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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fzo0ssnn0 8801 | Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.) |
..^ | ||
Theorem | fzo01 8802 | Expressing the singleton of as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ | ||
Theorem | fzo12sn 8803 | A 1-based half-open integer interval up to, but not including, 2 is a singleton. (Contributed by Alexander van der Vekens, 31-Jan-2018.) |
..^ | ||
Theorem | fzo0to2pr 8804 | A half-open integer range from 0 to 2 is an unordered pair. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
..^ | ||
Theorem | fzo0to3tp 8805 | A half-open integer range from 0 to 3 is an unordered triple. (Contributed by Alexander van der Vekens, 9-Nov-2017.) |
..^ | ||
Theorem | fzo0to42pr 8806 | A half-open integer range from 0 to 4 is a union of two unordered pairs. (Contributed by Alexander van der Vekens, 17-Nov-2017.) |
..^ | ||
Theorem | fzo0sn0fzo1 8807 | A half-open range of nonnegative integers is the union of the singleton set containing 0 and a half-open range of positive integers. (Contributed by Alexander van der Vekens, 18-May-2018.) |
..^ ..^ | ||
Theorem | fzoend 8808 | The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.) |
..^ ..^ | ||
Theorem | fzo0end 8809 | The endpoint of a zero-based half-open range. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
..^ | ||
Theorem | ssfzo12 8810 | Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 16-Mar-2018.) |
..^ ..^ | ||
Theorem | ssfzo12bi 8811 | Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 5-Nov-2018.) |
..^ ..^ | ||
Theorem | ubmelm1fzo 8812 | The result of subtracting 1 and an integer of a half-open range of nonnegative integers from the upper bound of this range is contained in this range. (Contributed by AV, 23-Mar-2018.) (Revised by AV, 30-Oct-2018.) |
..^ ..^ | ||
Theorem | fzofzp1 8813 | If a point is in a half-open range, the next point is in the closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ | ||
Theorem | fzofzp1b 8814 | If a point is in a half-open range, the next point is in the closed range. (Contributed by Mario Carneiro, 27-Sep-2015.) |
..^ | ||
Theorem | elfzom1b 8815 | An integer is a member of a 1-based finite set of sequential integers iff its predecessor is a member of the corresponding 0-based set. (Contributed by Mario Carneiro, 27-Sep-2015.) |
..^ ..^ | ||
Theorem | elfzonelfzo 8816 | If an element of a half-open integer range is not contained in the lower subrange, it must be in the upper subrange. (Contributed by Alexander van der Vekens, 30-Mar-2018.) |
..^ ..^ ..^ | ||
Theorem | elfzomelpfzo 8817 | An integer increased by another integer is an element of a half-open integer range if and only if the integer is contained in the half-open integer range with bounds decreased by the other integer. (Contributed by Alexander van der Vekens, 30-Mar-2018.) |
..^ ..^ | ||
Theorem | peano2fzor 8818 | A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.) |
..^ ..^ | ||
Theorem | fzosplitsn 8819 | Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ ..^ | ||
Theorem | fzosplitprm1 8820 | Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
..^ ..^ | ||
Theorem | fzosplitsni 8821 | Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ ..^ | ||
Theorem | fzisfzounsn 8822 | A finite interval of integers as union of a half-open integer range and a singleton. (Contributed by Alexander van der Vekens, 15-Jun-2018.) |
..^ | ||
Theorem | fzostep1 8823 | Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ ..^ | ||
Theorem | fzoshftral 8824* | Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 8700. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
..^ ..^ | ||
Theorem | fzind2 8825* | Induction on the integers from to inclusive. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Version of fzind 8089 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.) |
..^ | ||
Theorem | frec2uz0d 8826* | 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 8827* | The value of (see frec2uz0d 8826) is an integer. (Contributed by Jim Kingdon, 16-May-2020.) |
frec | ||
Theorem | frec2uzsucd 8828* | The value of (see frec2uz0d 8826) at a successor. (Contributed by Jim Kingdon, 16-May-2020.) |
frec | ||
Theorem | frec2uzuzd 8829* | The value (see frec2uz0d 8826) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.) |
frec | ||
Theorem | frec2uzltd 8830* | Less-than relation for (see frec2uz0d 8826). (Contributed by Jim Kingdon, 16-May-2020.) |
frec | ||
Theorem | frec2uzlt2d 8831* | The mapping (see frec2uz0d 8826) preserves order. (Contributed by Jim Kingdon, 16-May-2020.) |
frec | ||
Theorem | frec2uzrand 8832* | Range of (see frec2uz0d 8826). (Contributed by Jim Kingdon, 17-May-2020.) |
frec | ||
Theorem | frec2uzf1od 8833* | (see frec2uz0d 8826) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.) |
frec | ||
Theorem | frec2uzisod 8834* | (see frec2uz0d 8826) is an isomorphism from natural ordinals to upper integers. (Contributed by Jim Kingdon, 17-May-2020.) |
frec | ||
Theorem | frecuzrdgrrn 8835* | 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, 27-May-2020.) |
frec frec | ||
Theorem | frec2uzrdg 8836* | 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 8826 which describes and the index translation. (Contributed by Jim Kingdon, 24-May-2020.) |
frec frec | ||
Theorem | frecuzrdgrom 8837* | 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, 26-May-2020.) |
frec frec | ||
Theorem | frecuzrdglem 8838* | A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-May-2020.) |
frec frec | ||
Theorem | frecuzrdgfn 8839* | The recursive definition generator on upper integers is a function. See comment in frec2uz0d 8826 for the description of as the mapping from to . (Contributed by Jim Kingdon, 26-May-2020.) |
frec frec | ||
Theorem | frecuzrdgcl 8840* | Closure law for the recursive definition generator on upper integers. See comment in frec2uz0d 8826 for the description of as the mapping from to . (Contributed by Jim Kingdon, 31-May-2020.) |
frec frec | ||
Theorem | frecuzrdg0 8841* | Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 8826 for the description of as the mapping from to . (Contributed by Jim Kingdon, 27-May-2020.) |
frec frec | ||
Theorem | frecuzrdgsuc 8842* | Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 8826 for the description of as the mapping from to . (Contributed by Jim Kingdon, 28-May-2020.) |
frec frec | ||
Theorem | uzenom 8843 | An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Theorem | frecfzennn 8844 | The cardinality of a finite set of sequential integers. (See frec2uz0d 8826 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.) |
frec | ||
Theorem | frecfzen2 8845 | The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.) |
frec | ||
Theorem | frechashgf1o 8846 | maps one-to-one onto . (Contributed by Jim Kingdon, 19-May-2020.) |
frec | ||
Theorem | fzfig 8847 | A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.) |
Theorem | fzfigd 8848 | Deduction form of fzfig 8847. (Contributed by Jim Kingdon, 21-May-2020.) |
Theorem | fzofig 8849 | Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.) |
..^ | ||
Theorem | nn0ennn 8850 | The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.) |
Theorem | nnenom 8851 | 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 8852 | Extend class notation with recursive sequence builder. |
Definition | df-iseq 8853* |
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 8862 and at successors.
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 | iseqeq1 8854 | Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) |
Theorem | iseqeq2 8855 | Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) |
Theorem | iseqeq3 8856 | Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) |
Theorem | iseqeq4 8857 | Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) |
Theorem | nfiseq 8858 | Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) |
Theorem | iseqovex 8859* | 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 8860* | Value of the sequence builder function. (Contributed by Jim Kingdon, 30-May-2020.) |
frec | ||
Theorem | iseqfn 8861* | The sequence builder function is a function. (Contributed by Jim Kingdon, 30-May-2020.) |
Theorem | iseq1 8862* | Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 31-May-2020.) |
Theorem | iseqcl 8863* | Closure properties of the recursive sequence builder. (Contributed by Jim Kingdon, 1-Jun-2020.) |
Theorem | iseqp1 8864* | Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 31-May-2020.) |
Theorem | iseqfveq2 8865* | Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.) |
Theorem | iseqfeq2 8866* | Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.) |
Theorem | iseqfveq 8867* | Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.) |
Syntax | cexp 8868 | Extend class notation to include exponentiation of a complex number to an integer power. |
Definition | df-iexp 8869* | Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 8873 and expp1 8876 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 Kingodon, 7-Jun-2020.) |
Theorem | expivallem 8870 | Lemma for expival 8871. 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 Kingodon, 7-Jun-2020.) |
# # | ||
Theorem | expival 8871 | Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.) |
# | ||
Theorem | expinnval 8872 | Value of exponentiation to positive integer powers. (Contributed by Jim Kingdon, 8-Jun-2020.) |
Theorem | exp0 8873 | 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 8874 | (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 8875 | 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 8876 | 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 8877 | Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.) |
# | ||
Theorem | expineg2 8878 | Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.) |
# | ||
Theorem | expn1ap0 8879 | A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.) |
# | ||
Theorem | expcllem 8880* | Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.) |
Theorem | expcl2lemap 8881* | Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.) |
# # | ||
Theorem | nnexpcl 8882 | Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.) |
Theorem | nn0expcl 8883 | Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.) |
Theorem | zexpcl 8884 | Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.) |
Theorem | qexpcl 8885 | Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.) |
Theorem | reexpcl 8886 | Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.) |
Theorem | expcl 8887 | Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.) |
Theorem | rpexpcl 8888 | Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.) |
Theorem | reexpclzap 8889 | Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.) |
# | ||
Theorem | qexpclz 8890 | Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.) |
Theorem | m1expcl2 8891 | Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.) |
Theorem | m1expcl 8892 | Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.) |
Theorem | expclzaplem 8893* | Closure law for integer exponentiation. Lemma for expclzap 8894 and expap0i 8901. (Contributed by Jim Kingdon, 9-Jun-2020.) |
# # | ||
Theorem | expclzap 8894 | Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.) |
# | ||
Theorem | nn0expcli 8895 | Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.) |
Theorem | nn0sqcl 8896 | The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
Theorem | expm1t 8897 | Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.) |
Theorem | 1exp 8898 | Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.) |
Theorem | expap0 8899 | 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 8900 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 8900 | Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.) |
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