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Theorem List for Metamath Proof Explorer - 37801-37900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgbpart7 37801 The (weak) Goldbach partition of 7. (Contributed by AV, 20-Jul-2020.)
 |-  7  =  ( ( 2  +  2 )  +  3 )
 
Theoremgbpart8 37802 The Goldbach partition of 8. (Contributed by AV, 20-Jul-2020.)
 |-  8  =  ( 3  +  5 )
 
Theoremgbpart9 37803 The (strong) Goldbach partition of 9. (Contributed by AV, 26-Jul-2020.)
 |-  9  =  ( ( 3  +  3 )  +  3 )
 
Theoremgbpart11 37804 The (strong) Goldbach partition of 11. (Contributed by AV, 29-Jul-2020.)
 |- ; 1 1  =  ( ( 3  +  3 )  +  5 )
 
Theorem6gbe 37805 6 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
 |-  6  e. GoldbachEven
 
Theorem7gbo 37806 7 is an odd Goldbach number. (Contributed by AV, 20-Jul-2020.)
 |-  7  e. GoldbachOdd
 
Theorem8gbe 37807 8 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
 |-  8  e. GoldbachEven
 
Theorem9gboa 37808 9 is an odd Goldbach number. (Contributed by AV, 26-Jul-2020.)
 |-  9  e. GoldbachOddALTV
 
Theorem11gboa 37809 11 is an odd Goldbach number. (Contributed by AV, 29-Jul-2020.)
 |- ; 1 1  e. GoldbachOddALTV
 
Theoremstgoldbwt 37810 If the strong ternary Goldbach conjecture is valid, then the weak ternary Goldbach conjecture holds, too. (Contributed by AV, 27-Jul-2020.)
 |-  ( A. n  e. Odd  ( 7  <  n  ->  n  e. GoldbachOddALTV  )  ->  A. n  e. Odd  (
 5  <  n  ->  n  e. GoldbachOdd  ) )
 
Theorembgoldbwt 37811* If the binary Goldbach conjecture is valid, then the (weak) ternary Goldbach conjecture holds, too. (Contributed by AV, 20-Jul-2020.)
 |-  ( A. n  e. Even  ( 4  <  n  ->  n  e. GoldbachEven 
 )  ->  A. m  e. Odd 
 ( 5  <  m  ->  m  e. GoldbachOdd  ) )
 
Theorembgoldbst 37812* If the binary Goldbach conjecture is valid, then the (strong) ternary Goldbach conjecture holds, too. (Contributed by AV, 26-Jul-2020.)
 |-  ( A. n  e. Even  ( 4  <  n  ->  n  e. GoldbachEven 
 )  ->  A. m  e. Odd 
 ( 7  <  m  ->  m  e. GoldbachOddALTV  ) )
 
Theoremsgoldbaltlem1 37813 Lemma 1 for sgoldbalt 37815: If an even number greater than 4 is the sum of two primes, one of the prime summands must be odd, i.e. not 2. (Contributed by AV, 22-Jul-2020.)
 |-  (
 ( P  e.  Prime  /\  Q  e.  Prime )  ->  ( ( N  e. Even  /\  4  <  N  /\  N  =  ( P  +  Q ) )  ->  Q  e. Odd  ) )
 
Theoremsgoldbaltlem2 37814 Lemma 2 for sgoldbalt 37815: If an even number greater than 4 is the sum of two primes, the primes must be odd, i.e. not 2. (Contributed by AV, 22-Jul-2020.)
 |-  (
 ( P  e.  Prime  /\  Q  e.  Prime )  ->  ( ( N  e. Even  /\  4  <  N  /\  N  =  ( P  +  Q ) )  ->  ( P  e. Odd  /\  Q  e. Odd  ) ) )
 
Theoremsgoldbalt 37815* An alternate (the original) formulation of the binary Goldbach conjecture: Every even integer greater than 2 can be expressed as the sum of two primes. (Contributed by AV, 22-Jul-2020.)
 |-  ( A. n  e. Even  ( 4  <  n  ->  n  e. GoldbachEven 
 ) 
 <-> 
 A. n  e. Even  (
 2  <  n  ->  E. p  e.  Prime  E. q  e.  Prime  n  =  ( p  +  q ) ) )
 
Theoremnnsum3primes4 37816* 4 is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.)
 |-  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1
 ... d ) ) ( d  <_  3  /\  4  =  sum_ k  e.  ( 1 ... d ) ( f `
  k ) )
 
Theoremnnsum4primes4 37817* 4 is the sum of at most 4 (actually 2) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)
 |-  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1
 ... d ) ) ( d  <_  4  /\  4  =  sum_ k  e.  ( 1 ... d ) ( f `
  k ) )
 
Theoremnnsum3primesprm 37818* Every prime is "the sum of at most 3" (actually one - the prime itself) primes. (Contributed by AV, 2-Aug-2020.)
 |-  ( P  e.  Prime  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1
 ... d ) ) ( d  <_  3  /\  P  =  sum_ k  e.  ( 1 ... d
 ) ( f `  k ) ) )
 
Theoremnnsum4primesprm 37819* Every prime is "the sum of at most 4" (actually one - the prime itself) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)
 |-  ( P  e.  Prime  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1
 ... d ) ) ( d  <_  4  /\  P  =  sum_ k  e.  ( 1 ... d
 ) ( f `  k ) ) )
 
Theoremnnsum3primesgbe 37820* Any even Goldbach number is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.)
 |-  ( N  e. GoldbachEven  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1
 ... d ) ) ( d  <_  3  /\  N  =  sum_ k  e.  ( 1 ... d
 ) ( f `  k ) ) )
 
Theoremnnsum4primesgbe 37821* Any even Goldbach number is the sum of at most 4 (actually 2) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)
 |-  ( N  e. GoldbachEven  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1
 ... d ) ) ( d  <_  4  /\  N  =  sum_ k  e.  ( 1 ... d
 ) ( f `  k ) ) )
 
Theoremnnsum3primesle9 37822* Every integer greater than 1 and less than or equal to 8 is the sum of at most 3 primes. (Contributed by AV, 2-Aug-2020.)
 |-  (
 ( N  e.  ( ZZ>=
 `  2 )  /\  N  <_  8 )  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
 ) ) ( d 
 <_  3  /\  N  =  sum_
 k  e.  ( 1
 ... d ) ( f `  k ) ) )
 
Theoremnnsum4primesle9 37823* Every integer greater than 1 and less than or equal to 8 is the sum of at most 4 primes. (Contributed by AV, 24-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)
 |-  (
 ( N  e.  ( ZZ>=
 `  2 )  /\  N  <_  8 )  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
 ) ) ( d 
 <_  4  /\  N  =  sum_
 k  e.  ( 1
 ... d ) ( f `  k ) ) )
 
Theoremnnsum4primesodd 37824* If the (weak) ternary Goldbach conjecture is valid, then every odd integer greater than 5 is the sum of 3 primes. (Contributed by AV, 2-Jul-2020.)
 |-  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  ( ( N  e.  ( ZZ>= `  6 )  /\  N  e. Odd  ) 
 ->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) N  =  sum_ k  e.  (
 1 ... 3 ) ( f `  k ) ) )
 
Theoremnnsum4primesoddALTV 37825* If the (strong) ternary Goldbach conjecture is valid, then every odd integer greater than 7 is the sum of 3 primes. (Contributed by AV, 26-Jul-2020.)
 |-  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  ( ( N  e.  ( ZZ>= `  8
 )  /\  N  e. Odd  ) 
 ->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) N  =  sum_ k  e.  (
 1 ... 3 ) ( f `  k ) ) )
 
Theoremevengpop3 37826* If the (weak) ternary Goldbach conjecture is valid, then every even integer greater than 8 is the sum of an odd Goldbach number and 3. (Contributed by AV, 24-Jul-2020.)
 |-  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  ( ( N  e.  ( ZZ>= `  9 )  /\  N  e. Even  ) 
 ->  E. o  e. GoldbachOdd  N  =  ( o  +  3
 ) ) )
 
Theoremevengpoap3 37827* If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of an odd Goldbach number and 3. (Contributed by AV, 27-Jul-2020.)
 |-  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even  )  ->  E. o  e. GoldbachOddALTV  N  =  ( o  +  3 ) ) )
 
Theoremnnsum4primeseven 37828* If the (weak) ternary Goldbach conjecture is valid, then every even integer greater than 8 is the sum of 4 primes. (Contributed by AV, 25-Jul-2020.)
 |-  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  ( ( N  e.  ( ZZ>= `  9 )  /\  N  e. Even  ) 
 ->  E. f  e.  ( Prime  ^m  ( 1 ... 4 ) ) N  =  sum_ k  e.  (
 1 ... 4 ) ( f `  k ) ) )
 
Theoremnnsum4primesevenALTV 37829* If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of 4 primes. (Contributed by AV, 27-Jul-2020.)
 |-  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even  )  ->  E. f  e.  ( Prime  ^m  ( 1 ... 4
 ) ) N  =  sum_
 k  e.  ( 1
 ... 4 ) ( f `  k ) ) )
 
Theoremwtgoldbnnsum4prm 37830* If the (weak) ternary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 4 primes, showing that Schnirelmann's constant would be less than or equal to 4. See corollary 1.1 in [Helfgott] p. 4. (Contributed by AV, 25-Jul-2020.)
 |-  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  A. n  e.  ( ZZ>= `  2 ) E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
 ) ) ( d 
 <_  4  /\  n  = 
 sum_ k  e.  (
 1 ... d ) ( f `  k ) ) )
 
Theoremstgoldbnnsum4prm 37831* If the (strong) ternary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 4 primes. (Contributed by AV, 27-Jul-2020.)
 |-  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  A. n  e.  ( ZZ>=
 `  2 ) E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1
 ... d ) ) ( d  <_  4  /\  n  =  sum_ k  e.  ( 1 ... d ) ( f `
  k ) ) )
 
Theorembgoldbnnsum3prm 37832* If the binary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 3 primes, showing that Schnirelmann's constant would be equal to 3. (Contributed by AV, 2-Aug-2020.)
 |-  ( A. m  e. Even  ( 4  <  m  ->  m  e. GoldbachEven 
 )  ->  A. n  e.  ( ZZ>= `  2 ) E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
 ) ) ( d 
 <_  3  /\  n  = 
 sum_ k  e.  (
 1 ... d ) ( f `  k ) ) )
 
Theorembgoldbtbndlem1 37833 Lemma 1 for bgoldbtbnd 37837: the odd numbers between 7 and 13 (exclusive) are (strong) odd Goldbach numbers. (Contributed by AV, 29-Jul-2020.)
 |-  (
 ( N  e. Odd  /\  7  <  N  /\  N  e.  ( 7 [,); 1 3 ) ) 
 ->  N  e. GoldbachOddALTV  )
 
Theorembgoldbtbndlem2 37834* Lemma 2 for bgoldbtbnd 37837. (Contributed by AV, 1-Aug-2020.)
 |-  ( ph  ->  M  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  A. n  e. Even  (
 ( 4  <  n  /\  n  <  N ) 
 ->  n  e. GoldbachEven  ) )   &    |-  ( ph  ->  D  e.  ( ZZ>= `  3 )
 )   &    |-  ( ph  ->  F  e.  (RePart `  D )
 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ D ) ( ( F `  i )  e.  ( Prime  \  { 2 } )  /\  ( ( F `  ( i  +  1 ) )  -  ( F `  i ) )  < 
 ( N  -  4
 )  /\  4  <  ( ( F `  (
 i  +  1 ) )  -  ( F `
  i ) ) ) )   &    |-  ( ph  ->  ( F `  0 )  =  7 )   &    |-  ( ph  ->  ( F `  1 )  = ; 1 3 )   &    |-  ( ph  ->  M  <  ( F `  D ) )   &    |-  S  =  ( X  -  ( F `  ( I  -  1 ) ) )   =>    |-  ( ( ph  /\  X  e. Odd  /\  I  e.  (
 1..^ D ) ) 
 ->  ( ( X  e.  ( ( F `  I ) [,) ( F `  ( I  +  1 ) ) ) 
 /\  ( X  -  ( F `  I ) )  <_  4 )  ->  ( S  e. Even  /\  S  <  N  /\  4  <  S ) ) )
 
Theorembgoldbtbndlem3 37835* Lemma 3 for bgoldbtbnd 37837. (Contributed by AV, 1-Aug-2020.)
 |-  ( ph  ->  M  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  A. n  e. Even  (
 ( 4  <  n  /\  n  <  N ) 
 ->  n  e. GoldbachEven  ) )   &    |-  ( ph  ->  D  e.  ( ZZ>= `  3 )
 )   &    |-  ( ph  ->  F  e.  (RePart `  D )
 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ D ) ( ( F `  i )  e.  ( Prime  \  { 2 } )  /\  ( ( F `  ( i  +  1 ) )  -  ( F `  i ) )  < 
 ( N  -  4
 )  /\  4  <  ( ( F `  (
 i  +  1 ) )  -  ( F `
  i ) ) ) )   &    |-  ( ph  ->  ( F `  0 )  =  7 )   &    |-  ( ph  ->  ( F `  1 )  = ; 1 3 )   &    |-  ( ph  ->  M  <  ( F `  D ) )   &    |-  ( ph  ->  ( F `  D )  e.  RR )   &    |-  S  =  ( X  -  ( F `  I ) )   =>    |-  ( ( ph  /\  X  e. Odd  /\  I  e.  ( 1..^ D ) )  ->  ( ( X  e.  ( ( F `  I ) [,) ( F `  ( I  +  1 )
 ) )  /\  4  <  S )  ->  ( S  e. Even  /\  S  <  N 
 /\  4  <  S ) ) )
 
Theorembgoldbtbndlem4 37836* Lemma 4 for bgoldbtbnd 37837. (Contributed by AV, 1-Aug-2020.)
 |-  ( ph  ->  M  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  A. n  e. Even  (
 ( 4  <  n  /\  n  <  N ) 
 ->  n  e. GoldbachEven  ) )   &    |-  ( ph  ->  D  e.  ( ZZ>= `  3 )
 )   &    |-  ( ph  ->  F  e.  (RePart `  D )
 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ D ) ( ( F `  i )  e.  ( Prime  \  { 2 } )  /\  ( ( F `  ( i  +  1 ) )  -  ( F `  i ) )  < 
 ( N  -  4
 )  /\  4  <  ( ( F `  (
 i  +  1 ) )  -  ( F `
  i ) ) ) )   &    |-  ( ph  ->  ( F `  0 )  =  7 )   &    |-  ( ph  ->  ( F `  1 )  = ; 1 3 )   &    |-  ( ph  ->  M  <  ( F `  D ) )   &    |-  ( ph  ->  ( F `  D )  e.  RR )   =>    |-  ( ( ( ph  /\  I  e.  ( 1..^ D ) )  /\  X  e. Odd  )  ->  ( ( X  e.  (
 ( F `  I
 ) [,) ( F `  ( I  +  1
 ) ) )  /\  ( X  -  ( F `  I ) ) 
 <_  4 )  ->  E. p  e.  Prime  E. q  e.  Prime  E. r  e.  Prime  (
 ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd 
 )  /\  X  =  ( ( p  +  q )  +  r
 ) ) ) )
 
Theorembgoldbtbnd 37837* If the binary Goldbach conjecture is valid up to an integer  N, and there is a series ("ladder") of primes with a difference of at most  N up to an integer  M, then the strong ternary Goldbach conjecture is valid up to  M, see section 1.2.2 in [Helfgott] p. 4 with N = 4 x 10^18, taken from [OeSilva], and M = 8.875 x 10^30. (Contributed by AV, 1-Aug-2020.)
 |-  ( ph  ->  M  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 ` ; 1 1 ) )   &    |-  ( ph  ->  A. n  e. Even  (
 ( 4  <  n  /\  n  <  N ) 
 ->  n  e. GoldbachEven  ) )   &    |-  ( ph  ->  D  e.  ( ZZ>= `  3 )
 )   &    |-  ( ph  ->  F  e.  (RePart `  D )
 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ D ) ( ( F `  i )  e.  ( Prime  \  { 2 } )  /\  ( ( F `  ( i  +  1 ) )  -  ( F `  i ) )  < 
 ( N  -  4
 )  /\  4  <  ( ( F `  (
 i  +  1 ) )  -  ( F `
  i ) ) ) )   &    |-  ( ph  ->  ( F `  0 )  =  7 )   &    |-  ( ph  ->  ( F `  1 )  = ; 1 3 )   &    |-  ( ph  ->  M  <  ( F `  D ) )   &    |-  ( ph  ->  ( F `  D )  e.  RR )   =>    |-  ( ph  ->  A. n  e. Odd  ( ( 7  < 
 n  /\  n  <  M )  ->  n  e. GoldbachOddALTV  )
 )
 
21.33.5  Proth's theorem
 
Theoremmodexp2m1d 37838 The square of an integer which is -1 modulo a number greater than 1 is 1 modulo the same modulus. (Contributed by AV, 5-Jul-2020.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  1  <  E )   &    |-  ( ph  ->  ( A  mod  E )  =  ( -u 1  mod  E ) )   =>    |-  ( ph  ->  ( ( A ^ 2 )  mod  E )  =  1 )
 
Theoremproththdlem 37839 Lemma for proththd 37840. (Contributed by AV, 4-Jul-2020.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  P  =  ( ( K  x.  ( 2 ^ N ) )  +  1 ) )   =>    |-  ( ph  ->  ( P  e.  NN  /\  1  <  P  /\  (
 ( P  -  1
 )  /  2 )  e.  NN ) )
 
Theoremproththd 37840* Proth's theorem (1878). If P is a Proth number, i.e. a number of the form k2^n+1 with k less than 2^n, and if there exists an integer x for which x^((P-1)/2) is -1 modulo P, then P is prime. Such a prime is called a Proth prime. Like Pocklington's theorem (see pockthg 14525), Proth's theorem allows for a convenient method for verifying large primes. (Contributed by AV, 5-Jul-2020.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  P  =  ( ( K  x.  ( 2 ^ N ) )  +  1 ) )   &    |-  ( ph  ->  K  <  (
 2 ^ N ) )   &    |-  ( ph  ->  E. x  e.  ZZ  (
 ( x ^ (
 ( P  -  1
 )  /  2 )
 )  mod  P )  =  ( -u 1  mod  P ) )   =>    |-  ( ph  ->  P  e.  Prime )
 
Theorem5tcu2e40 37841 5 times the cube of 2 is 40. (Contributed by AV, 4-Jul-2020.)
 |-  (
 5  x.  ( 2 ^ 3 ) )  = ; 4 0
 
Theorem3exp4mod41 37842 3 to the fourth power is -1 modulo 41. (Contributed by AV, 5-Jul-2020.)
 |-  (
 ( 3 ^ 4
 )  mod ; 4 1 )  =  ( -u 1  mod ; 4 1 )
 
Theorem41prothprmlem1 37843 Lemma 1 for 41prothprm 37845. (Contributed by AV, 4-Jul-2020.)
 |-  P  = ; 4 1   =>    |-  ( ( P  -  1 )  /  2
 )  = ; 2 0
 
Theorem41prothprmlem2 37844 Lemma 2 for 41prothprm 37845. (Contributed by AV, 5-Jul-2020.)
 |-  P  = ; 4 1   =>    |-  ( ( 3 ^
 ( ( P  -  1 )  /  2
 ) )  mod  P )  =  ( -u 1  mod  P )
 
Theorem41prothprm 37845 41 is a Proth prime. (Contributed by AV, 5-Jul-2020.)
 |-  P  = ; 4 1   =>    |-  ( P  =  ( ( 5  x.  (
 2 ^ 3 ) )  +  1 ) 
 /\  P  e.  Prime )
 
21.33.6  Words over a set (extension)
 
21.33.6.1  Last symbol of a word (extension)
 
Theoremlswn0 37846 The last symbol of a not empty word exists. The empty set must be excluded as symbol, because otherwise, it cannot be distinguished between valid cases ( (/) is the last symbol) and invalid cases ( (/) means that no last symbol exists. This is because of the special definition of a function in set.mm. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 |-  (
 ( W  e. Word  V  /\  (/)  e/  V  /\  ( # `  W )  =/=  0 )  ->  ( lastS  `  W )  =/=  (/) )
 
21.33.6.2  Concatenations with singleton words (extension)
 
Theoremccatw2s1cl 37847 The concatenation of a word with two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
 |-  (
 ( W  e. Word  V  /\  X  e.  V  /\  Y  e.  V )  ->  ( ( W ++  <" X "> ) ++  <" Y "> )  e. Word  V )
 
21.33.6.3  Prefixes of a word

In https://www.allacronyms.com/prefix/abbreviated, "pfx" is proposed as abbreviation for "prefix". Regarding the meaning of "prefix", it is different in computer science (automata theory/formal languages) compared with linguistics: in linguistics, a prefix has a meaning (see Wikipedia "Prefix" https://en.wikipedia.org/wiki/Prefix), whereas in computer science, a prefix is an arbitrary substring/subword starting at the beginning of a string/word (see Wikipedia "Substring" https://en.wikipedia.org/wiki/Substring#Prefix), or https://math.stackexchange.com/questions/2190559/ is-there-standard-terminology-notation-for-the-prefix-of-a-word ).

 
Syntaxcpfx 37848 Syntax for the prefix operator.
 class prefix
 
Definitiondf-pfx 37849* Define an operation which extracts prefixes of words, i.e. subwords starting at the beginning of a word. Definition in section 9.1 of [AhoHopUll] p. 318. "pfx" is used as label fragment. (Contributed by AV, 2-May-2020.)
 |- prefix  =  ( s  e.  _V ,  l  e.  NN0  |->  ( s substr  <. 0 ,  l >. ) )
 
Theorempfxval 37850 Value of a prefix. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e.  V  /\  L  e.  NN0 )  ->  ( S prefix  L )  =  ( S substr  <. 0 ,  L >. ) )
 
Theorempfx00 37851 A zero length prefix. (Contributed by AV, 2-May-2020.)
 |-  ( S prefix  0 )  =  (/)
 
Theorempfx0 37852 A prefix of an empty set is always the empty set. (Contributed by AV, 3-May-2020.)
 |-  ( (/) prefix  L )  =  (/)
 
Theorempfxcl 37853 Closure of the prefix extractor. (Contributed by AV, 2-May-2020.)
 |-  ( S  e. Word  A  ->  ( S prefix  L )  e. Word  A )
 
Theorempfxmpt 37854* Value of the prefix extractor as mapping. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e. Word  A  /\  L  e.  ( 0
 ... ( # `  S ) ) )  ->  ( S prefix  L )  =  ( x  e.  (
 0..^ L )  |->  ( S `  x ) ) )
 
Theorempfxres 37855 Value of the prefix extractor as restriction. Could replace swrd0val 12609. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e. Word  A  /\  L  e.  ( 0
 ... ( # `  S ) ) )  ->  ( S prefix  L )  =  ( S  |`  ( 0..^ L ) ) )
 
Theorempfxf 37856 A prefix of a word is a function from a half-open range of nonnegative integers of the same length as the prefix to the set of symbols for the original word. Could replace swrd0f 12615. (Contributed by AV, 2-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 0
 ... ( # `  W ) ) )  ->  ( W prefix  L ) : ( 0..^ L ) --> V )
 
Theorempfxfn 37857 Value of the prefix extractor as function with domain. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e. Word  V  /\  L  e.  ( 0
 ... ( # `  S ) ) )  ->  ( S prefix  L )  Fn  ( 0..^ L ) )
 
Theorempfxlen 37858 Length of a prefix. Could replace swrd0len 12610. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e. Word  A  /\  L  e.  ( 0
 ... ( # `  S ) ) )  ->  ( # `  ( S prefix  L ) )  =  L )
 
Theorempfxid 37859 A word is a prefix of itself. (Contributed by AV, 2-May-2020.)
 |-  ( S  e. Word  A  ->  ( S prefix  ( # `  S ) )  =  S )
 
Theorempfxrn 37860 The range of a prefix of a word is a subset of the set of symbols for the word. (Contributed by AV, 2-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 0
 ... ( # `  W ) ) )  ->  ran  ( W prefix  L )  C_  V )
 
Theorempfxn0 37861 A prefix consisting of at least one symbol is not empty. Could replace swrdn0 12618. (Contributed by AV, 2-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  NN  /\  L  <_  ( # `  W ) )  ->  ( W prefix  L )  =/=  (/) )
 
Theorempfxnd 37862 The value of the prefix extractor is the empty set (undefined) if the argument is not within the range of the word. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  NN0  /\  ( # `
  W )  <  L )  ->  ( W prefix  L )  =  (/) )
 
Theorempfxlen0 37863 Length of a prefix of a word reduced by a single symbol. Could replace swrd0len0 12624. TODO-AV: Really useful? swrd0len0 12624 is only used in wwlknred 25021. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  NN0  /\  ( # `
  W )  =  ( L  +  1 ) )  ->  ( # `
  ( W prefix  L ) )  =  L )
 
Theoremaddlenrevpfx 37864 The sum of the lengths of two reversed parts of a word is the length of the word. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( # `  ( W substr 
 <. M ,  ( # `  W ) >. ) )  +  ( # `  ( W prefix  M ) ) )  =  ( # `  W ) )
 
Theoremaddlenpfx 37865 The sum of the lengths of two parts of a word is the length of the word. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( # `  ( W prefix  M ) )  +  ( # `  ( W substr  <. M ,  ( # `  W ) >. ) ) )  =  ( # `  W ) )
 
Theorempfxfv 37866 A symbol in a prefix of a word, indexed using the prefix' indices. Could replace swrd0fv 12627. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 0
 ... ( # `  W ) )  /\  I  e.  ( 0..^ L ) )  ->  ( ( W prefix  L ) `  I
 )  =  ( W `
  I ) )
 
Theorempfxfv0 37867 The first symbol in a prefix of a word. Could replace swrd0fv0 12628. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 1
 ... ( # `  W ) ) )  ->  ( ( W prefix  L ) `  0 )  =  ( W `  0
 ) )
 
Theorempfxtrcfv 37868 A symbol in a word truncated by one symbol. Could replace swrdtrcfv 12629. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  W  =/=  (/)  /\  I  e.  ( 0..^ ( ( # `  W )  -  1 ) ) ) 
 ->  ( ( W prefix  (
 ( # `  W )  -  1 ) ) `
  I )  =  ( W `  I
 ) )
 
Theorempfxtrcfv0 37869 The first symbol in a word truncated by one symbol. Could replace swrdtrcfv0 12630. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  2  <_  ( # `  W ) )  ->  ( ( W prefix  (
 ( # `  W )  -  1 ) ) `
  0 )  =  ( W `  0
 ) )
 
Theorempfxfvlsw 37870 The last symbol in a (not empty) prefix of a word. Could replace swrd0fvlsw 12631. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 1
 ... ( # `  W ) ) )  ->  ( lastS  `  ( W prefix  L ) )  =  ( W `  ( L  -  1 ) ) )
 
Theorempfxeq 37871* The prefixes of two words are equal iff they have the same length and the same symbols at each position. Could replace swrdeq 12632. (Contributed by AV, 4-May-2020.)
 |-  (
 ( ( W  e. Word  V 
 /\  U  e. Word  V )  /\  ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `
  W )  /\  N  <_  ( # `  U ) ) )  ->  ( ( W prefix  M )  =  ( U prefix  N )  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( W `  i
 )  =  ( U `
  i ) ) ) )
 
Theorempfxtrcfvl 37872 The last symbol in a word truncated by one symbol. Could replace swrdtrcfvl 12638. (Contributed by AV, 5-May-2020.)
 |-  (
 ( W  e. Word  V  /\  2  <_  ( # `  W ) )  ->  ( lastS  `  ( W prefix  (
 ( # `  W )  -  1 ) ) )  =  ( W `
  ( ( # `  W )  -  2
 ) ) )
 
Theorempfxsuffeqwrdeq 37873 Two words are equal if and only if they have the same prefix and the same suffix. Could replace 2swrdeqwrdeq 12641. (Contributed by AV, 5-May-2020.)
 |-  (
 ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W ) ) )  ->  ( W  =  S  <->  ( ( # `  W )  =  ( # `  S )  /\  ( ( W prefix  I
 )  =  ( S prefix  I )  /\  ( W substr  <. I ,  ( # `  W ) >. )  =  ( S substr  <. I ,  ( # `  W )
 >. ) ) ) ) )
 
Theorempfxsuff1eqwrdeq 37874 Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. Could replace 2swrd1eqwrdeq 12642. (Contributed by AV, 6-May-2020.)
 |-  (
 ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W ) )  ->  ( W  =  U  <->  ( ( # `  W )  =  ( # `  U )  /\  ( ( W prefix  (
 ( # `  W )  -  1 ) )  =  ( U prefix  (
 ( # `  W )  -  1 ) ) 
 /\  ( lastS  `  W )  =  ( lastS  `  U ) ) ) ) )
 
Theoremdisjwrdpfx 37875* Sets of words are disjoint if each set contains extensions of distinct words of a fixed length. Could replace disjxwrd 12643. (Contributed by AV, 6-May-2020.)
 |- Disj  y  e.  W  { x  e. Word  V  |  ( x prefix  N )  =  y }
 
Theoremccatpfx 37876 Joining a prefix with an adjacent subword makes a longer prefix. (Contributed by AV, 7-May-2020.)
 |-  (
 ( S  e. Word  A  /\  Y  e.  ( 0
 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) )  ->  ( ( S prefix  Y ) ++  ( S substr  <. Y ,  Z >. ) )  =  ( S prefix  Z )
 )
 
Theorempfxccat1 37877 Recover the left half of a concatenated word. Could replace swrdccat1 12645. (Contributed by AV, 6-May-2020.)
 |-  (
 ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S ++  T ) prefix  ( # `  S ) )  =  S )
 
Theorempfx1 37878 A prefix of length 1. (Contributed by AV, 15-May-2020.)
 |-  (
 ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( W prefix  1 )  =  <" ( W `
  0 ) "> )
 
Theorempfx2 37879 A prefix of length 2. (Contributed by AV, 15-May-2020.)
 |-  (
 ( W  e. Word  V  /\  2  <_  ( # `  W ) )  ->  ( W prefix  2 )  =  <" ( W `
  0 ) ( W `  1 ) "> )
 
Theorempfxswrd 37880 A prefix of a subword. Could replace swrd0swrd 12649. (Contributed by AV, 8-May-2020.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) )  /\  M  e.  ( 0 ... N ) )  ->  ( L  e.  ( 0 ... ( N  -  M ) )  ->  ( ( W substr  <. M ,  N >. ) prefix  L )  =  ( W substr  <. M ,  ( M  +  L ) >. ) ) )
 
Theoremswrdpfx 37881 A subword of a prefix. Could replace swrdswrd0 12650. (Contributed by AV, 8-May-2020.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( K  e.  ( 0 ... N )  /\  L  e.  ( K ... N ) ) 
 ->  ( ( W prefix  N ) substr 
 <. K ,  L >. )  =  ( W substr  <. K ,  L >. ) ) )
 
Theorempfxpfx 37882 A prefix of a prefix. Could replace swrd0swrd0 12651. (Contributed by AV, 8-May-2020.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) )  /\  L  e.  ( 0 ... N ) )  ->  ( ( W prefix  N ) prefix  L )  =  ( W prefix  L ) )
 
Theorempfxpfxid 37883 A prefix of a prefix with the same length is the prefix. Could replace swrd0swrdid 12652. (Contributed by AV, 8-May-2020.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( W prefix  N ) prefix  N )  =  ( W prefix  N ) )
 
Theorempfxcctswrd 37884 The concatenation of the prefix of a word and the rest of the word yields the word itself. Could replace wrdcctswrd 12653. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( W prefix  M ) ++  ( W substr  <. M ,  ( # `  W )
 >. ) )  =  W )
 
Theoremlenpfxcctswrd 37885 The length of the concatenation of the prefix of a word and the rest of the word is the length of the word. Could replace lencctswrd 12654. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( # `  ( ( W prefix  M ) ++  ( W substr  <. M ,  ( # `  W ) >. ) ) )  =  ( # `  W ) )
 
Theoremlenrevpfxcctswrd 37886 The length of the concatenation of the rest of a word and the prefix of the word is the length of the word. Could replace lenrevcctswrd 12655. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( # `  ( ( W substr  <. M ,  ( # `
  W ) >. ) ++  ( W prefix  M )
 ) )  =  ( # `  W ) )
 
Theorempfxlswccat 37887 Reconstruct a nonempty word from its prefix and last symbol. Could replace wrdeqcats1OLD 12662 resp. swrdccatwrd 12656. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( ( W prefix  (
 ( # `  W )  -  1 ) ) ++ 
 <" ( lastS  `  W ) "> )  =  W )
 
Theoremccats1pfxeq 37888 The last symbol of a word concatenated with the word with the last symbol removed having results in the word itself. Could replace ccats1swrdeq 12657. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  ->  ( W  =  ( U prefix  ( # `  W ) ) 
 ->  U  =  ( W ++ 
 <" ( lastS  `  U ) "> ) ) )
 
Theoremccats1pfxeqrex 37889* There exists a symbol such that its concatenation with the prefix obtained by deleting the last symbol of a nonempty word results in the word itself. Could replace ccats1swrdeqrex 12667. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  ->  ( W  =  ( U prefix  ( # `  W ) ) 
 ->  E. s  e.  V  U  =  ( W ++  <" s "> ) ) )
 
Theorempfxccatin12lem1 37890 Lemma 1 for pfxccatin12 37892. Could replace swrdccatin12lem2b 12674. (Contributed by AV, 9-May-2020.)
 |-  (
 ( M  e.  (
 0 ... L )  /\  N  e.  ( L ... X ) )  ->  ( ( K  e.  ( 0..^ ( N  -  M ) )  /\  -.  K  e.  ( 0..^ ( L  -  M ) ) )  ->  ( K  -  ( L  -  M ) )  e.  ( 0..^ ( N  -  L ) ) ) )
 
Theorempfxccatin12lem2 37891 Lemma 2 for pfxccatin12 37892. Could replace swrdccatin12lem2 12677. (Contributed by AV, 9-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ...
 L )  /\  N  e.  ( L ... ( L  +  ( # `  B ) ) ) ) )  ->  ( ( K  e.  ( 0..^ ( N  -  M ) )  /\  -.  K  e.  ( 0..^ ( L  -  M ) ) )  ->  ( (
 ( A ++  B ) substr  <. M ,  N >. ) `
  K )  =  ( ( B prefix  ( N  -  L ) ) `
  ( K  -  ( # `  ( A substr  <. M ,  L >. ) ) ) ) ) )
 
Theorempfxccatin12 37892 The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12 12679. (Contributed by AV, 9-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ...
 L )  /\  N  e.  ( L ... ( L  +  ( # `  B ) ) ) ) 
 ->  ( ( A ++  B ) substr 
 <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) ++  ( B prefix  ( N  -  L ) ) ) ) )
 
Theorempfxccat3 37893 The subword of a concatenation is either a subword of the first concatenated word or a subword of the second concatenated word or a concatenation of a suffix of the first word with a prefix of the second word. Could replace swrdccat3 12680. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ...
 N )  /\  N  e.  ( 0 ... ( L  +  ( # `  B ) ) ) ) 
 ->  ( ( A ++  B ) substr 
 <. M ,  N >. )  =  if ( N 
 <_  L ,  ( A substr  <. M ,  N >. ) ,  if ( L 
 <_  M ,  ( B substr  <. ( M  -  L ) ,  ( N  -  L ) >. ) ,  ( ( A substr  <. M ,  L >. ) ++  ( B prefix 
 ( N  -  L ) ) ) ) ) ) )
 
Theorempfxccatpfx1 37894 A prefix of a concatenation being a prefix of the first concatenated word. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V  /\  N  e.  ( 0
 ... L ) ) 
 ->  ( ( A ++  B ) prefix  N )  =  ( A prefix  N ) )
 
Theorempfxccatpfx2 37895 A prefix of a concatenation of two words being the first word concatenated with a prefix of the second word. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   &    |-  M  =  ( # `  B )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V  /\  N  e.  ( ( L  +  1 )
 ... ( L  +  M ) ) ) 
 ->  ( ( A ++  B ) prefix  N )  =  ( A ++  ( B prefix  ( N  -  L ) ) ) )
 
Theorempfxccat3a 37896 A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. Could replace swrdccat3a 12682. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   &    |-  M  =  ( # `  B )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V )  ->  ( N  e.  ( 0 ... ( L  +  M )
 )  ->  ( ( A ++  B ) prefix  N )  =  if ( N 
 <_  L ,  ( A prefix  N ) ,  ( A ++  ( B prefix  ( N  -  L ) ) ) ) ) )
 
Theorempfxccatid 37897 A prefix of a concatenation of length of the first concatenated word is the first word itself. Could replace swrdccatid 12685. (Contributed by AV, 10-May-2020.)
 |-  (
 ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A ) )  ->  ( ( A ++  B ) prefix  N )  =  A )
 
Theoremccats1pfxeqbi 37898 A word is a prefix of a word with length greater by 1 than the first word iff the second word is the first word concatenated with the last symbol of the second word. Could replace ccats1swrdeqbi 12686. (Contributed by AV, 10-May-2020.)
 |-  (
 ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  ->  ( W  =  ( U prefix  ( # `  W ) )  <->  U  =  ( W ++  <" ( lastS  `  U ) "> ) ) )
 
Theorempfxccatin12d 37899 The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12d 12689. (Contributed by AV, 10-May-2020.)
 |-  ( ph  ->  ( # `  A )  =  L )   &    |-  ( ph  ->  ( A  e. Word  V 
 /\  B  e. Word  V ) )   &    |-  ( ph  ->  M  e.  ( 0 ...
 L ) )   &    |-  ( ph  ->  N  e.  ( L ... ( L  +  ( # `  B ) ) ) )   =>    |-  ( ph  ->  ( ( A ++  B ) substr  <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) ++  ( B prefix  ( N  -  L ) ) ) )
 
Theoremreuccatpfxs1lem 37900* Lemma for reuccatpfxs1 37901. Could replace reuccats1lem 12668. (Contributed by AV, 9-May-2020.)
 |-  (
 ( ( W  e. Word  V 
 /\  U  e.  X )  /\  A. s  e.  V  ( ( W ++ 
 <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x )  =  ( ( # `  W )  +  1 ) ) )  ->  ( W  =  ( U prefix  ( # `  W ) )  ->  U  =  ( W ++  <" S "> ) ) )
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