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Theorem List for Metamath Proof Explorer - 11201-11300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmoddifz 11201 The modulo operation differs from by an integer multiple of . (Contributed by Mario Carneiro, 15-Jul-2014.)

Theoremmodfrac 11202 The fractional part of a number is the number modulo 1. (Contributed by NM, 11-Nov-2008.)

Theoremflmod 11203 The floor function expressed in terms of the modulo operation. (Contributed by NM, 11-Nov-2008.)

Theoremintfrac 11204 Break a number into its integer part and its fractional part. (Contributed by NM, 31-Dec-2008.)

Theoremzmod10 11205 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremmodmulnn 11206 Move a natural number in and out of a floor in the first argument of a modulo operation. (Contributed by NM, 2-Jan-2009.)

Theoremzmodcl 11207 Closure law for the modulo operation restricted to integers. (Contributed by NM, 27-Nov-2008.)

Theoremzmodcld 11208 Closure law for the modulo operation restricted to integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzmodfz 11209 An integer mod lies in the first nonnegative integers. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremzmodfzo 11210 An integer mod lies in the first nonnegative integers. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^

Theoremmodid 11211 Identity law for modulo. (Contributed by NM, 29-Dec-2008.)

Theoremmodid2 11212 Identity law for modulo. (Contributed by NM, 29-Dec-2008.)

Theorem0mod 11213 Special case: 0 modulo a positive real number is 0. (Contributed by Mario Carneiro, 22-Feb-2014.)

Theorem1mod 11214 Special case: 1 modulo a real number greater than 1 is 1. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremmodabs 11215 Absorption law for modulo. (Contributed by NM, 29-Dec-2008.)

Theoremmodabs2 11216 Absorption law for modulo. (Contributed by NM, 29-Dec-2008.)

Theoremmodcyc 11217 The modulo operation is periodic. (Contributed by NM, 10-Nov-2008.)

Theoremmodcyc2 11218 The modulo operation is periodic. (Contributed by NM, 12-Nov-2008.)

Theoremmodadd1 11219 Addition property of the modulo operation. (Contributed by NM, 12-Nov-2008.)

Theoremmodmul1 11220 Multiplication property of the modulo operation. Note that the multiplier must be an integer. (Contributed by NM, 12-Nov-2008.)

Theoremmodmul12d 11221 Multiplication property of the modulo operation. (Contributed by Mario Carneiro, 5-Feb-2015.)

Theoremmodnegd 11222 Negation property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremmodadd12d 11223 Additive property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremmodsub12d 11224 Subtraction property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremmoddi 11225 Distribute multiplication over a modulo operation. (Contributed by NM, 11-Nov-2008.)

Theoremmodsubdir 11226 Distribute the modulo operation over a subtraction. (Contributed by NM, 30-Dec-2008.)

Theoremmodirr 11227 A number modulo an irrational multiple of it is nonzero. (Contributed by NM, 11-Nov-2008.)

5.6.3  The infinite sequence builder "seq"

Theoremom2uz0i 11228* 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. (This series of theorems generalizes an earlier series for contributed by Raph Levien, 10-Apr-2004.) (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzsuci 11229* The value of (see om2uz0i 11228) at a successor. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzuzi 11230* The value (see om2uz0i 11228) at an ordinal natural number is in the upper integers. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzlti 11231* Less-than relation for (see om2uz0i 11228). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzlt2i 11232* The mapping (see om2uz0i 11228) preserves order. (Contributed by NM, 4-May-2005.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzrani 11233* Range of (see om2uz0i 11228). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzf1oi 11234* (see om2uz0i 11228) is a one-to-one onto mapping. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzisoi 11235* (see om2uz0i 11228) is an isomorphism from natural ordinals to upper integers. (Contributed by NM, 9-Oct-2008.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzoi 11236* An alternative definition of in terms of df-oi 7426. (Contributed by Mario Carneiro, 2-Jun-2015.)
OrdIso

Theoremom2uzrdg 11237* A helper lemma for the value of a recursive definition generator on upper integers (typically either or ) with characteristic function and initial value . Normally is a function on the partition, and is a member of the partition. See also comment in om2uz0i 11228. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)

Theoremuzrdglem 11238* A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)

Theoremuzrdgfni 11239* The recursive definition generator on upper integers is a function. See comment in om2uzrdg 11237. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 4-May-2015.)

Theoremuzrdg0i 11240* Initial value of a recursive definition generator on upper integers. See comment in om2uzrdg 11237. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)

Theoremuzrdgsuci 11241* Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 11237. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremltweuz 11242 is a well-founded relation on any sequence of upper integers. (Contributed by Andrew Salmon, 13-Nov-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Theoremltwenn 11243 Less than well-orders the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)

Theoremltwefz 11244 Less than well-orders a set of finite integers. (Contributed by Scott Fenton, 8-Aug-2013.)

Theoremuzenom 11245 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremuzinf 11246 An upper integer set is infinite. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)

Theoremuzrdgxfr 11247* Transfer the value of the recursive sequence builder from one base to another. (Contributed by Mario Carneiro, 1-Apr-2014.)

Theoremfzennn 11248 The cardinality of a finite set of sequential integers. (See om2uz0i 11228 for a description of the hypothesis.) (Contributed by Mario Carneiro, 12-Feb-2013.) (Revised by Mario Carneiro, 7-Mar-2014.)

Theoremfzen2 11249 The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Mario Carneiro, 13-Feb-2014.)

Theoremcardfz 11250 The cardinality of a finite set of sequential integers. (See om2uz0i 11228 for a description of the hypothesis.) (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhashgf1o 11251 maps one-to-one onto . (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremfzfi 11252 A finite interval of integers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)

Theoremfzfid 11253 Commonly used special case of fzfi 11252. (Contributed by Mario Carneiro, 25-May-2014.)

Theoremfzofi 11254 Half-open integer sets are finite. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremfsequb 11255* The values of a finite real sequence have an upper bound. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfsequb2 11256* The values of a finite real sequence have an upper bound. (Contributed by NM, 20-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfseqsupcl 11257 The values of a finite real sequence have a supremum. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfseqsupubi 11258 The values of a finite real sequence are bounded by their supremum. (Contributed by NM, 20-Sep-2005.)

Theoremnn0ennn 11259 The nonnegative integers are equinumerous to the natural numbers. (Contributed by NM, 19-Jul-2004.)

Theoremnnenom 11260 The set of natural numbers (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.)

Theoremuzindi 11261* Indirect strong induction on the upper integers. (Contributed by Stefan O'Rear, 25-Aug-2015.)
..^

Theoremaxdc4uzlem 11262* Lemma for axdc4uz 11263. (Contributed by Mario Carneiro, 8-Jan-2014.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremaxdc4uz 11263* A version of axdc4 8283 that works on a set of upper integers instead of . (Contributed by Mario Carneiro, 8-Jan-2014.)

Syntaxcseq 11264 Extend class notation with recursive sequence builder.

Definitiondf-seq 11265* Define a general-purpose operation that builds a recursive sequence (i.e. a function on the natural numbers or some other upper integer set) 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 seq1 11277 and seqp1 11279. 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 12287), by climdm 12289 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 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 is exactly the sequence we want, and we just extract the range (restricted to omega) and throw away the domain.

This definition has its roots in a series of theorems from om2uz0i 11228 through om2uzf1oi 11234, originally proved by Raph Levien for use with df-exp 11324 and later generalized for arbitrary recursive sequences. Definition df-sum 12421 extracts the summation values from partial (finite) and complete (infinite) series. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 4-Sep-2013.)

Theoremseqex 11266 Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Theoremseqeq1 11267 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Theoremseqeq2 11268 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Theoremseqeq3 11269 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Theoremseqeq1d 11270 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)

Theoremseqeq2d 11271 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)

Theoremseqeq3d 11272 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)

Theoremseqeq123d 11273 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)

Theoremnfseq 11274 Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremseqval 11275* Value of the sequence builder function. (Contributed by Mario Carneiro, 24-Jun-2013.)

Theoremseqfn 11276 The sequence builder function is a function. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremseq1 11277 Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremseq1i 11278 Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 30-Apr-2014.)

Theoremseqp1 11279 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremseqp1i 11280 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 30-Apr-2014.)

Theoremseqm1 11281 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.)

Theoremseqcl2 11282* Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqf2 11283* Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqcl 11284* Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqf 11285* Range of the recursive sequence builder (special case of seqf2 11283). (Contributed by Mario Carneiro, 24-Jun-2013.)

Theoremseqfveq2 11286* Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqfeq2 11287* Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqfveq 11288* Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqfeq 11289* Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqshft2 11290* Shifting the index set of a sequence. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqres 11291 Restricting its characteristic function to does not affect the function. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremserf 11292* 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 11293* 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 11294* Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)

Theoremmonoord2 11295* Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)

Theoremsermono 11296* The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-Jun-2013.)

Theoremseqsplit 11297* Split a sequence into two sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseq1p 11298* Removing the first term from a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqcaopr3 11299* Lemma for seqcaopr2 11300. (Contributed by Mario Carneiro, 25-Apr-2016.)
..^

Theoremseqcaopr2 11300* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.)

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