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Theorem List for Intuitionistic Logic Explorer - 10301-10400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremzmodcld 10301 Closure law for the modulo operation restricted to integers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  NN )   =>    |-  ( ph  ->  ( A  mod  B )  e.  NN0 )
 
Theoremzmodfz 10302 An integer mod  B lies in the first  B nonnegative integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  mod  B )  e.  ( 0
 ... ( B  -  1 ) ) )
 
Theoremzmodfzo 10303 An integer mod  B lies in the first  B nonnegative integers. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  mod  B )  e.  ( 0..^ B ) )
 
Theoremzmodfzp1 10304 An integer mod  B lies in the first  B  +  1 nonnegative integers. (Contributed by AV, 27-Oct-2018.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  mod  B )  e.  ( 0
 ... B ) )
 
Theoremmodqid 10305 Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  (
 0  <_  A  /\  A  <  B ) ) 
 ->  ( A  mod  B )  =  A )
 
Theoremmodqid0 10306 A positive real number modulo itself is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( N  e.  QQ  /\  0  <  N )  ->  ( N  mod  N )  =  0 )
 
Theoremmodqid2 10307 Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( ( A  mod  B )  =  A  <->  ( 0  <_  A  /\  A  <  B ) ) )
 
Theoremzmodid2 10308 Identity law for modulo restricted to integers. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( M 
 mod  N )  =  M  <->  M  e.  ( 0 ... ( N  -  1
 ) ) ) )
 
Theoremzmodidfzo 10309 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
 |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( M 
 mod  N )  =  M  <->  M  e.  ( 0..^ N ) ) )
 
Theoremzmodidfzoimp 10310 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
 |-  ( M  e.  (
 0..^ N )  ->  ( M  mod  N )  =  M )
 
Theoremq0mod 10311 Special case: 0 modulo a positive real number is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( N  e.  QQ  /\  0  <  N )  ->  ( 0  mod 
 N )  =  0 )
 
Theoremq1mod 10312 Special case: 1 modulo a real number greater than 1 is 1. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( N  e.  QQ  /\  1  <  N )  ->  ( 1  mod 
 N )  =  1 )
 
Theoremmodqabs 10313 Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  0  <  B )   &    |-  ( ph  ->  C  e.  QQ )   &    |-  ( ph  ->  B 
 <_  C )   =>    |-  ( ph  ->  (
 ( A  mod  B )  mod  C )  =  ( A  mod  B ) )
 
Theoremmodqabs2 10314 Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( ( A  mod  B )  mod  B )  =  ( A  mod  B ) )
 
Theoremmodqcyc 10315 The modulo operation is periodic. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  <  B ) )  ->  ( ( A  +  ( N  x.  B ) )  mod  B )  =  ( A  mod  B ) )
 
Theoremmodqcyc2 10316 The modulo operation is periodic. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  <  B ) )  ->  ( ( A  -  ( B  x.  N ) )  mod  B )  =  ( A  mod  B ) )
 
Theoremmodqadd1 10317 Addition property of the modulo operation. (Contributed by Jim Kingdon, 22-Oct-2021.)
 |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  C  e.  QQ )   &    |-  ( ph  ->  D  e.  QQ )   &    |-  ( ph  ->  0  <  D )   &    |-  ( ph  ->  ( A  mod  D )  =  ( B 
 mod  D ) )   =>    |-  ( ph  ->  ( ( A  +  C )  mod  D )  =  ( ( B  +  C )  mod  D ) )
 
Theoremmodqaddabs 10318 Absorption law for modulo. (Contributed by Jim Kingdon, 22-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( ( ( A 
 mod  C )  +  ( B  mod  C ) ) 
 mod  C )  =  ( ( A  +  B )  mod  C ) )
 
Theoremmodqaddmod 10319 The sum of a number modulo a modulus and another number equals the sum of the two numbers modulo the same modulus. (Contributed by Jim Kingdon, 23-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( ( A 
 mod  M )  +  B )  mod  M )  =  ( ( A  +  B )  mod  M ) )
 
Theoremmulqaddmodid 10320 The sum of a positive rational number less than an upper bound and the product of an integer and the upper bound is the positive rational number modulo the upper bound. (Contributed by Jim Kingdon, 23-Oct-2021.)
 |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M ) ) )  ->  ( ( ( N  x.  M )  +  A )  mod  M )  =  A )
 
Theoremmulp1mod1 10321 The product of an integer and an integer greater than 1 increased by 1 is 1 modulo the integer greater than 1. (Contributed by AV, 15-Jul-2021.)
 |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>=
 `  2 ) ) 
 ->  ( ( ( N  x.  A )  +  1 )  mod  N )  =  1 )
 
Theoremmodqmuladd 10322* Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  B  e.  (
 0 [,) M ) )   &    |-  ( ph  ->  M  e.  QQ )   &    |-  ( ph  ->  0  <  M )   =>    |-  ( ph  ->  ( ( A  mod  M )  =  B  <->  E. k  e.  ZZ  A  =  ( (
 k  x.  M )  +  B ) ) )
 
Theoremmodqmuladdim 10323* Implication of a decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
 |-  ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M ) 
 ->  ( ( A  mod  M )  =  B  ->  E. k  e.  ZZ  A  =  ( ( k  x.  M )  +  B ) ) )
 
Theoremmodqmuladdnn0 10324* Implication of a decomposition of a nonnegative integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
 |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  ( ( A  mod  M )  =  B  ->  E. k  e.  NN0  A  =  ( ( k  x.  M )  +  B ) ) )
 
Theoremqnegmod 10325 The negation of a number modulo a positive number is equal to the difference of the modulus and the number modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N ) 
 ->  ( -u A  mod  N )  =  ( ( N  -  A )  mod  N ) )
 
Theoremm1modnnsub1 10326 Minus one modulo a positive integer is equal to the integer minus one. (Contributed by AV, 14-Jul-2021.)
 |-  ( M  e.  NN  ->  ( -u 1  mod  M )  =  ( M  -  1 ) )
 
Theoremm1modge3gt1 10327 Minus one modulo an integer greater than two is greater than one. (Contributed by AV, 14-Jul-2021.)
 |-  ( M  e.  ( ZZ>=
 `  3 )  -> 
 1  <  ( -u 1  mod  M ) )
 
Theoremaddmodid 10328 The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by Alexander van der Vekens, 30-Oct-2018.) (Proof shortened by AV, 5-Jul-2020.)
 |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  ( ( M  +  A )  mod  M )  =  A )
 
Theoremaddmodidr 10329 The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by AV, 19-Mar-2021.)
 |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  ( ( A  +  M )  mod  M )  =  A )
 
Theoremmodqadd2mod 10330 The sum of a number modulo a modulus and another number equals the sum of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( B  +  ( A  mod  M ) )  mod  M )  =  ( ( B  +  A )  mod  M ) )
 
Theoremmodqm1p1mod0 10331 If a number modulo a modulus equals the modulus decreased by 1, the first number increased by 1 modulo the modulus equals 0. (Contributed by Jim Kingdon, 24-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M ) 
 ->  ( ( A  mod  M )  =  ( M  -  1 )  ->  ( ( A  +  1 )  mod  M )  =  0 ) )
 
Theoremmodqltm1p1mod 10332 If a number modulo a modulus is less than the modulus decreased by 1, the first number increased by 1 modulo the modulus equals the first number modulo the modulus, increased by 1. (Contributed by Jim Kingdon, 24-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  ( A  mod  M )  < 
 ( M  -  1
 ) )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( A  +  1 )  mod  M )  =  ( ( A 
 mod  M )  +  1 ) )
 
Theoremmodqmul1 10333 Multiplication property of the modulo operation. Note that the multiplier  C must be an integer. (Contributed by Jim Kingdon, 24-Oct-2021.)
 |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  D  e.  QQ )   &    |-  ( ph  ->  0  <  D )   &    |-  ( ph  ->  ( A  mod  D )  =  ( B 
 mod  D ) )   =>    |-  ( ph  ->  ( ( A  x.  C )  mod  D )  =  ( ( B  x.  C )  mod  D ) )
 
Theoremmodqmul12d 10334 Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Jim Kingdon, 24-Oct-2021.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  D  e.  ZZ )   &    |-  ( ph  ->  E  e.  QQ )   &    |-  ( ph  ->  0  <  E )   &    |-  ( ph  ->  ( A  mod  E )  =  ( B  mod  E ) )   &    |-  ( ph  ->  ( C  mod  E )  =  ( D  mod  E ) )   =>    |-  ( ph  ->  (
 ( A  x.  C )  mod  E )  =  ( ( B  x.  D )  mod  E ) )
 
Theoremmodqnegd 10335 Negation property of the modulo operation. (Contributed by Jim Kingdon, 24-Oct-2021.)
 |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  C  e.  QQ )   &    |-  ( ph  ->  0  <  C )   &    |-  ( ph  ->  ( A  mod  C )  =  ( B  mod  C ) )   =>    |-  ( ph  ->  ( -u A  mod  C )  =  ( -u B  mod  C ) )
 
Theoremmodqadd12d 10336 Additive property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  C  e.  QQ )   &    |-  ( ph  ->  D  e.  QQ )   &    |-  ( ph  ->  E  e.  QQ )   &    |-  ( ph  ->  0  <  E )   &    |-  ( ph  ->  ( A  mod  E )  =  ( B  mod  E ) )   &    |-  ( ph  ->  ( C  mod  E )  =  ( D  mod  E ) )   =>    |-  ( ph  ->  (
 ( A  +  C )  mod  E )  =  ( ( B  +  D )  mod  E ) )
 
Theoremmodqsub12d 10337 Subtraction property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  C  e.  QQ )   &    |-  ( ph  ->  D  e.  QQ )   &    |-  ( ph  ->  E  e.  QQ )   &    |-  ( ph  ->  0  <  E )   &    |-  ( ph  ->  ( A  mod  E )  =  ( B  mod  E ) )   &    |-  ( ph  ->  ( C  mod  E )  =  ( D  mod  E ) )   =>    |-  ( ph  ->  (
 ( A  -  C )  mod  E )  =  ( ( B  -  D )  mod  E ) )
 
Theoremmodqsubmod 10338 The difference of a number modulo a modulus and another number equals the difference of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( ( A 
 mod  M )  -  B )  mod  M )  =  ( ( A  -  B )  mod  M ) )
 
Theoremmodqsubmodmod 10339 The difference of a number modulo a modulus and another number modulo the same modulus equals the difference of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( ( A 
 mod  M )  -  ( B  mod  M ) ) 
 mod  M )  =  ( ( A  -  B )  mod  M ) )
 
Theoremq2txmodxeq0 10340 Two times a positive number modulo the number is zero. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ( X  e.  QQ  /\  0  <  X )  ->  ( ( 2  x.  X )  mod  X )  =  0 )
 
Theoremq2submod 10341 If a number is between a modulus and twice the modulus, the first number modulo the modulus equals the first number minus the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B 
 <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  mod  B )  =  ( A  -  B ) )
 
Theoremmodifeq2int 10342 If a nonnegative integer is less than twice a positive integer, the nonnegative integer modulo the positive integer equals the nonnegative integer or the nonnegative integer minus the positive integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  ->  ( A  mod  B )  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
 
Theoremmodaddmodup 10343 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.)
 |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  ->  ( ( B  +  ( A  mod  M ) )  -  M )  =  ( ( B  +  A )  mod  M ) ) )
 
Theoremmodaddmodlo 10344 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.)
 |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  ->  ( B  +  ( A  mod  M ) )  =  ( ( B  +  A )  mod  M ) ) )
 
Theoremmodqmulmod 10345 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.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( ( A 
 mod  M )  x.  B )  mod  M )  =  ( ( A  x.  B )  mod  M ) )
 
Theoremmodqmulmodr 10346 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.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( A  x.  ( B  mod  M ) )  mod  M )  =  ( ( A  x.  B )  mod  M ) )
 
Theoremmodqaddmulmod 10347 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.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( A  +  ( ( B  mod  M )  x.  C ) )  mod  M )  =  ( ( A  +  ( B  x.  C ) )  mod  M ) )
 
Theoremmodqdi 10348 Distribute multiplication over a modulo operation. (Contributed by Jim Kingdon, 26-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( A  x.  ( B  mod  C ) )  =  ( ( A  x.  B )  mod  ( A  x.  C ) ) )
 
Theoremmodqsubdir 10349 Distribute the modulo operation over a subtraction. (Contributed by Jim Kingdon, 26-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( ( B  mod  C )  <_  ( A  mod  C )  <->  ( ( A  -  B )  mod  C )  =  ( ( A  mod  C )  -  ( B  mod  C ) ) ) )
 
Theoremmodqeqmodmin 10350 A rational number equals the difference of the rational number and a modulus modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M ) 
 ->  ( A  mod  M )  =  ( ( A  -  M )  mod  M ) )
 
Theoremmodfzo0difsn 10351* 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.)
 |-  ( ( J  e.  ( 0..^ N )  /\  K  e.  ( (
 0..^ N )  \  { J } ) ) 
 ->  E. i  e.  (
 1..^ N ) K  =  ( ( i  +  J )  mod  N ) )
 
Theoremmodsumfzodifsn 10352 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.)
 |-  ( ( J  e.  ( 0..^ N )  /\  K  e.  ( 1..^ N ) )  ->  ( ( K  +  J )  mod  N )  e.  ( ( 0..^ N )  \  { J } ) )
 
Theoremmodlteq 10353 Two nonnegative integers less than the modulus are equal iff they are equal modulo the modulus. (Contributed by AV, 14-Mar-2021.)
 |-  ( ( I  e.  ( 0..^ N ) 
 /\  J  e.  (
 0..^ N ) ) 
 ->  ( ( I  mod  N )  =  ( J 
 mod  N )  <->  I  =  J ) )
 
Theoremaddmodlteq 10354 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.)
 |-  ( ( I  e.  ( 0..^ N ) 
 /\  J  e.  (
 0..^ N )  /\  S  e.  ZZ )  ->  ( ( ( I  +  S )  mod  N )  =  ( ( J  +  S ) 
 mod  N )  <->  I  =  J ) )
 
4.6.3  Miscellaneous theorems about integers
 
Theoremfrec2uz0d 10355* 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.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  ( G `  (/) )  =  C )
 
Theoremfrec2uzzd 10356* The value of  G (see frec2uz0d 10355) is an integer. (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   =>    |-  ( ph  ->  ( G `  A )  e. 
 ZZ )
 
Theoremfrec2uzsucd 10357* The value of  G (see frec2uz0d 10355) at a successor. (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   =>    |-  ( ph  ->  ( G `  suc  A )  =  ( ( G `
  A )  +  1 ) )
 
Theoremfrec2uzuzd 10358* The value  G (see frec2uz0d 10355) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   =>    |-  ( ph  ->  ( G `  A )  e.  ( ZZ>= `  C )
 )
 
Theoremfrec2uzltd 10359* Less-than relation for  G (see frec2uz0d 10355). (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   &    |-  ( ph  ->  B  e.  om )   =>    |-  ( ph  ->  ( A  e.  B  ->  ( G `  A )  <  ( G `  B ) ) )
 
Theoremfrec2uzlt2d 10360* The mapping  G (see frec2uz0d 10355) preserves order. (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   &    |-  ( ph  ->  B  e.  om )   =>    |-  ( ph  ->  ( A  e.  B  <->  ( G `  A )  <  ( G `
  B ) ) )
 
Theoremfrec2uzrand 10361* Range of  G (see frec2uz0d 10355). (Contributed by Jim Kingdon, 17-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  ran  G  =  ( ZZ>= `  C )
 )
 
Theoremfrec2uzf1od 10362*  G (see frec2uz0d 10355) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  G : om
 -1-1-onto-> ( ZZ>= `  C )
 )
 
Theoremfrec2uzisod 10363*  G (see frec2uz0d 10355) is an isomorphism from natural ordinals to upper integers. (Contributed by Jim Kingdon, 17-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  G  Isom  _E  ,  <  ( om ,  ( ZZ>= `  C ) ) )
 
Theoremfrecuzrdgrrn 10364* 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, 28-Mar-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   =>    |-  ( ( ph  /\  D  e.  om )  ->  ( R `  D )  e.  ( ( ZZ>= `  C )  X.  S ) )
 
Theoremfrec2uzrdg 10365* A helper lemma for the value of a recursive definition generator on upper integers (typically either  NN or  NN0) with characteristic function 
F ( x ,  y ) and initial value  A. 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 10355 which describes  G and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  B  e.  om )   =>    |-  ( ph  ->  ( R `  B )  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B ) ) >. )
 
Theoremfrecuzrdgrcl 10366* 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, 1-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   =>    |-  ( ph  ->  R : om --> ( ( ZZ>= `  C )  X.  S ) )
 
Theoremfrecuzrdglem 10367* A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  B  e.  ( ZZ>= `  C ) )   =>    |-  ( ph  ->  <. B ,  ( 2nd `  ( R `  ( `' G `  B ) ) )
 >.  e.  ran  R )
 
Theoremfrecuzrdgtcl 10368* The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10355 for the description of  G as the mapping from  om to  ( ZZ>= `  C
). (Contributed by Jim Kingdon, 26-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  T  =  ran  R )   =>    |-  ( ph  ->  T :
 ( ZZ>= `  C ) --> S )
 
Theoremfrecuzrdg0 10369* Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 10355 for the description of  G as the mapping from  om to  ( ZZ>= `  C
). (Contributed by Jim Kingdon, 27-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  T  =  ran  R )   =>    |-  ( ph  ->  ( T `  C )  =  A )
 
Theoremfrecuzrdgsuc 10370* Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10355 for the description of  G as the mapping from 
om to  ( ZZ>= `  C
). (Contributed by Jim Kingdon, 28-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  T  =  ran  R )   =>    |-  ( ( ph  /\  B  e.  ( ZZ>= `  C )
 )  ->  ( T `  ( B  +  1 ) )  =  ( B F ( T `
  B ) ) )
 
Theoremfrecuzrdgrclt 10371* The function  R (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of  S. Similar to frecuzrdgrcl 10366 except that  S and  T need not be the same. (Contributed by Jim Kingdon, 22-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   =>    |-  ( ph  ->  R : om --> ( ( ZZ>= `  C )  X.  S ) )
 
Theoremfrecuzrdgg 10372* Lemma for other theorems involving the the recursive definition generator on upper integers. Evaluating  R at a natural number gives an ordered pair whose first element is the mapping of that natural number via  G. (Contributed by Jim Kingdon, 23-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  N  e.  om )   &    |-  G  = frec ( ( x  e. 
 ZZ  |->  ( x  +  1 ) ) ,  C )   =>    |-  ( ph  ->  ( 1st `  ( R `  N ) )  =  ( G `  N ) )
 
Theoremfrecuzrdgdomlem 10373* The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  dom  ran  R  =  ( ZZ>= `  C ) )
 
Theoremfrecuzrdgdom 10374* The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   =>    |-  ( ph  ->  dom  ran  R  =  ( ZZ>= `  C ) )
 
Theoremfrecuzrdgfunlem 10375* The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  Fun  ran  R )
 
Theoremfrecuzrdgfun 10376* The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   =>    |-  ( ph  ->  Fun  ran  R )
 
Theoremfrecuzrdgtclt 10377* The recursive definition generator on upper integers is a function. (Contributed by Jim Kingdon, 22-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  P  =  ran  R )   =>    |-  ( ph  ->  P :
 ( ZZ>= `  C ) --> S )
 
Theoremfrecuzrdg0t 10378* Initial value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 28-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  P  =  ran  R )   =>    |-  ( ph  ->  ( P `  C )  =  A )
 
Theoremfrecuzrdgsuctlem 10379* Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10355 for the description of  G as the mapping from  om to  ( ZZ>= `  C ). (Contributed by Jim Kingdon, 29-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  P  =  ran  R )   =>    |-  ( ( ph  /\  B  e.  ( ZZ>= `  C ) )  ->  ( P `  ( B  +  1 ) )  =  ( B F ( P `  B ) ) )
 
Theoremfrecuzrdgsuct 10380* Successor value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 29-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  P  =  ran  R )   =>    |-  ( ( ph  /\  B  e.  ( ZZ>= `  C )
 )  ->  ( P `  ( B  +  1 ) )  =  ( B F ( P `
  B ) ) )
 
Theoremuzenom 10381 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  Z  ~~  om )
 
Theoremfrecfzennn 10382 The cardinality of a finite set of sequential integers. (See frec2uz0d 10355 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   =>    |-  ( N  e.  NN0  ->  ( 1 ... N ) 
 ~~  ( `' G `  N ) )
 
Theoremfrecfzen2 10383 The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( M ... N ) 
 ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
 
Theoremfrechashgf1o 10384  G maps  om one-to-one onto  NN0. (Contributed by Jim Kingdon, 19-May-2020.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   =>    |-  G : om -1-1-onto-> NN0
 
Theoremfrec2uzled 10385* The mapping  G (see frec2uz0d 10355) preserves order. (Contributed by Jim Kingdon, 24-Feb-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   &    |-  ( ph  ->  B  e.  om )   =>    |-  ( ph  ->  ( A  C_  B  <->  ( G `  A )  <_  ( G `
  B ) ) )
 
Theoremfzfig 10386 A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N )  e.  Fin )
 
Theoremfzfigd 10387 Deduction form of fzfig 10386. (Contributed by Jim Kingdon, 21-May-2020.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( M ... N )  e.  Fin )
 
Theoremfzofig 10388 Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M..^ N )  e.  Fin )
 
Theoremnn0ennn 10389 The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
 |- 
 NN0  ~~  NN
 
Theoremnnenom 10390 The set of positive integers (as a subset of complex numbers) is equinumerous to omega (the set of natural numbers as ordinals). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |- 
 NN  ~~  om
 
Theoremnnct 10391  NN is dominated by  om. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |- 
 NN  ~<_  om
 
Theoremuzennn 10392 An upper integer set is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 30-Jul-2023.)
 |-  ( M  e.  ZZ  ->  ( ZZ>= `  M )  ~~  NN )
 
Theoremfnn0nninf 10393* A function from  NN0 into ℕ. (Contributed by Jim Kingdon, 16-Jul-2022.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   &    |-  F  =  ( n  e.  om  |->  ( i  e. 
 om  |->  if ( i  e.  n ,  1o ,  (/) ) ) )   =>    |-  ( F  o.  `' G ) : NN0 -->
 
Theoremfxnn0nninf 10394* A function from NN0* into ℕ. (Contributed by Jim Kingdon, 16-Jul-2022.) TODO: use infnninf 7100 instead of infnninfOLD 7101. More generally, this theorem and most theorems in this section could use an extended  G defined by  G  =  (frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )  u.  <. om , +oo >. ) and  F  =  ( n  e.  suc  om  |->  ( i  e.  om  |->  if ( i  e.  n ,  1o ,  (/) ) ) ) as in nnnninf2 7103.
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   &    |-  F  =  ( n  e.  om  |->  ( i  e. 
 om  |->  if ( i  e.  n ,  1o ,  (/) ) ) )   &    |-  I  =  ( ( F  o.  `' G )  u.  { <. +oo ,  ( om  X. 
 { 1o } ) >. } )   =>    |-  I :NN0* -->
 
Theorem0tonninf 10395* The mapping of zero into ℕ is the sequence of all zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   &    |-  F  =  ( n  e.  om  |->  ( i  e. 
 om  |->  if ( i  e.  n ,  1o ,  (/) ) ) )   &    |-  I  =  ( ( F  o.  `' G )  u.  { <. +oo ,  ( om  X. 
 { 1o } ) >. } )   =>    |-  ( I `  0
 )  =  ( x  e.  om  |->  (/) )
 
Theorem1tonninf 10396* The mapping of one into ℕ is a sequence which is a one followed by zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   &    |-  F  =  ( n  e.  om  |->  ( i  e. 
 om  |->  if ( i  e.  n ,  1o ,  (/) ) ) )   &    |-  I  =  ( ( F  o.  `' G )  u.  { <. +oo ,  ( om  X. 
 { 1o } ) >. } )   =>    |-  ( I `  1
 )  =  ( x  e.  om  |->  if ( x  =  (/) ,  1o ,  (/) ) )
 
Theoreminftonninf 10397* The mapping of +oo into ℕ is the sequence of all ones. (Contributed by Jim Kingdon, 17-Jul-2022.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   &    |-  F  =  ( n  e.  om  |->  ( i  e. 
 om  |->  if ( i  e.  n ,  1o ,  (/) ) ) )   &    |-  I  =  ( ( F  o.  `' G )  u.  { <. +oo ,  ( om  X. 
 { 1o } ) >. } )   =>    |-  ( I ` +oo )  =  ( x  e.  om  |->  1o )
 
4.6.4  Strong induction over upper sets of integers
 
Theoremuzsinds 10398* Strong (or "total") induction principle over an upper set of integers. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  N  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  ( ZZ>= `  M )  ->  ( A. y  e.  ( M ... ( x  -  1
 ) ) ps  ->  ph ) )   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  ch )
 
Theoremnnsinds 10399* Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  N  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  NN  ->  (
 A. y  e.  (
 1 ... ( x  -  1 ) ) ps 
 ->  ph ) )   =>    |-  ( N  e.  NN  ->  ch )
 
Theoremnn0sinds 10400* Strong (or "total") induction principle over the nonnegative integers. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  N  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  NN0  ->  ( A. y  e.  (
 0 ... ( x  -  1 ) ) ps 
 ->  ph ) )   =>    |-  ( N  e.  NN0 
 ->  ch )
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