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Theorem List for Intuitionistic Logic Explorer - 9301-9400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremzmodid2 9301 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 9302 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 9303 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
 |-  ( M  e.  (
 0..^ N )  ->  ( M  mod  N )  =  M )
 
Theoremq0mod 9304 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 9305 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 9306 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 9307 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 9308 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 9309 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 9310 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 9311 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 9312 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 9313 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 9314 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 9315* 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 9316* 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 9317* 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 9318 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 9319 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 9320 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 9321 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 9322 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 9323 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 9324 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 9325 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 9326 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 9327 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 9328 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 9329 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 9330 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 9331 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 9332 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 9333 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 9334 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 9335 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 9336 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 9337 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 9338 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 9339 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 9340 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 9341 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 9342 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 9343 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 9344* 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 9345 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 9346 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 9347 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 ) )
 
3.6.3  Miscellaneous theorems about integers
 
Theoremfrec2uz0d 9348* 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 9349* The value of  G (see frec2uz0d 9348) 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 9350* The value of  G (see frec2uz0d 9348) 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 9351* The value  G (see frec2uz0d 9348) 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 9352* Less-than relation for  G (see frec2uz0d 9348). (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 9353* The mapping  G (see frec2uz0d 9348) 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 9354* Range of  G (see frec2uz0d 9348). (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 9355*  G (see frec2uz0d 9348) 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 9356*  G (see frec2uz0d 9348) 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 9357* 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.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( 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 9358* 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 9348 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  ->  S  e.  V )   &    |-  ( 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 ) ) >. )
 
Theoremfrecuzrdgrom 9359* 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.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( 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  Fn  om )
 
Theoremfrecuzrdglem 9360* 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  ->  S  e.  V )   &    |-  ( 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 )
 
Theoremfrecuzrdgfn 9361* The recursive definition generator on upper integers is a function. See comment in frec2uz0d 9348 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  ->  S  e.  V )   &    |-  ( 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  Fn  ( ZZ>= `  C )
 )
 
Theoremfrecuzrdgcl 9362* Closure law for the recursive definition generator on upper integers. See comment in frec2uz0d 9348 for the description of  G as the mapping from 
om to  ( ZZ>= `  C
). (Contributed by Jim Kingdon, 31-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( 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 )  e.  S )
 
Theoremfrecuzrdg0 9363* Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 9348 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  ->  S  e.  V )   &    |-  ( 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 9364* Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 9348 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  ->  S  e.  V )   &    |-  ( 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 ) ) )
 
Theoremuzenom 9365 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  Z  ~~  om )
 
Theoremfrecfzennn 9366 The cardinality of a finite set of sequential integers. (See frec2uz0d 9348 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 9367 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 9368  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
 
Theoremfzfig 9369 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 9370 Deduction form of fzfig 9369. (Contributed by Jim Kingdon, 21-May-2020.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( M ... N )  e.  Fin )
 
Theoremfzofig 9371 Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M..^ N )  e.  Fin )
 
Theoremnn0ennn 9372 The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
 |- 
 NN0  ~~  NN
 
Theoremnnenom 9373 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
 
3.6.4  The infinite sequence builder "seq"
 
Syntaxcseq 9374 Extend class notation with recursive sequence builder.
 class  seq M (  .+  ,  F ,  S )
 
Definitiondf-iseq 9375* 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 9385 and iseqp1 9388. 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 ,  QQ ) with values 1, 3/2, 7/4, 15/8,.., so that  (  seq 1
(  +  ,  F ,  QQ ) `  1
)  =  1,  (  seq 1
(  +  ,  F ,  QQ ) `  2
)  = 3/2, etc. In other words,  seq M (  +  ,  F ,  QQ ) 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 ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. ( x  +  1 ) ,  (
 y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )
 
Theoremiseqex 9376 Existence of the sequence builder operation. (Contributed by Jim Kingdon, 20-Aug-2021.)
 |- 
 seq M (  .+  ,  F ,  S )  e.  _V
 
Theoremiseqeq1 9377 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 9378 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 9379 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 9380 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 9381 Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
 |-  F/_ x M   &    |-  F/_ x  .+   &    |-  F/_ x F   &    |-  F/_ x S   =>    |-  F/_ x  seq M ( 
 .+  ,  F ,  S )
 
Theoremiseqovex 9382* 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.)
 |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  ->  ( x ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `
  ( z  +  1 ) ) ) ) y )  e.  S )
 
Theoremiseqval 9383* Value of the sequence builder function. (Contributed by Jim Kingdon, 30-May-2020.)
 |-  R  = frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. ( x  +  1 ) ,  ( x ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `
  ( z  +  1 ) ) ) ) y ) >. ) ,  <. M ,  ( F `  M ) >. )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  seq M (  .+  ,  F ,  S )  =  ran  R )
 
Theoremiseqfn 9384* The sequence builder function is a function. (Contributed by Jim Kingdon, 30-May-2020.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  S  e.  V )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  seq M (  .+  ,  F ,  S )  Fn  ( ZZ>=
 `  M ) )
 
Theoremiseq1 9385* Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 31-May-2020.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  S  e.  V )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `  M )  =  ( F `  M ) )
 
Theoremiseqcl 9386* Closure properties of the recursive sequence builder. (Contributed by Jim Kingdon, 1-Jun-2020.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( F `  x )  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ,  S ) `
  N )  e.  S )
 
Theoremiseqf 9387* Range of the recursive sequence builder. (Contributed by Jim Kingdon, 23-Jul-2021.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  x  e.  Z )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x 
 .+  y )  e.  S )   =>    |-  ( ph  ->  seq M (  .+  ,  F ,  S ) : Z --> S )
 
Theoremiseqp1 9388* Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 31-May-2020.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( F `  x )  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ,  S ) `
  ( N  +  1 ) )  =  ( (  seq M (  .+  ,  F ,  S ) `  N )  .+  ( F `  ( N  +  1
 ) ) ) )
 
Theoremiseqss 9389* Specifying a larger universe for 
seq. As long as  F and  .+ are closed over  S, then any set which contains  S can be used as the last argument to  seq. This theorem does not allow  T to be a proper class, however. It also currently requires that  .+ be closed over  T (as well as  S). (Contributed by Jim Kingdon, 18-Aug-2021.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  T  e.  V )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  T  /\  y  e.  T )
 )  ->  ( x  .+  y )  e.  T )   =>    |-  ( ph  ->  seq M (  .+  ,  F ,  S )  =  seq M (  .+  ,  F ,  T ) )
 
Theoremiseqm1 9390* Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  ( M  +  1 )
 ) )   &    |-  ( ph  ->  S  e.  V )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `  N )  =  ( (  seq M ( 
 .+  ,  F ,  S ) `  ( N  -  1 ) ) 
 .+  ( F `  N ) ) )
 
Theoremiseqfveq2 9391* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
 |-  ( ph  ->  K  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  ( G `  K ) )   &    |-  ( ph  ->  S  e.  V )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  K )
 )  ->  ( G `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  K )
 )   &    |-  ( ( ph  /\  k  e.  ( ( K  +  1 ) ... N ) )  ->  ( F `
  k )  =  ( G `  k
 ) )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `  N )  =  ( 
 seq K (  .+  ,  G ,  S ) `
  N ) )
 
Theoremiseqfeq2 9392* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
 |-  ( ph  ->  K  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  ( G `  K ) )   &    |-  ( ph  ->  S  e.  V )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  K )
 )  ->  ( G `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  ( K  +  1 ) ) )  ->  ( F `  k )  =  ( G `  k ) )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ,  S )  |`  ( ZZ>= `  K )
 )  =  seq K (  .+  ,  G ,  S ) )
 
Theoremiseqfveq 9393* Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  =  ( G `  k
 ) )   &    |-  ( ph  ->  S  e.  V )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( G `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `  N )  =  ( 
 seq M (  .+  ,  G ,  S ) `
  N ) )
 
Theoremiseqfeq 9394* Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  S  e.  V )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( F `  k )  =  ( G `  k ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x 
 .+  y )  e.  S )   =>    |-  ( ph  ->  seq M (  .+  ,  F ,  S )  =  seq M (  .+  ,  G ,  S ) )
 
Theoremiseqshft2 9395* Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( F `  k
 )  =  ( G `
  ( k  +  K ) ) )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( F `  x )  e.  S )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  ( M  +  K ) ) ) 
 ->  ( G `  x )  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ,  S ) `
  N )  =  (  seq ( M  +  K ) ( 
 .+  ,  G ,  S ) `  ( N  +  K )
 ) )
 
Theoremiserf 9396* An infinite series of complex terms is a function from  NN to  CC. (Contributed by Jim Kingdon, 15-Aug-2021.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   =>    |-  ( ph  ->  seq M (  +  ,  F ,  CC ) : Z --> CC )
 
Theoremiserfre 9397* An infinite series of real numbers is a function from  NN to  RR. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   =>    |-  ( ph  ->  seq M (  +  ,  F ,  RR ) : Z --> RR )
 
Theoremmonoord 9398* Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  ( M
 ... ( N  -  1 ) ) ) 
 ->  ( F `  k
 )  <_  ( F `  ( k  +  1 ) ) )   =>    |-  ( ph  ->  ( F `  M ) 
 <_  ( F `  N ) )
 
Theoremmonoord2 9399* Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  ( M
 ... ( N  -  1 ) ) ) 
 ->  ( F `  (
 k  +  1 ) )  <_  ( F `  k ) )   =>    |-  ( ph  ->  ( F `  N ) 
 <_  ( F `  M ) )
 
Theoremisermono 9400* The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)
 |-  ( ph  ->  K  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  K )
 )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  RR )   &    |-  ( ( ph  /\  x  e.  ( ( K  +  1 ) ... N ) )  ->  0  <_  ( F `  x ) )   =>    |-  ( ph  ->  (  seq M (  +  ,  F ,  RR ) `  K )  <_  (  seq M (  +  ,  F ,  RR ) `  N ) )
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