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Theorem List for Metamath Proof Explorer - 32501-32600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
21.25.3.2  Definitions and basic properties
 
Syntaxceven 32501 Extend the definition of a class to include the set of even numbers.
 class Even
 
Syntaxcodd 32502 Extend the definition of a class to include the set of odd numbers.
 class Odd
 
Definitiondf-even 32503 Define the set of even numbers. (Contributed by AV, 14-Jun-2020.)
 |- Even  =  {
 z  e.  ZZ  |  ( z  /  2
 )  e.  ZZ }
 
Definitiondf-odd 32504 Define the set of odd numbers. (Contributed by AV, 14-Jun-2020.)
 |- Odd  =  {
 z  e.  ZZ  |  ( ( z  +  1 )  /  2
 )  e.  ZZ }
 
Theoremiseven 32505 The predicate "is an even number". An even number is an integer which is divisible by 2, i.e. the result of dividing the even integer by 2 is still an integer. (Contributed by AV, 14-Jun-2020.)
 |-  ( Z  e. Even  <->  ( Z  e.  ZZ  /\  ( Z  / 
 2 )  e.  ZZ ) )
 
Theoremisodd 32506 The predicate "is an odd number". An odd number is an integer which is not divisible by 2, i.e. the result of dividing the odd integer increased by 1 and then divided by 2 is still an integer. (Contributed by AV, 14-Jun-2020.)
 |-  ( Z  e. Odd  <->  ( Z  e.  ZZ  /\  ( ( Z  +  1 )  / 
 2 )  e.  ZZ ) )
 
Theoremevenz 32507 An even number is an integer. (Contributed by AV, 14-Jun-2020.)
 |-  ( Z  e. Even  ->  Z  e.  ZZ )
 
Theoremoddz 32508 An odd number is an integer. (Contributed by AV, 14-Jun-2020.)
 |-  ( Z  e. Odd  ->  Z  e.  ZZ )
 
Theoremevendiv2z 32509 The result of dividing an even number by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
 |-  ( Z  e. Even  ->  ( Z 
 /  2 )  e. 
 ZZ )
 
Theoremoddp1div2z 32510 The result of dividing an odd number increased by 1 and then divided by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
 |-  ( Z  e. Odd  ->  ( ( Z  +  1 ) 
 /  2 )  e. 
 ZZ )
 
Theoremzob 32511 Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.)
 |-  ( N  e.  ZZ  ->  ( ( ( N  +  1 )  /  2
 )  e.  ZZ  <->  ( ( N  -  1 )  / 
 2 )  e.  ZZ ) )
 
Theoremoddm1div2z 32512 The result of dividing an odd number decreased by 1 and then divided by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
 |-  ( Z  e. Odd  ->  ( ( Z  -  1 ) 
 /  2 )  e. 
 ZZ )
 
Theoremisodd2 32513 The predicate "is an odd number". An odd number is an integer which is not divisible by 2, i.e. the result of dividing the odd number decreased by 1 and then divided by 2 is still an integer. (Contributed by AV, 15-Jun-2020.)
 |-  ( Z  e. Odd  <->  ( Z  e.  ZZ  /\  ( ( Z  -  1 )  / 
 2 )  e.  ZZ ) )
 
Theoremdfodd2 32514 Alternate definition for odd numbers. (Contributed by AV, 15-Jun-2020.)
 |- Odd  =  {
 z  e.  ZZ  |  ( ( z  -  1 )  /  2
 )  e.  ZZ }
 
Theoremdfodd6 32515* Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
 |- Odd  =  {
 z  e.  ZZ  |  E. i  e.  ZZ  z  =  ( (
 2  x.  i )  +  1 ) }
 
Theoremdfeven4 32516* Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
 |- Even  =  {
 z  e.  ZZ  |  E. i  e.  ZZ  z  =  ( 2  x.  i ) }
 
Theoremevenm1odd 32517 The predecessor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
 |-  ( Z  e. Even  ->  ( Z  -  1 )  e. Odd 
 )
 
Theoremevenp1odd 32518 The successor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
 |-  ( Z  e. Even  ->  ( Z  +  1 )  e. Odd 
 )
 
Theoremoddp1eveni 32519 The successor of an odd number is even. (Contributed by AV, 16-Jun-2020.)
 |-  ( Z  e. Odd  ->  ( Z  +  1 )  e. Even 
 )
 
Theoremoddm1eveni 32520 The predecessor of an odd number is even. (Contributed by AV, 6-Jul-2020.)
 |-  ( Z  e. Odd  ->  ( Z  -  1 )  e. Even 
 )
 
Theoremevennodd 32521 An even number is not an odd number. (Contributed by AV, 16-Jun-2020.)
 |-  ( Z  e. Even  ->  -.  Z  e. Odd  )
 
Theoremoddneven 32522 An odd number is not an even number. (Contributed by AV, 16-Jun-2020.)
 |-  ( Z  e. Odd  ->  -.  Z  e. Even  )
 
Theoremenege 32523 The negative of an even number is even. (Contributed by AV, 20-Jun-2020.)
 |-  ( A  e. Even  ->  -u A  e. Even  )
 
Theoremonego 32524 The negative of an odd number is odd. (Contributed by AV, 20-Jun-2020.)
 |-  ( A  e. Odd  ->  -u A  e. Odd  )
 
Theoremm1expevenALTV 32525 Exponentiation of -1 by an even power. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 6-Jul-2020.)
 |-  ( N  e. Even  ->  ( -u 1 ^ N )  =  1 )
 
Theoremm1expoddALTV 32526 Exponentiation of -1 by an odd power. (Contributed by AV, 6-Jul-2020.)
 |-  ( N  e. Odd  ->  ( -u 1 ^ N )  =  -u 1 )
 
21.25.3.3  Alternate definitions using the "divides" relation
 
Theoremdfeven2 32527 Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
 |- Even  =  {
 z  e.  ZZ  | 
 2  ||  z }
 
Theoremdfodd3 32528 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
 |- Odd  =  {
 z  e.  ZZ  |  -.  2  ||  z }
 
Theoremiseven2 32529 The predicate "is an even number". An even number is an integer which is divisible by 2. (Contributed by AV, 18-Jun-2020.)
 |-  ( Z  e. Even  <->  ( Z  e.  ZZ  /\  2  ||  Z ) )
 
Theoremisodd3 32530 The predicate "is an odd number". An odd number is an integer which is not divisible by 2. (Contributed by AV, 18-Jun-2020.)
 |-  ( Z  e. Odd  <->  ( Z  e.  ZZ  /\  -.  2  ||  Z ) )
 
Theorem2dvdseven 32531 2 divides an even number. (Contributed by AV, 18-Jun-2020.)
 |-  ( Z  e. Even  ->  2  ||  Z )
 
Theorem2ndvdsodd 32532 2 does not divide an odd number. (Contributed by AV, 18-Jun-2020.)
 |-  ( Z  e. Odd  ->  -.  2  ||  Z )
 
Theorem2dvdsoddp1 32533 2 divides an odd number increased by 1. (Contributed by AV, 18-Jun-2020.)
 |-  ( Z  e. Odd  ->  2  ||  ( Z  +  1
 ) )
 
Theorem2dvdsoddm1 32534 2 divides an odd number decreased by 1. (Contributed by AV, 18-Jun-2020.)
 |-  ( Z  e. Odd  ->  2  ||  ( Z  -  1
 ) )
 
21.25.3.4  Alternate definitions using the "modulo" operation
 
Theoremdfeven3 32535 Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
 |- Even  =  {
 z  e.  ZZ  |  ( z  mod  2 )  =  0 }
 
Theoremdfodd4 32536 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
 |- Odd  =  {
 z  e.  ZZ  |  ( z  mod  2 )  =  1 }
 
Theoremdfodd5 32537 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
 |- Odd  =  {
 z  e.  ZZ  |  ( z  mod  2 )  =/=  0 }
 
Theoremzefldiv2ALTV 32538 The floor of an even number divided by 2 is equal to the even number divided by 2. (Contributed by AV, 7-Jun-2020.) (Revised by AV, 18-Jun-2020.)
 |-  ( N  e. Even  ->  ( |_ `  ( N  /  2
 ) )  =  ( N  /  2 ) )
 
Theoremzofldiv2ALTV 32539 The floor of an odd numer divided by 2 is equal to the odd number first decreased by 1 and then divided by 2. (Contributed by AV, 7-Jun-2020.) (Revised by AV, 18-Jun-2020.)
 |-  ( N  e. Odd  ->  ( |_ `  ( N  /  2
 ) )  =  ( ( N  -  1
 )  /  2 )
 )
 
TheoremoddflALTV 32540 Odd number representation by using the floor function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 18-Jun-2020.)
 |-  ( K  e. Odd  ->  K  =  ( ( 2  x.  ( |_ `  ( K  /  2 ) ) )  +  1 ) )
 
21.25.3.5  Alternate definitions using the "gcd" operation
 
Theoremgcdzeq 32541 A positive integer  A is equal to its gcd with an integer  B if and only if  A divides  B. Generalization of gcdeq 14211. (Contributed by AV, 1-Jul-2020.)
 |-  (
 ( A  e.  NN  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  =  A  <->  A  ||  B ) )
 
Theoremiseven5 32542 The predicate "is an even number". An even number and 2 have 2 as greatest common divisor. (Contributed by AV, 1-Jul-2020.)
 |-  ( Z  e. Even  <->  ( Z  e.  ZZ  /\  ( 2  gcd 
 Z )  =  2 ) )
 
Theoremisodd7 32543 The predicate "is an odd number". An odd number and 2 have 1 as greatest common divisor. (Contributed by AV, 1-Jul-2020.)
 |-  ( Z  e. Odd  <->  ( Z  e.  ZZ  /\  ( 2  gcd 
 Z )  =  1 ) )
 
Theoremdfeven5 32544 Alternate definition for even numbers. (Contributed by AV, 1-Jul-2020.)
 |- Even  =  {
 z  e.  ZZ  |  ( 2  gcd  z
 )  =  2 }
 
Theoremdfodd7 32545 Alternate definition for odd numbers. (Contributed by AV, 1-Jul-2020.)
 |- Odd  =  {
 z  e.  ZZ  |  ( 2  gcd  z
 )  =  1 }
 
21.25.3.6  Theorems of part 5 revised
 
TheoremzneoALTV 32546 No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Revised by AV, 16-Jun-2020.)
 |-  (
 ( A  e. Even  /\  B  e. Odd  )  ->  A  =/=  B )
 
TheoremzeoALTV 32547 An integer is even or odd. (Contributed by NM, 1-Jan-2006.) (Revised by AV, 16-Jun-2020.)
 |-  ( Z  e.  ZZ  ->  ( Z  e. Even  \/  Z  e. Odd  ) )
 
Theoremzeo2ALTV 32548 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.) (Revised by AV, 16-Jun-2020.)
 |-  ( Z  e.  ZZ  ->  ( Z  e. Even  <->  -.  Z  e. Odd  )
 )
 
TheoremnneoALTV 32549 A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Revised by AV, 19-Jun-2020.)
 |-  ( N  e.  NN  ->  ( N  e. Even  <->  -.  N  e. Odd  )
 )
 
TheoremnneoiALTV 32550 A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.) (Revised by AV, 19-Jun-2020.)
 |-  N  e.  NN   =>    |-  ( N  e. Even  <->  -.  N  e. Odd  )
 
21.25.3.7  Theorems of part 6 revised
 
Theoremodd2np1ALTV 32551* An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by AV, 19-Jun-2020.)
 |-  ( N  e.  ZZ  ->  ( N  e. Odd  <->  E. n  e.  ZZ  ( ( 2  x.  n )  +  1 )  =  N ) )
 
Theoremoddm1evenALTV 32552 An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
 |-  ( N  e.  ZZ  ->  ( N  e. Odd  <->  ( N  -  1 )  e. Even  ) )
 
Theoremoddp1evenALTV 32553 An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
 |-  ( N  e.  ZZ  ->  ( N  e. Odd  <->  ( N  +  1 )  e. Even  ) )
 
TheoremoexpnegALTV 32554 The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.) (Revised by AV, 19-Jun-2020.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN  /\  N  e. Odd  )  ->  (
 -u A ^ N )  =  -u ( A ^ N ) )
 
Theoremoexpnegnz 32555 The exponential of the negative of a number not being 0, when the exponent is odd. (Contributed by AV, 19-Jun-2020.)
 |-  (
 ( A  e.  CC  /\  A  =/=  0  /\  N  e. Odd  )  ->  (
 -u A ^ N )  =  -u ( A ^ N ) )
 
Theorembits0ALTV 32556 Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
 |-  ( N  e.  ZZ  ->  ( 0  e.  (bits `  N )  <->  N  e. Odd  ) )
 
Theorembits0eALTV 32557 The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
 |-  ( N  e. Even  ->  -.  0  e.  (bits `  N )
 )
 
Theorembits0oALTV 32558 The zeroth bit of an odd number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
 |-  ( N  e. Odd  ->  0  e.  (bits `  N )
 )
 
TheoremdivgcdoddALTV 32559 Either  A  /  ( A  gcd  B ) is odd or  B  /  ( A  gcd  B ) is odd. (Contributed by Scott Fenton, 19-Apr-2014.) (Revised by AV, 21-Jun-2020.)
 |-  (
 ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  /  ( A  gcd  B ) )  e. Odd  \/  ( B  /  ( A  gcd  B ) )  e. Odd  )
 )
 
TheoremopoeALTV 32560 The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
 |-  (
 ( A  e. Odd  /\  B  e. Odd  )  ->  ( A  +  B )  e. Even  )
 
TheoremopeoALTV 32561 The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
 |-  (
 ( A  e. Odd  /\  B  e. Even  )  ->  ( A  +  B )  e. Odd  )
 
TheoremomoeALTV 32562 The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
 |-  (
 ( A  e. Odd  /\  B  e. Odd  )  ->  ( A  -  B )  e. Even  )
 
TheoremomeoALTV 32563 The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
 |-  (
 ( A  e. Odd  /\  B  e. Even  )  ->  ( A  -  B )  e. Odd  )
 
TheoremoddprmALTV 32564 A prime not equal to  2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by AV, 21-Jun-2020.)
 |-  ( N  e.  ( Prime  \  { 2 } )  ->  N  e. Odd  )
 
21.25.3.8  Theorems of AV's mathbox revised
 
Theorem0evenALTV 32565 0 is an even number. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 17-Jun-2020.)
 |-  0  e. Even
 
Theorem0noddALTV 32566 0 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 17-Jun-2020.)
 |-  0  e/ Odd
 
Theorem1oddALTV 32567 1 is an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
 |-  1  e. Odd
 
Theorem1nevenALTV 32568 1 is not an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
 |-  1  e/ Even
 
Theorem2evenALTV 32569 2 is an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
 |-  2  e. Even
 
Theorem2noddALTV 32570 2 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
 |-  2  e/ Odd
 
Theoremnn0o1gt2ALTV 32571 An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 21-Jun-2020.)
 |-  (
 ( N  e.  NN0  /\  N  e. Odd  )  ->  ( N  =  1  \/  2  <  N ) )
 
TheoremnnoALTV 32572 An alternate characterization of an odd number greater than 1. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 21-Jun-2020.)
 |-  (
 ( N  e.  ( ZZ>=
 `  2 )  /\  N  e. Odd  )  ->  ( ( N  -  1
 )  /  2 )  e.  NN )
 
Theoremnn0oALTV 32573 An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Revised by AV, 21-Jun-2020.)
 |-  (
 ( N  e.  NN0  /\  N  e. Odd  )  ->  ( ( N  -  1 )  /  2
 )  e.  NN0 )
 
Theoremnn0e 32574 An alternate characterization of an even nonnegative integer. (Contributed by AV, 22-Jun-2020.)
 |-  (
 ( N  e.  NN0  /\  N  e. Even  )  ->  ( N  /  2 )  e.  NN0 )
 
Theoremnn0onn0exALTV 32575* For each odd nonnegative integer there is a nonnegative integer which, multiplied by 2 and increased by 1, results in the odd nonnegative integer. (Contributed by AV, 30-May-2020.) (Revised by AV, 22-Jun-2020.)
 |-  (
 ( N  e.  NN0  /\  N  e. Odd  )  ->  E. m  e.  NN0  N  =  ( ( 2  x.  m )  +  1 ) )
 
Theoremnn0enn0exALTV 32576* For each even nonnegative integer there is a nonnegative integer which, multiplied by 2, results in the even nonnegative integer. (Contributed by AV, 30-May-2020.) (Revised by AV, 22-Jun-2020.)
 |-  (
 ( N  e.  NN0  /\  N  e. Even  )  ->  E. m  e.  NN0  N  =  ( 2  x.  m ) )
 
Theoremnnpw2evenALTV 32577 2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 20-Jun-2020.)
 |-  ( N  e.  NN  ->  ( 2 ^ N )  e. Even  )
 
21.25.3.9  Perfect Number Theorem (revised)
 
TheoremperfectALTVlem1 32578 Lemma for perfectALTV 32580. (Contributed by Mario Carneiro, 7-Jun-2016.) (Revised by AV, 1-Jul-2020.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  B  e. Odd  )   &    |-  ( ph  ->  ( 1  sigma  ( (
 2 ^ A )  x.  B ) )  =  ( 2  x.  ( ( 2 ^ A )  x.  B ) ) )   =>    |-  ( ph  ->  ( ( 2 ^ ( A  +  1 )
 )  e.  NN  /\  ( ( 2 ^
 ( A  +  1 ) )  -  1
 )  e.  NN  /\  ( B  /  (
 ( 2 ^ ( A  +  1 )
 )  -  1 ) )  e.  NN )
 )
 
TheoremperfectALTVlem2 32579 Lemma for perfectALTV 32580. (Contributed by Mario Carneiro, 17-May-2016.) (Revised by AV, 1-Jul-2020.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  B  e. Odd  )   &    |-  ( ph  ->  ( 1  sigma  ( (
 2 ^ A )  x.  B ) )  =  ( 2  x.  ( ( 2 ^ A )  x.  B ) ) )   =>    |-  ( ph  ->  ( B  e.  Prime  /\  B  =  ( ( 2 ^
 ( A  +  1 ) )  -  1
 ) ) )
 
TheoremperfectALTV 32580* The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer  N is a perfect number (that is, its divisor sum is  2 N) if and only if it is of the form  2 ^ ( p  - 
1 )  x.  (
2 ^ p  - 
1 ), where  2 ^ p  -  1 is prime (a Mersenne prime). (It follows from this that  p is also prime.) This is Metamath 100 proof #70. (Contributed by Mario Carneiro, 17-May-2016.) (Revised by AV, 1-Jul-2020.) (Proof modification is discouraged.)
 |-  (
 ( N  e.  NN  /\  N  e. Even  )  ->  ( ( 1  sigma  N )  =  ( 2  x.  N )  <->  E. p  e.  ZZ  ( ( ( 2 ^ p )  -  1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  -  1 ) ) ) ) )
 
21.25.3.10  Proth's theorem
 
Theoremmodexp2m1d 32581 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 32582 Lemma for proththd 32583. (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 32583* 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 14445), 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 32584 5 times the cube of 2 is 40. (Contributed by AV, 4-Jul-2020.)
 |-  (
 5  x.  ( 2 ^ 3 ) )  = ; 4 0
 
Theorem3exp4mod41 32585 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 32586 Lemma 1 for 41prothprm 32588. (Contributed by AV, 4-Jul-2020.)
 |-  P  = ; 4 1   =>    |-  ( ( P  -  1 )  /  2
 )  = ; 2 0
 
Theorem41prothprmlem2 32587 Lemma 2 for 41prothprm 32588. (Contributed by AV, 5-Jul-2020.)
 |-  P  = ; 4 1   =>    |-  ( ( 3 ^
 ( ( P  -  1 )  /  2
 ) )  mod  P )  =  ( -u 1  mod  P )
 
Theorem41prothprm 32588 41 is a Proth prime. (Contributed by AV, 5-Jul-2020.)
 |-  P  = ; 4 1   =>    |-  ( P  =  ( ( 5  x.  (
 2 ^ 3 ) )  +  1 ) 
 /\  P  e.  Prime )
 
21.25.4  Words over a set (extension)
 
21.25.4.1  Last symbol of a word (extension)
 
Theoremlswn0 32589 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 )  =/=  (/) )
 
21.25.4.2  Concatenations with singleton words (extension)
 
Theoremccatw2s1cl 32590 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 )
 
21.25.4.3  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 32591 Syntax for the prefix operator.
 class prefix
 
Definitiondf-pfx 32592* 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 32593 Value of a prefix. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e.  V  /\  L  e.  NN0 )  ->  ( S prefix  L )  =  ( S substr  <. 0 ,  L >. ) )
 
Theorempfx00 32594 A zero length prefix. (Contributed by AV, 2-May-2020.)
 |-  ( S prefix  0 )  =  (/)
 
Theorempfx0 32595 A prefix of an empty set is always the empty set. (Contributed by AV, 3-May-2020.)
 |-  ( (/) prefix  L )  =  (/)
 
Theorempfxcl 32596 Closure of the prefix extractor. (Contributed by AV, 2-May-2020.)
 |-  ( S  e. Word  A  ->  ( S prefix  L )  e. Word  A )
 
Theorempfxmpt 32597* 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 32598 Value of the prefix extractor as restriction. Could replace swrd0val 12576. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e. Word  A  /\  L  e.  ( 0
 ... ( # `  S ) ) )  ->  ( S prefix  L )  =  ( S  |`  ( 0..^ L ) ) )
 
Theorempfxf 32599 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 12582. (Contributed by AV, 2-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 0
 ... ( # `  W ) ) )  ->  ( W prefix  L ) : ( 0..^ L ) --> V )
 
Theorempfxfn 32600 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 ) )
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