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Theorem List for Intuitionistic Logic Explorer - 8801-8900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfzo0ssnn0 8801 Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.)
 0..^ N  C_  NN0
 
Theoremfzo01 8802 Expressing the singleton of  0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 0..^ 1  { 0 }
 
Theoremfzo12sn 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.)
 1..^ 2  { 1 }
 
Theoremfzo0to2pr 8804 A half-open integer range from 0 to 2 is an unordered pair. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 0..^ 2  { 0 ,  1 }
 
Theoremfzo0to3tp 8805 A half-open integer range from 0 to 3 is an unordered triple. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
 0..^ 3  { 0 ,  1 ,  2 }
 
Theoremfzo0to42pr 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.)
 0..^ 4  { 0 ,  1 }  u.  { 2 ,  3 }
 
Theoremfzo0sn0fzo1 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.)
 N  NN  0..^ N  { 0 }  u.  1..^ N
 
Theoremfzoend 8808 The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^  -  1 ..^
 
Theoremfzo0end 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.)
 NN  -  1  0..^
 
Theoremssfzo12 8810 Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
 K  ZZ  L  ZZ  K  <  L  K..^ L  C_  M..^ N  M  <_  K  L  <_  N
 
Theoremssfzo12bi 8811 Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 5-Nov-2018.)
 K  ZZ  L  ZZ  M  ZZ  N  ZZ  K  <  L  K..^ L  C_  M..^ N  M 
 <_  K  L  <_  N
 
Theoremubmelm1fzo 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.)
 K 
 0..^ N  N  -  K  -  1  0..^ N
 
Theoremfzofzp1 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.)
 C ..^  C  +  1  ...
 
Theoremfzofzp1b 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.)
 C  ZZ>=
 `  C ..^  C  +  1  ...
 
Theoremelfzom1b 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.)
 K  ZZ  N  ZZ  K  1..^ N  K  -  1  0..^ N  -  1
 
Theoremelfzonelfzo 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.)
 N  ZZ  K  M..^ R  K  M..^ N 
 K  N..^ R
 
Theoremelfzomelpfzo 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.)
 M  ZZ  N  ZZ  K  ZZ  L  ZZ  K  M  -  L..^ N  -  L  K  +  L  M..^ N
 
Theorempeano2fzor 8818 A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.)
 K  ZZ>= `  M  K  +  1  M..^ N 
 K  M..^ N
 
Theoremfzosplitsn 8819 Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 ZZ>=
 ` ..^  +  1 ..^  u.  { }
 
Theoremfzosplitprm1 8820 Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
 ZZ  ZZ  < ..^  +  1 ..^ 
 -  1  u.  { 
 -  1 ,  }
 
Theoremfzosplitsni 8821 Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 ZZ>=
 `  C ..^  +  1  C ..^  C
 
Theoremfzisfzounsn 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.)
 ZZ>=
 `  ... ..^  u.  { }
 
Theoremfzostep1 8823 Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^ C  +  1 ..^ C  +  1  C
 
Theoremfzoshftral 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.)
 M  ZZ  N  ZZ  K  ZZ  j  M..^ N  k  M  +  K..^ N  +  K [. k  -  K  j ].
 
Theoremfzind2 8825* Induction on the integers from  M to  N 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.)
 M    &       &     +  1    &     K    &     N  ZZ>=
 `  M    &     M..^ N    =>     K  M ... N
 
3.5.7  Miscellaneous theorems about integers
 
Theoremfrec2uz0d 8826* The mapping  G is a one-to-one mapping from  om onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number  C (normally 0 for the upper integers  NN0 or 1 for the upper integers  NN), 1 maps to  C + 1, etc. This theorem shows the value of  G at ordinal natural number zero. (Contributed by Jim Kingdon, 16-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   =>     G `
  (/)  C
 
Theoremfrec2uzzd 8827* The value of  G (see frec2uz0d 8826) is an integer. (Contributed by Jim Kingdon, 16-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   &     om   =>     G ` 
 ZZ
 
Theoremfrec2uzsucd 8828* The value of  G (see frec2uz0d 8826) at a successor. (Contributed by Jim Kingdon, 16-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   &     om   =>     G `  suc  G `
  +  1
 
Theoremfrec2uzuzd 8829* The value  G (see frec2uz0d 8826) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   &     om   =>     G `  ZZ>= `  C
 
Theoremfrec2uzltd 8830* Less-than relation for  G (see frec2uz0d 8826). (Contributed by Jim Kingdon, 16-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   &     om   &     om   =>     G `  <  G `
 
Theoremfrec2uzlt2d 8831* The mapping  G (see frec2uz0d 8826) preserves order. (Contributed by Jim Kingdon, 16-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   &     om   &     om   =>     G `
  <  G `
 
Theoremfrec2uzrand 8832* Range of  G (see frec2uz0d 8826). (Contributed by Jim Kingdon, 17-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   =>     ran  G  ZZ>= `  C
 
Theoremfrec2uzf1od 8833*  G (see frec2uz0d 8826) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   =>     G : om
 -1-1-onto-> ZZ>= `  C
 
Theoremfrec2uzisod 8834*  G (see frec2uz0d 8826) is an isomorphism from natural ordinals to upper integers. (Contributed by Jim Kingdon, 17-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   =>     G  Isom  _E 
 ,  <  om ,  ZZ>= `  C
 
Theoremfrecuzrdgrrn 8835* The function  R (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of 
S. (Contributed by Jim Kingdon, 27-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   &     S  V   &     S   &     ZZ>= `  C  S  F  S   &     R frec  ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. ,  <. C ,  >.   =>     D  om  R `  D  ZZ>= `  C  X.  S
 
Theoremfrec2uzrdg 8836* A helper lemma for the value of a recursive definition generator on upper integers (typically either  NN or  NN0) with characteristic function 
F , and initial value . This lemma shows that evaluating  R at an element of  om gives an ordered pair whose first element is the index (translated from  om to  ZZ>= `  C). See comment in frec2uz0d 8826 which describes  G and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   &     S  V   &     S   &     ZZ>= `  C  S  F  S   &     R frec  ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. ,  <. C ,  >.   &     om   =>     R ` 
 <. G `  ,  2nd `  R `  >.
 
Theoremfrecuzrdgrom 8837* The function  R (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.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   &     S  V   &     S   &     ZZ>= `  C  S  F  S   &     R frec  ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. ,  <. C ,  >.   =>     R  Fn  om
 
Theoremfrecuzrdglem 8838* A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   &     S  V   &     S   &     ZZ>= `  C  S  F  S   &     R frec  ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. ,  <. C ,  >.   &     ZZ>= `  C   =>     <. ,  2nd `  R `  `' G ` 
 >.  ran  R
 
Theoremfrecuzrdgfn 8839* The recursive definition generator on upper integers is a function. See comment in frec2uz0d 8826 for the description of  G as the mapping from  om to  ZZ>= `  C. (Contributed by Jim Kingdon, 26-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   &     S  V   &     S   &     ZZ>= `  C  S  F  S   &     R frec  ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. ,  <. C ,  >.   &     T  ran  R   =>     T  Fn  ZZ>= `  C
 
Theoremfrecuzrdgcl 8840* Closure law for the recursive definition generator on upper integers. See comment in frec2uz0d 8826 for the description of  G as the mapping from 
om to  ZZ>= `  C. (Contributed by Jim Kingdon, 31-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   &     S  V   &     S   &     ZZ>= `  C  S  F  S   &     R frec  ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. ,  <. C ,  >.   &     T  ran  R   =>     ZZ>= `  C  T `  S
 
Theoremfrecuzrdg0 8841* Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 8826 for the description of  G as the mapping from 
om to  ZZ>= `  C. (Contributed by Jim Kingdon, 27-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   &     S  V   &     S   &     ZZ>= `  C  S  F  S   &     R frec  ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. ,  <. C ,  >.   &     T  ran  R   =>     T `  C
 
Theoremfrecuzrdgsuc 8842* Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 8826 for the description of  G as the mapping from 
om to  ZZ>= `  C. (Contributed by Jim Kingdon, 28-May-2020.)
 C  ZZ   &     G frec  ZZ  |->  +  1 ,  C   &     S  V   &     S   &     ZZ>= `  C  S  F  S   &     R frec  ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. ,  <. C ,  >.   &     T  ran  R   =>     ZZ>= `  C  T `  +  1  F T `
 
Theoremuzenom 8843 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
 Z  ZZ>= `  M   =>     M  ZZ  Z  ~~  om
 
Theoremfrecfzennn 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.)
 G frec  ZZ  |->  +  1 ,  0   =>     N  NN0  1 ... N 
 ~~  `' G `  N
 
Theoremfrecfzen2 8845 The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.)
 G frec  ZZ  |->  +  1 ,  0   =>     N  ZZ>=
 `  M  M ... N 
 ~~  `' G `  N  +  1  -  M
 
Theoremfrechashgf1o 8846  G maps  om one-to-one onto  NN0. (Contributed by Jim Kingdon, 19-May-2020.)
 G frec  ZZ  |->  +  1 ,  0   =>     G : om -1-1-onto-> NN0
 
Theoremfzfig 8847 A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.)
 M  ZZ  N  ZZ  M ... N  Fin
 
Theoremfzfigd 8848 Deduction form of fzfig 8847. (Contributed by Jim Kingdon, 21-May-2020.)
 M  ZZ   &     N  ZZ   =>     M ... N 
 Fin
 
Theoremfzofig 8849 Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.)
 M  ZZ  N  ZZ  M..^ N  Fin
 
Theoremnn0ennn 8850 The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)

 NN0  ~~  NN
 
Theoremnnenom 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.)

 NN  ~~  om
 
3.5.8  The infinite sequence builder "seq"
 
Syntaxcseq 8852 Extend class notation with recursive sequence builder.
 seq M  .+  ,  F ,  S
 
Definitiondf-iseq 8853* Define a general-purpose operation that builds a recursive sequence (i.e. a function on the positive integers  NN 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  F with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence  seq 1  +  ,  F with values 1, 3/2, 7/4, 15/8,.., so that  seq 1  +  ,  F `  1  1,  seq 1  +  ,  F `  2 3/2, etc. In other words, 
seq M  +  ,  F transforms a sequence  F into an infinite series.

Internally, the frec function generates as its values a set of ordered pairs starting at 
<. M ,  F `
 M >., 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.)


 seq M  .+  ,  F ,  S  ran frec  ZZ>= `  M ,  S  |->  <.  +  1 , 
 .+  F `  +  1 >. ,  <. M ,  F `
  M >.
 
Theoremiseqeq1 8854 Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
 M  N  seq M  .+  ,  F ,  S  seq N 
 .+  ,  F ,  S
 
Theoremiseqeq2 8855 Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
 .+  Q  seq M  .+  ,  F ,  S  seq M Q ,  F ,  S
 
Theoremiseqeq3 8856 Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
 F  G  seq M  .+  ,  F ,  S  seq M 
 .+  ,  G ,  S
 
Theoremiseqeq4 8857 Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
 S  T  seq M  .+  ,  F ,  S  seq M 
 .+  ,  F ,  T
 
Theoremnfiseq 8858 Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
 F/_ M   &     F/_  .+   &     F/_ F   &     F/_ S   =>     F/_  seq M 
 .+  ,  F ,  S
 
Theoremiseqovex 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.)
 ZZ>= `  M  F `  S   &     S  S  .+  S   =>     ZZ>= `  M  S  ZZ>=
 `  M ,  S  |-> 
 .+  F `  +  1  S
 
Theoremiseqval 8860* Value of the sequence builder function. (Contributed by Jim Kingdon, 30-May-2020.)
 R frec  ZZ>= `  M ,  S  |->  <.  +  1 ,  ZZ>=
 `  M ,  S  |-> 
 .+  F `  +  1 >. ,  <. M ,  F `  M >.   &     ZZ>= `  M  F `  S   &     S  S  .+  S   =>     seq M  .+  ,  F ,  S  ran  R
 
Theoremiseqfn 8861* The sequence builder function is a function. (Contributed by Jim Kingdon, 30-May-2020.)
 M  ZZ   &     S  V   &     ZZ>= `  M  F `  S   &     S  S  .+  S   =>     seq M  .+  ,  F ,  S  Fn  ZZ>=
 `  M
 
Theoremiseq1 8862* Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 31-May-2020.)
 M  ZZ   &     S  V   &     ZZ>= `  M  F `  S   &     S  S  .+  S   =>     seq M  .+  ,  F ,  S `  M  F `  M
 
Theoremiseqcl 8863* Closure properties of the recursive sequence builder. (Contributed by Jim Kingdon, 1-Jun-2020.)
 N  ZZ>= `  M   &     S  V   &     ZZ>= `  M  F `  S   &     S  S  .+  S   =>     seq M  .+  ,  F ,  S `  N  S
 
Theoremiseqp1 8864* Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 31-May-2020.)
 N  ZZ>= `  M   &     S  V   &     ZZ>= `  M  F `  S   &     S  S  .+  S   =>     seq M  .+  ,  F ,  S `  N  +  1  seq M 
 .+  ,  F ,  S `  N 
 .+  F `  N  +  1
 
Theoremiseqfveq2 8865* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
 K  ZZ>= `  M   &     seq M  .+  ,  F ,  S `  K  G `  K   &     S  V   &     ZZ>= `  M  F `  S   &     ZZ>= `  K  G `  S   &     S  S  .+  S   &     N  ZZ>= `  K   &     k  K  +  1 ... N  F `
  k  G `  k   =>     seq M  .+  ,  F ,  S `  N 
 seq K  .+  ,  G ,  S
 `  N
 
Theoremiseqfeq2 8866* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
 K  ZZ>= `  M   &     seq M  .+  ,  F ,  S `  K  G `  K   &     S  V   &     ZZ>= `  M  F `  S   &     ZZ>= `  K  G `  S   &     S  S  .+  S   &     k  ZZ>= `  K  +  1  F `
  k  G `  k   =>     seq M  .+  ,  F ,  S  |`  ZZ>= `  K  seq K 
 .+  ,  G ,  S
 
Theoremiseqfveq 8867* Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)
 N  ZZ>= `  M   &     k  M ... N  F `
  k  G `  k   &     S  V   &     ZZ>= `  M  F `  S   &     ZZ>= `  M  G `  S   &     S  S  .+  S   =>     seq M  .+  ,  F ,  S `  N 
 seq M  .+  ,  G ,  S
 `  N
 
3.5.9  Integer powers
 
Syntaxcexp 8868 Extend class notation to include exponentiation of a complex number to an integer power.
 ^
 
Definitiondf-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  0 ^ 0  1 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  0 ,  <  0 gives the value  1  0, so we will avoid this case in our theorems. (Contributed by Jim Kingodon, 7-Jun-2020.)

 ^  CC ,  ZZ  |->  if  0 ,  1 ,  if 0  <  ,  seq 1  x.  ,  NN  X.  { } ,  CC `  , 
 1  seq 1  x.  ,  NN  X.  { } ,  CC `  -u
 
Theoremexpivallem 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.)
 CC #  0  N  NN  seq 1  x.  ,  NN 
 X.  { } ,  CC `  N #  0
 
Theoremexpival 8871 Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)
 CC  N  ZZ #  0  0  <_  N  ^ N  if N  0 ,  1 ,  if 0  <  N ,  seq 1  x.  ,  NN  X.  { } ,  CC `  N ,  1  seq 1  x.  ,  NN  X.  { } ,  CC `  -u N
 
Theoremexpinnval 8872 Value of exponentiation to positive integer powers. (Contributed by Jim Kingdon, 8-Jun-2020.)
 CC  N  NN  ^ N  seq 1  x.  ,  NN  X.  { } ,  CC `  N
 
Theoremexp0 8873 Value of a complex number raised to the 0th power. Note that under our definition,  0 ^ 0  1, 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.)
 CC  ^ 0  1
 
Theorem0exp0e1 8874  0 ^
0  1 (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.)
 0 ^ 0  1
 
Theoremexp1 8875 Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.)
 CC  ^ 1
 
Theoremexpp1 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.)
 CC  N  NN0  ^ N  +  1  ^ N  x.
 
Theoremexpnegap0 8877 Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
 CC #  0  N  NN0  ^ -u N  1  ^ N
 
Theoremexpineg2 8878 Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
 CC #  0  N  CC  -u N  NN0  ^ N  1  ^ -u N
 
Theoremexpn1ap0 8879 A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.)
 CC #  0  ^ -u 1  1
 
Theoremexpcllem 8880* Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)
 F  C_  CC   &     F  F  x.  F   &     1  F   =>     F  NN0  ^  F
 
Theoremexpcl2lemap 8881* Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.)
 F  C_  CC   &     F  F  x.  F   &     1  F   &     F #  0  1  F   =>     F #  0  ZZ  ^  F
 
Theoremnnexpcl 8882 Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
 NN  N  NN0  ^ N  NN
 
Theoremnn0expcl 8883 Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
 NN0  N  NN0  ^ N  NN0
 
Theoremzexpcl 8884 Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
 ZZ  N  NN0  ^ N  ZZ
 
Theoremqexpcl 8885 Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
 QQ  N  NN0  ^ N  QQ
 
Theoremreexpcl 8886 Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
 RR  N  NN0  ^ N  RR
 
Theoremexpcl 8887 Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
 CC  N  NN0  ^ N  CC
 
Theoremrpexpcl 8888 Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
 RR+  N  ZZ  ^ N  RR+
 
Theoremreexpclzap 8889 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
 RR #  0  N  ZZ  ^ N  RR
 
Theoremqexpclz 8890 Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
 QQ  =/=  0  N  ZZ  ^ N  QQ
 
Theoremm1expcl2 8891 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
 N  ZZ  -u 1 ^ N  { -u 1 ,  1 }
 
Theoremm1expcl 8892 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
 N  ZZ  -u 1 ^ N  ZZ
 
Theoremexpclzaplem 8893* Closure law for integer exponentiation. Lemma for expclzap 8894 and expap0i 8901. (Contributed by Jim Kingdon, 9-Jun-2020.)
 CC #  0  N  ZZ  ^ N  { 
 CC  | #  0 }
 
Theoremexpclzap 8894 Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
 CC #  0  N  ZZ  ^ N  CC
 
Theoremnn0expcli 8895 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN0   &     N  NN0   =>     ^ N  NN0
 
Theoremnn0sqcl 8896 The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 NN0  ^ 2  NN0
 
Theoremexpm1t 8897 Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
 CC  N  NN  ^ N  ^ N  -  1  x.
 
Theorem1exp 8898 Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
 N  ZZ  1 ^ N  1
 
Theoremexpap0 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.)
 CC  N  NN 
 ^ N #  0 #  0
 
Theoremexpeq0 8900 Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.)
 CC  N  NN 
 ^ N  0  0
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