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Theorem List for Intuitionistic Logic Explorer - 10001-10100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmodaddmodup 10001 The sum of an integer modulo a positive integer and another integer minus the positive integer equals the sum of the two integers modulo the positive integer if the other integer is in the upper part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
..^

Theoremmodaddmodlo 10002 The sum of an integer modulo a positive integer and another integer equals the sum of the two integers modulo the positive integer if the other integer is in the lower part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
..^

Theoremmodqmulmod 10003 The product of a rational number modulo a modulus and an integer equals the product of the rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)

Theoremmodqmulmodr 10004 The product of an integer and a rational number modulo a modulus equals the product of the integer and the rational number modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)

Theoremmodqaddmulmod 10005 The sum of a rational number and the product of a second rational number modulo a modulus and an integer equals the sum of the rational number and the product of the other rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)

Theoremmodqdi 10006 Distribute multiplication over a modulo operation. (Contributed by Jim Kingdon, 26-Oct-2021.)

Theoremmodqsubdir 10007 Distribute the modulo operation over a subtraction. (Contributed by Jim Kingdon, 26-Oct-2021.)

Theoremmodqeqmodmin 10008 A rational number equals the difference of the rational number and a modulus modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)

Theoremmodfzo0difsn 10009* 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.)
..^ ..^ ..^

Theoremmodsumfzodifsn 10010 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.)
..^ ..^ ..^

Theoremmodlteq 10011 Two nonnegative integers less than the modulus are equal iff they are equal modulo the modulus. (Contributed by AV, 14-Mar-2021.)
..^ ..^

Theoremaddmodlteq 10012 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.)
..^ ..^

Theoremfrec2uz0d 10013* 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

Theoremfrec2uzzd 10014* The value of (see frec2uz0d 10013) is an integer. (Contributed by Jim Kingdon, 16-May-2020.)
frec

Theoremfrec2uzsucd 10015* The value of (see frec2uz0d 10013) at a successor. (Contributed by Jim Kingdon, 16-May-2020.)
frec

Theoremfrec2uzuzd 10016* The value (see frec2uz0d 10013) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.)
frec

Theoremfrec2uzltd 10017* Less-than relation for (see frec2uz0d 10013). (Contributed by Jim Kingdon, 16-May-2020.)
frec

Theoremfrec2uzlt2d 10018* The mapping (see frec2uz0d 10013) preserves order. (Contributed by Jim Kingdon, 16-May-2020.)
frec

Theoremfrec2uzrand 10019* Range of (see frec2uz0d 10013). (Contributed by Jim Kingdon, 17-May-2020.)
frec

Theoremfrec2uzf1od 10020* (see frec2uz0d 10013) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.)
frec

Theoremfrec2uzisod 10021* (see frec2uz0d 10013) is an isomorphism from natural ordinals to upper integers. (Contributed by Jim Kingdon, 17-May-2020.)
frec

Theoremfrecuzrdgrrn 10022* 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

Theoremfrec2uzrdg 10023* 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 10013 which describes and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)
frec                      frec

Theoremfrecuzrdgrcl 10024* 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

Theoremfrecuzrdglem 10025* A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-May-2020.)
frec                      frec

Theoremfrecuzrdgtcl 10026* The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10013 for the description of as the mapping from to . (Contributed by Jim Kingdon, 26-May-2020.)
frec                      frec

Theoremfrecuzrdg0 10027* Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 10013 for the description of as the mapping from to . (Contributed by Jim Kingdon, 27-May-2020.)
frec                      frec

Theoremfrecuzrdgsuc 10028* Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10013 for the description of as the mapping from to . (Contributed by Jim Kingdon, 28-May-2020.)
frec                      frec

Theoremfrecuzrdgrclt 10029* 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 10024 except that and need not be the same. (Contributed by Jim Kingdon, 22-Apr-2022.)
frec

Theoremfrecuzrdgg 10030* 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

Theoremfrecuzrdgdomlem 10031* The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
frec        frec

Theoremfrecuzrdgdom 10032* The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
frec

Theoremfrecuzrdgfunlem 10033* The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
frec        frec

Theoremfrecuzrdgfun 10034* The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
frec

Theoremfrecuzrdgtclt 10035* The recursive definition generator on upper integers is a function. (Contributed by Jim Kingdon, 22-Apr-2022.)
frec

Theoremfrecuzrdg0t 10036* Initial value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 28-Apr-2022.)
frec

Theoremfrecuzrdgsuctlem 10037* Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10013 for the description of as the mapping from to . (Contributed by Jim Kingdon, 29-Apr-2022.)
frec        frec

Theoremfrecuzrdgsuct 10038* Successor value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 29-Apr-2022.)
frec

Theoremuzenom 10039 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremfrecfzennn 10040 The cardinality of a finite set of sequential integers. (See frec2uz0d 10013 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)
frec

Theoremfrecfzen2 10041 The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.)
frec

Theoremfrechashgf1o 10042 maps one-to-one onto . (Contributed by Jim Kingdon, 19-May-2020.)
frec

Theoremfrec2uzled 10043* The mapping (see frec2uz0d 10013) preserves order. (Contributed by Jim Kingdon, 24-Feb-2022.)
frec

Theoremfzfig 10044 A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.)

Theoremfzfigd 10045 Deduction form of fzfig 10044. (Contributed by Jim Kingdon, 21-May-2020.)

Theoremfzofig 10046 Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.)
..^

Theoremnn0ennn 10047 The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)

Theoremnnenom 10048 The set of positive integers (as a subset of complex numbers) is equinumerous to omega (the set of finite ordinal numbers). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremnnct 10049 is dominated by . (Contributed by Thierry Arnoux, 29-Dec-2016.)

Theoremuzennn 10050 An upper integer set is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 30-Jul-2023.)

Theoremfnn0nninf 10051* A function from into ℕ. (Contributed by Jim Kingdon, 16-Jul-2022.)
frec

Theoremfxnn0nninf 10052* A function from NN0* into ℕ. (Contributed by Jim Kingdon, 16-Jul-2022.)
frec                      NN0*

Theorem0tonninf 10053* The mapping of zero into ℕ is the sequence of all zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
frec

Theorem1tonninf 10054* The mapping of one into ℕ is a sequence which is a one followed by zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
frec

Theoreminftonninf 10055* The mapping of into ℕ is the sequence of all ones. (Contributed by Jim Kingdon, 17-Jul-2022.)
frec

3.6.4  Strong induction over upper sets of integers

Theoremuzsinds 10056* Strong (or "total") induction principle over an upper set of integers. (Contributed by Scott Fenton, 16-May-2014.)

Theoremnnsinds 10057* Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.)

Theoremnn0sinds 10058* Strong (or "total") induction principle over the nonnegative integers. (Contributed by Scott Fenton, 16-May-2014.)

3.6.5  The infinite sequence builder "seq"

Syntaxcseq 10059 Extend class notation with recursive sequence builder.

Definitiondf-seqfrec 10060* 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 10075, seq3-1 10074 and seq3p1 10076. 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 10901), by climdm 10903 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

Theoremseqex 10061 Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Theoremseqeq1 10062 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Theoremseqeq2 10063 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Theoremseqeq3 10064 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Theoremseqeq1d 10065 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)

Theoremseqeq2d 10066 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)

Theoremseqeq3d 10067 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)

Theoremseqeq123d 10068 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)

Theoremnfseq 10069 Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremiseqovex 10070* 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.)

Theoremiseqvalcbv 10071* Changing the bound variables in an expression which appears in some related proofs. (Contributed by Jim Kingdon, 28-Apr-2022.)
frec frec

Theoremseq3val 10072* Value of the sequence builder function. This helps expand the definition although there should be little need for it once we have proved seqf 10075, seq3-1 10074 and seq3p1 10076, 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

Theoremseqvalcd 10073* Value of the sequence builder function. Similar to seq3val 10072 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

Theoremseq3-1 10074* Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 3-Oct-2022.)

Theoremseqf 10075* Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.)

Theoremseq3p1 10076* Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 30-Apr-2022.)

Theoremseqovcd 10077* A closure law for the recursive sequence builder. This is a lemma for theorems such as seqf2 10078 and seq1cd 10079 and is unlikely to be needed once such theorems are proved. (Contributed by Jim Kingdon, 20-Jul-2023.)

Theoremseqf2 10078* Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 7-Jul-2023.)

Theoremseq1cd 10079* Initial value of the recursive sequence builder. A version of seq3-1 10074 which provides two classes and for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 19-Jul-2023.)

Theoremseqp1cd 10080* Value of the sequence builder function at a successor. A version of seq3p1 10076 which provides two classes and for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 20-Jul-2023.)

Theoremseq3clss 10081* Closure property of the recursive sequence builder. (Contributed by Jim Kingdon, 28-Sep-2022.)

Theoremseq3m1 10082* Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 3-Nov-2022.)

Theoremseq3fveq2 10083* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)

Theoremseq3feq2 10084* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)

Theoremseq3fveq 10085* Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)

Theoremseq3feq 10086* Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)

Theoremseq3shft2 10087* Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)

Theoremserf 10088* An infinite series of complex terms is a function from to . (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremserfre 10089* An infinite series of real numbers is a function from to . (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremmonoord 10090* Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)

Theoremmonoord2 10091* Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)

Theoremser3mono 10092* 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.)

Theoremseq3split 10093* Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.) (Revised by Jim Kingdon, 21-Oct-2022.)

Theoremseq3-1p 10094* Removing the first term from a sequence. (Contributed by Jim Kingdon, 16-Aug-2021.)

Theoremseq3caopr3 10095* Lemma for seq3caopr2 10096. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.)
..^

Theoremseq3caopr2 10096* 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.)

Theoremseq3caopr 10097* 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.)

Theoremiseqf1olemkle 10098* Lemma for seq3f1o 10118. (Contributed by Jim Kingdon, 21-Aug-2022.)
..^

Theoremiseqf1olemklt 10099* Lemma for seq3f1o 10118. (Contributed by Jim Kingdon, 21-Aug-2022.)
..^

Theoremiseqf1olemqcl 10100 Lemma for seq3f1o 10118. (Contributed by Jim Kingdon, 27-Aug-2022.)

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