtraction. (Contributed by Jim Kingdon, 26-Oct-2021.) |
Theorem | modqeqmodmin 10302 | A rational number equals the difference of the rational number and a modulus modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.) |
Theorem | modfzo0difsn 10303* | For a number within a half-open range of nonnegative integers with one excluded integer there is a positive integer so that the number is equal to the sum of the positive integer and the excluded integer modulo the upper bound of the range. (Contributed by AV, 19-Mar-2021.) |
..^ ..^ ..^ | ||
Theorem | modsumfzodifsn 10304 | The sum of a number within a half-open range of positive integers is an element of the corresponding open range of nonnegative integers with one excluded integer modulo the excluded integer. (Contributed by AV, 19-Mar-2021.) |
..^ ..^ ..^ | ||
Theorem | modlteq 10305 | Two nonnegative integers less than the modulus are equal iff they are equal modulo the modulus. (Contributed by AV, 14-Mar-2021.) |
..^ ..^ | ||
Theorem | addmodlteq 10306 | Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. (Contributed by AV, 20-Mar-2021.) |
..^ ..^ | ||
Theorem | frec2uz0d 10307* | The mapping is a one-to-one mapping from onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number (normally 0 for the upper integers or 1 for the upper integers ), 1 maps to + 1, etc. This theorem shows the value of at ordinal natural number zero. (Contributed by Jim Kingdon, 16-May-2020.) |
frec | ||
Theorem | frec2uzzd 10308* | The value of (see frec2uz0d 10307) is an integer. (Contributed by Jim Kingdon, 16-May-2020.) |
frec | ||
Theorem | frec2uzsucd 10309* | The value of (see frec2uz0d 10307) at a successor. (Contributed by Jim Kingdon, 16-May-2020.) |
frec | ||
Theorem | frec2uzuzd 10310* | The value (see frec2uz0d 10307) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.) |
frec | ||
Theorem | frec2uzltd 10311* | Less-than relation for (see frec2uz0d 10307). (Contributed by Jim Kingdon, 16-May-2020.) |
frec | ||
Theorem | frec2uzlt2d 10312* | The mapping (see frec2uz0d 10307) preserves order. (Contributed by Jim Kingdon, 16-May-2020.) |
frec | ||
Theorem | frec2uzrand 10313* | Range of (see frec2uz0d 10307). (Contributed by Jim Kingdon, 17-May-2020.) |
frec | ||
Theorem | frec2uzf1od 10314* | (see frec2uz0d 10307) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.) |
frec | ||
Theorem | frec2uzisod 10315* | (see frec2uz0d 10307) is an isomorphism from natural ordinals to upper integers. (Contributed by Jim Kingdon, 17-May-2020.) |
frec | ||
Theorem | frecuzrdgrrn 10316* | The function (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of . (Contributed by Jim Kingdon, 28-Mar-2022.) |
frec frec | ||
Theorem | frec2uzrdg 10317* | A helper lemma for the value of a recursive definition generator on upper integers (typically either or ) with characteristic function and initial value . This lemma shows that evaluating at an element of gives an ordered pair whose first element is the index (translated from to ). See comment in frec2uz0d 10307 which describes and the index translation. (Contributed by Jim Kingdon, 24-May-2020.) |
frec frec | ||
Theorem | frecuzrdgrcl 10318* | The function (used in the definition of the recursive definition generator on upper integers) is a function defined for all natural numbers. (Contributed by Jim Kingdon, 1-Apr-2022.) |
frec frec | ||
Theorem | frecuzrdglem 10319* | A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-May-2020.) |
frec frec | ||
Theorem | frecuzrdgtcl 10320* | The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10307 for the description of as the mapping from to . (Contributed by Jim Kingdon, 26-May-2020.) |
frec frec | ||
Theorem | frecuzrdg0 10321* | Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 10307 for the description of as the mapping from to . (Contributed by Jim Kingdon, 27-May-2020.) |
frec frec | ||
Theorem | frecuzrdgsuc 10322* | Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10307 for the description of as the mapping from to . (Contributed by Jim Kingdon, 28-May-2020.) |
frec frec | ||
Theorem | frecuzrdgrclt 10323* | The function (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of . Similar to frecuzrdgrcl 10318 except that and need not be the same. (Contributed by Jim Kingdon, 22-Apr-2022.) |
frec | ||
Theorem | frecuzrdgg 10324* | Lemma for other theorems involving the the recursive definition generator on upper integers. Evaluating at a natural number gives an ordered pair whose first element is the mapping of that natural number via . (Contributed by Jim Kingdon, 23-Apr-2022.) |
frec frec | ||
Theorem | frecuzrdgdomlem 10325* | The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.) |
frec frec | ||
Theorem | frecuzrdgdom 10326* | The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.) |
frec | ||
Theorem | frecuzrdgfunlem 10327* | The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.) |
frec frec | ||
Theorem | frecuzrdgfun 10328* | The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.) |
frec | ||
Theorem | frecuzrdgtclt 10329* | The recursive definition generator on upper integers is a function. (Contributed by Jim Kingdon, 22-Apr-2022.) |
frec | ||
Theorem | frecuzrdg0t 10330* | Initial value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 28-Apr-2022.) |
frec | ||
Theorem | frecuzrdgsuctlem 10331* | Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10307 for the description of as the mapping from to . (Contributed by Jim Kingdon, 29-Apr-2022.) |
frec frec | ||
Theorem | frecuzrdgsuct 10332* | Successor value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 29-Apr-2022.) |
frec | ||
Theorem | uzenom 10333 | An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Theorem | frecfzennn 10334 | The cardinality of a finite set of sequential integers. (See frec2uz0d 10307 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.) |
frec | ||
Theorem | frecfzen2 10335 | The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.) |
frec | ||
Theorem | frechashgf1o 10336 | maps one-to-one onto . (Contributed by Jim Kingdon, 19-May-2020.) |
frec | ||
Theorem | frec2uzled 10337* | The mapping (see frec2uz0d 10307) preserves order. (Contributed by Jim Kingdon, 24-Feb-2022.) |
frec | ||
Theorem | fzfig 10338 | A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.) |
Theorem | fzfigd 10339 | Deduction form of fzfig 10338. (Contributed by Jim Kingdon, 21-May-2020.) |
Theorem | fzofig 10340 | Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.) |
..^ | ||
Theorem | nn0ennn 10341 | The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.) |
Theorem | nnenom 10342 | The set of positive integers (as a subset of complex numbers) is equinumerous to omega (the set of natural numbers as ordinals). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.) |
Theorem | nnct 10343 | is dominated by . (Contributed by Thierry Arnoux, 29-Dec-2016.) |
Theorem | uzennn 10344 | An upper integer set is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 30-Jul-2023.) |
Theorem | fnn0nninf 10345* | A function from into ℕ∞. (Contributed by Jim Kingdon, 16-Jul-2022.) |
frec ℕ∞ | ||
Theorem | fxnn0nninf 10346* | A function from NN0* into ℕ∞. (Contributed by Jim Kingdon, 16-Jul-2022.) TODO: use infnninf 7069 instead of infnninfOLD 7070. More generally, this theorem and most theorems in this section could use an extended defined by frec and as in nnnninf2 7072. |
frec NN0*ℕ∞ | ||
Theorem | 0tonninf 10347* | The mapping of zero into ℕ∞ is the sequence of all zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.) |
frec | ||
Theorem | 1tonninf 10348* | The mapping of one into ℕ∞ is a sequence which is a one followed by zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.) |
frec | ||
Theorem | inftonninf 10349* | The mapping of into ℕ∞ is the sequence of all ones. (Contributed by Jim Kingdon, 17-Jul-2022.) |
frec | ||
Theorem | uzsinds 10350* | Strong (or "total") induction principle over an upper set of integers. (Contributed by Scott Fenton, 16-May-2014.) |
Theorem | nnsinds 10351* | Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.) |
Theorem | nn0sinds 10352* | Strong (or "total") induction principle over the nonnegative integers. (Contributed by Scott Fenton, 16-May-2014.) |
Syntax | cseq 10353 | Extend class notation with recursive sequence builder. |
Definition | df-seqfrec 10354* |
Define a general-purpose operation that builds a recursive sequence
(i.e., a function on an upper integer set such as or )
whose value at an index is a function of its previous value and the
value of an input sequence at that index. This definition is
complicated, but fortunately it is not intended to be used directly.
Instead, the only purpose of this definition is to provide us with an
object that has the properties expressed by seqf 10369, seq3-1 10368 and
seq3p1 10370. Typically, those are the main theorems
that would be used in
practice.
The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation , an input sequence with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence with values 1, 3/2, 7/4, 15/8,.., so that , 3/2, etc. In other words, transforms a sequence into an infinite series. means "the sum of F(n) from n = M to infinity is 2." Since limits are unique (climuni 11201), by climdm 11203 the "sum of F(n) from n = 1 to infinity" can be expressed as (provided the sequence converges) and evaluates to 2 in this example. Internally, the frec function generates as its values a set of ordered pairs starting at , with the first member of each pair incremented by one in each successive value. So, the range of frec is exactly the sequence we want, and we just extract the range and throw away the domain. (Contributed by NM, 18-Apr-2005.) (Revised by Jim Kingdon, 4-Nov-2022.) |
frec | ||
Theorem | seqex 10355 | Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
Theorem | seqeq1 10356 | Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
Theorem | seqeq2 10357 | Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
Theorem | seqeq3 10358 | Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
Theorem | seqeq1d 10359 | Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.) |
Theorem | seqeq2d 10360 | Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.) |
Theorem | seqeq3d 10361 | Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.) |
Theorem | seqeq123d 10362 | Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.) |
Theorem | nfseq 10363 | Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | iseqovex 10364* | Closure of a function used in proving sequence builder theorems. This can be thought of as a lemma for the small number of sequence builder theorems which need it. (Contributed by Jim Kingdon, 31-May-2020.) |
Theorem | iseqvalcbv 10365* | Changing the bound variables in an expression which appears in some related proofs. (Contributed by Jim Kingdon, 28-Apr-2022.) |
frec frec | ||
Theorem | seq3val 10366* | Value of the sequence builder function. This helps expand the definition although there should be little need for it once we have proved seqf 10369, seq3-1 10368 and seq3p1 10370, as further development can be done in terms of those. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 4-Nov-2022.) |
frec | ||
Theorem | seqvalcd 10367* | Value of the sequence builder function. Similar to seq3val 10366 but the classes (type of each term) and (type of the value we are accumulating) do not need to be the same. (Contributed by Jim Kingdon, 9-Jul-2023.) |
frec | ||
Theorem | seq3-1 10368* | Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 3-Oct-2022.) |
Theorem | seqf 10369* | Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) |
Theorem | seq3p1 10370* | Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 30-Apr-2022.) |
Theorem | seqovcd 10371* | A closure law for the recursive sequence builder. This is a lemma for theorems such as seqf2 10372 and seq1cd 10373 and is unlikely to be needed once such theorems are proved. (Contributed by Jim Kingdon, 20-Jul-2023.) |
Theorem | seqf2 10372* | Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 7-Jul-2023.) |
Theorem | seq1cd 10373* | Initial value of the recursive sequence builder. A version of seq3-1 10368 which provides two classes and for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 19-Jul-2023.) |
Theorem | seqp1cd 10374* | Value of the sequence builder function at a successor. A version of seq3p1 10370 which provides two classes and for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 20-Jul-2023.) |
Theorem | seq3clss 10375* | Closure property of the recursive sequence builder. (Contributed by Jim Kingdon, 28-Sep-2022.) |
Theorem | seq3m1 10376* | Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 3-Nov-2022.) |
Theorem | seq3fveq2 10377* | Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.) |
Theorem | seq3feq2 10378* | Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.) |
Theorem | seq3fveq 10379* | Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.) |
Theorem | seq3feq 10380* | Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.) |
Theorem | seq3shft2 10381* | Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.) |
Theorem | serf 10382* | An infinite series of complex terms is a function from to . (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.) |
Theorem | serfre 10383* | An infinite series of real numbers is a function from to . (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.) |
Theorem | monoord 10384* | Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.) |
Theorem | monoord2 10385* | Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.) |
Theorem | ser3mono 10386* | The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.) |
Theorem | seq3split 10387* | Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.) (Revised by Jim Kingdon, 21-Oct-2022.) |
Theorem | seq3-1p 10388* | Removing the first term from a sequence. (Contributed by Jim Kingdon, 16-Aug-2021.) |
Theorem | seq3caopr3 10389* | Lemma for seq3caopr2 10390. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.) |
..^ | ||
Theorem | seq3caopr2 10390* | The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.) |
Theorem | seq3caopr 10391* | The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.) |
Theorem | iseqf1olemkle 10392* | Lemma for seq3f1o 10412. (Contributed by Jim Kingdon, 21-Aug-2022.) |
..^ | ||
Theorem | iseqf1olemklt 10393* | Lemma for seq3f1o 10412. (Contributed by Jim Kingdon, 21-Aug-2022.) |
..^ | ||
Theorem | iseqf1olemqcl 10394 | Lemma for seq3f1o 10412. (Contributed by Jim Kingdon, 27-Aug-2022.) |
Theorem | iseqf1olemqval 10395* | Lemma for seq3f1o 10412. Value of the function . (Contributed by Jim Kingdon, 28-Aug-2022.) |
Theorem | iseqf1olemnab 10396* | Lemma for seq3f1o 10412. (Contributed by Jim Kingdon, 27-Aug-2022.) |
Theorem | iseqf1olemab 10397* | Lemma for seq3f1o 10412. (Contributed by Jim Kingdon, 27-Aug-2022.) |
Theorem | iseqf1olemnanb 10398* | Lemma for seq3f1o 10412. (Contributed by Jim Kingdon, 27-Aug-2022.) |
Theorem | iseqf1olemqf 10399* | Lemma for seq3f1o 10412. Domain and codomain of . (Contributed by Jim Kingdon, 26-Aug-2022.) |
Theorem | iseqf1olemmo 10400* | Lemma for seq3f1o 10412. Showing that is one-to-one. (Contributed by Jim Kingdon, 27-Aug-2022.) |
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