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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | frecuzrdgsuc 9201* | Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 9185 for the description of as the mapping from to . (Contributed by Jim Kingdon, 28-May-2020.) |
frec frec | ||
Theorem | uzenom 9202 | An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Theorem | frecfzennn 9203 | The cardinality of a finite set of sequential integers. (See frec2uz0d 9185 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.) |
frec | ||
Theorem | frecfzen2 9204 | The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.) |
frec | ||
Theorem | frechashgf1o 9205 | maps one-to-one onto . (Contributed by Jim Kingdon, 19-May-2020.) |
frec | ||
Theorem | fzfig 9206 | A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.) |
Theorem | fzfigd 9207 | Deduction form of fzfig 9206. (Contributed by Jim Kingdon, 21-May-2020.) |
Theorem | fzofig 9208 | Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.) |
..^ | ||
Theorem | nn0ennn 9209 | The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.) |
Theorem | nnenom 9210 | 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.) |
Syntax | cseq 9211 | Extend class notation with recursive sequence builder. |
Definition | df-iseq 9212* |
Define a general-purpose operation that builds a recursive sequence
(i.e. a function on the positive integers 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 iseq1 9222 and iseqp1 9225.
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. 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 Jim Kingdon, 29-May-2020.) |
frec | ||
Theorem | iseqex 9213 | Existence of the sequence builder operation. (Contributed by Jim Kingdon, 20-Aug-2021.) |
Theorem | iseqeq1 9214 | Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) |
Theorem | iseqeq2 9215 | Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) |
Theorem | iseqeq3 9216 | Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) |
Theorem | iseqeq4 9217 | Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) |
Theorem | nfiseq 9218 | Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) |
Theorem | iseqovex 9219* | 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 | iseqval 9220* | Value of the sequence builder function. (Contributed by Jim Kingdon, 30-May-2020.) |
frec | ||
Theorem | iseqfn 9221* | The sequence builder function is a function. (Contributed by Jim Kingdon, 30-May-2020.) |
Theorem | iseq1 9222* | Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 31-May-2020.) |
Theorem | iseqcl 9223* | Closure properties of the recursive sequence builder. (Contributed by Jim Kingdon, 1-Jun-2020.) |
Theorem | iseqf 9224* | Range of the recursive sequence builder. (Contributed by Jim Kingdon, 23-Jul-2021.) |
Theorem | iseqp1 9225* | Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 31-May-2020.) |
Theorem | iseqss 9226* | Specifying a larger universe for . As long as and are closed over , then any set which contains can be used as the last argument to . This theorem does not allow to be a proper class, however. It also currently requires that be closed over (as well as ). (Contributed by Jim Kingdon, 18-Aug-2021.) |
Theorem | iseqm1 9227* | Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) |
Theorem | iseqfveq2 9228* | Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.) |
Theorem | iseqfeq2 9229* | Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.) |
Theorem | iseqfveq 9230* | Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.) |
Theorem | iseqfeq 9231* | Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) |
Theorem | iseqshft2 9232* | Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) |
Theorem | iserf 9233* | An infinite series of complex terms is a function from to . (Contributed by Jim Kingdon, 15-Aug-2021.) |
Theorem | iserfre 9234* | 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 9235* | Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.) |
Theorem | monoord2 9236* | Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.) |
Theorem | isermono 9237* | The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) |
Theorem | iseqsplit 9238* | Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.) |
Theorem | iseq1p 9239* | Removing the first term from a sequence. (Contributed by Jim Kingdon, 16-Aug-2021.) |
Theorem | iseqcaopr3 9240* | Lemma for iseqcaopr2 . (Contributed by Jim Kingdon, 16-Aug-2021.) |
..^ | ||
Theorem | iseqcaopr2 9241* | The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) |
Theorem | iseqcaopr 9242* | The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Jim Kingdon, 17-Aug-2021.) |
Theorem | iseradd 9243* | The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 26-May-2014.) |
Theorem | isersub 9244* | The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.) |
Theorem | iseqid3 9245* | A sequence that consists entirely of zeroes (or whatever the identity is for operation ) sums to zero. (Contributed by Jim Kingdon, 18-Aug-2021.) |
Theorem | iseqid3s 9246* | A sequence that consists of zeroes up to sums to zero at . In this case by "zero" we mean whatever the identity is for the operation ). (Contributed by Jim Kingdon, 18-Aug-2021.) |
Theorem | iseqid 9247* | Discard the first few terms of a sequence that starts with all zeroes (or whatever the identity is for operation ). (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.) |
Theorem | iseqhomo 9248* | Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 21-Aug-2021.) |
Theorem | iseqdistr 9249* | The distributive property for series. (Contributed by Jim Kingdon, 21-Aug-2021.) |
Theorem | iser0 9250 | The value of the partial sums in a zero-valued infinite series. (Contributed by Jim Kingdon, 19-Aug-2021.) |
Theorem | iser0f 9251 | A zero-valued infinite series is equal to the constant zero function. (Contributed by Jim Kingdon, 19-Aug-2021.) |
Theorem | serige0 9252* | A finite sum of nonnegative terms is nonnegative. (Contributed by Jim Kingdon, 22-Aug-2021.) |
Theorem | serile 9253* | Comparison of partial sums of two infinite series of reals. (Contributed by Jim Kingdon, 22-Aug-2021.) |
Syntax | cexp 9254 | Extend class notation to include exponentiation of a complex number to an integer power. |
Definition | df-iexp 9255* | Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 9259 and expp1 9262 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts. 10-Jun-2005: The definition was extended to include zero exponents, so that per the convention of Definition 10-4.1 of [Gleason] p. 134. 4-Jun-2014: The definition was extended to include negative integer exponents. The case gives the value , so we will avoid this case in our theorems. (Contributed by Jim Kingdon, 7-Jun-2020.) |
Theorem | expivallem 9256 | Lemma for expival 9257. If we take a complex number apart from zero and raise it to a positive integer power, the result is apart from zero. (Contributed by Jim Kingdon, 7-Jun-2020.) |
# # | ||
Theorem | expival 9257 | Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.) |
# | ||
Theorem | expinnval 9258 | Value of exponentiation to positive integer powers. (Contributed by Jim Kingdon, 8-Jun-2020.) |
Theorem | exp0 9259 | Value of a complex number raised to the 0th power. Note that under our definition, , following the convention used by Gleason. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.) |
Theorem | 0exp0e1 9260 | (common case). This is our convention. It follows the convention used by Gleason; see Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | exp1 9261 | Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.) |
Theorem | expp1 9262 | Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.) |
Theorem | expnegap0 9263 | Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.) |
# | ||
Theorem | expineg2 9264 | Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.) |
# | ||
Theorem | expn1ap0 9265 | A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.) |
# | ||
Theorem | expcllem 9266* | Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.) |
Theorem | expcl2lemap 9267* | Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.) |
# # | ||
Theorem | nnexpcl 9268 | Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.) |
Theorem | nn0expcl 9269 | Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.) |
Theorem | zexpcl 9270 | Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.) |
Theorem | qexpcl 9271 | Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.) |
Theorem | reexpcl 9272 | Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.) |
Theorem | expcl 9273 | Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.) |
Theorem | rpexpcl 9274 | Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.) |
Theorem | reexpclzap 9275 | Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.) |
# | ||
Theorem | qexpclz 9276 | Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.) |
Theorem | m1expcl2 9277 | Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.) |
Theorem | m1expcl 9278 | Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.) |
Theorem | expclzaplem 9279* | Closure law for integer exponentiation. Lemma for expclzap 9280 and expap0i 9287. (Contributed by Jim Kingdon, 9-Jun-2020.) |
# # | ||
Theorem | expclzap 9280 | Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.) |
# | ||
Theorem | nn0expcli 9281 | Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.) |
Theorem | nn0sqcl 9282 | The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
Theorem | expm1t 9283 | Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.) |
Theorem | 1exp 9284 | Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.) |
Theorem | expap0 9285 | Positive integer exponentiation is apart from zero iff its mantissa is apart from zero. That it is easier to prove this first, and then prove expeq0 9286 in terms of it, rather than the other way around, is perhaps an illustration of the maxim "In constructive analysis, the apartness is more basic [ than ] equality." ([Geuvers], p. 1). (Contributed by Jim Kingdon, 10-Jun-2020.) |
# # | ||
Theorem | expeq0 9286 | Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.) |
Theorem | expap0i 9287 | Integer exponentiation is apart from zero if its mantissa is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.) |
# # | ||
Theorem | expgt0 9288 | Nonnegative integer exponentiation with a positive mantissa is positive. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.) |
Theorem | expnegzap 9289 | Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.) |
# | ||
Theorem | 0exp 9290 | Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.) |
Theorem | expge0 9291 | Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.) |
Theorem | expge1 9292 | Nonnegative integer exponentiation with a mantissa greater than or equal to 1 is greater than or equal to 1. (Contributed by NM, 21-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.) |
Theorem | expgt1 9293 | Positive integer exponentiation with a mantissa greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.) |
Theorem | mulexp 9294 | Positive integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 13-Feb-2005.) |
Theorem | mulexpzap 9295 | Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.) |
# # | ||
Theorem | exprecap 9296 | Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.) |
# | ||
Theorem | expadd 9297 | Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.) |
Theorem | expaddzaplem 9298 | Lemma for expaddzap 9299. (Contributed by Jim Kingdon, 10-Jun-2020.) |
# | ||
Theorem | expaddzap 9299 | Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.) |
# | ||
Theorem | expmul 9300 | Product of exponents law for positive integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 4-Jan-2006.) |
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