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Theorem List for Metamath Proof Explorer - 15001-15100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdecexp2 15001 Calculate a power of two. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  M  e.  NN0   &    |-  ( M  +  2 )  =  N   =>    |-  ( ( 4  x.  ( 2 ^ M ) )  +  0 )  =  (
 2 ^ N )
 
Theoremnumexp0 15002 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   =>    |-  ( A ^ 0
 )  =  1
 
Theoremnumexp1 15003 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   =>    |-  ( A ^ 1
 )  =  A
 
Theoremnumexpp1 15004 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   &    |-  M  e.  NN0   &    |-  ( M  +  1 )  =  N   &    |-  (
 ( A ^ M )  x.  A )  =  C   =>    |-  ( A ^ N )  =  C
 
Theoremnumexp2x 15005 Double an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   &    |-  M  e.  NN0   &    |-  ( 2  x.  M )  =  N   &    |-  ( A ^ M )  =  D   &    |-  ( D  x.  D )  =  C   =>    |-  ( A ^ N )  =  C
 
Theoremdecsplit0b 15006 Split a decimal number into two parts. Base case:  N  =  0. (Contributed by Mario Carneiro, 16-Jul-2015.)
 |-  A  e.  NN0   =>    |-  ( ( A  x.  ( 10 ^ 0 ) )  +  B )  =  ( A  +  B )
 
Theoremdecsplit0 15007 Split a decimal number into two parts. Base case:  N  =  0. (Contributed by Mario Carneiro, 16-Jul-2015.)
 |-  A  e.  NN0   =>    |-  ( ( A  x.  ( 10 ^ 0 ) )  +  0 )  =  A
 
Theoremdecsplit1 15008 Split a decimal number into two parts. Base case:  N  =  1. (Contributed by Mario Carneiro, 16-Jul-2015.)
 |-  A  e.  NN0   =>    |-  ( ( A  x.  ( 10 ^ 1 ) )  +  B )  = ; A B
 
Theoremdecsplit 15009 Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  D  e.  NN0   &    |-  M  e.  NN0   &    |-  ( M  +  1 )  =  N   &    |-  (
 ( A  x.  ( 10 ^ M ) )  +  B )  =  C   =>    |-  ( ( A  x.  ( 10 ^ N ) )  + ; B D )  = ; C D
 
Theoremkaratsuba 15010 The Karatsuba multiplication algorithm. If  X and 
Y are decomposed into two groups of digits of length  M (only the lower group is known to be this size but the algorithm is most efficient when the partition is chosen near the middle of the digit string), then  X Y can be written in three groups of digits, where each group needs only one multiplication. Thus, we can halve both inputs with only three multiplications on the smaller operands, yielding an asymptotic improvement of n^(log2 3) instead of n^2 for the "naive" algorithm decmul1c 11098. (Contributed by Mario Carneiro, 16-Jul-2015.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   &    |-  C  e.  NN0   &    |-  D  e.  NN0   &    |-  S  e.  NN0   &    |-  M  e.  NN0   &    |-  ( A  x.  C )  =  R   &    |-  ( B  x.  D )  =  T   &    |-  (
 ( A  +  B )  x.  ( C  +  D ) )  =  ( ( R  +  S )  +  T )   &    |-  ( ( A  x.  ( 10 ^ M ) )  +  B )  =  X   &    |-  ( ( C  x.  ( 10 ^ M ) )  +  D )  =  Y   &    |-  (
 ( R  x.  ( 10 ^ M ) )  +  S )  =  W   &    |-  ( ( W  x.  ( 10 ^ M ) )  +  T )  =  Z   =>    |-  ( X  x.  Y )  =  Z
 
Theorem2exp4 15011 Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 2 ^ 4
 )  = ; 1 6
 
Theorem2exp6 15012 Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
 |-  ( 2 ^ 6
 )  = ; 6 4
 
Theorem2exp6OLD 15013 Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of 2exp6 15012 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( 2 ^ 6
 )  = ; 6 4
 
Theorem2exp8 15014 Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 2 ^ 8
 )  = ;; 2 5 6
 
Theorem2exp16 15015 Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 2 ^; 1 6 )  = ;;;; 6 5 5 3 6
 
Theorem3exp3 15016 Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 3 ^ 3
 )  = ; 2 7
 
Theorem2expltfac 15017 The factorial grows faster than two to the power  N. (Contributed by Mario Carneiro, 15-Sep-2016.)
 |-  ( N  e.  ( ZZ>=
 `  4 )  ->  ( 2 ^ N )  <  ( ! `  N ) )
 
6.2.18  Cyclical shifts of words (cont.)
 
Theoremcshwsidrepsw 15018 If cyclically shifting a word of length being a prime number by a number of positions which is not divisible by the prime number results in the word itself, the word is a "repeated symbol word". (Contributed by AV, 18-May-2018.) (Revised by AV, 10-Nov-2018.)
 |-  ( ( W  e. Word  V 
 /\  ( # `  W )  e.  Prime )  ->  ( ( L  e.  ZZ  /\  ( L  mod  ( # `  W ) )  =/=  0  /\  ( W cyclShift  L )  =  W )  ->  W  =  ( ( W `  0 ) repeatS  ( # `  W ) ) ) )
 
Theoremcshwsidrepswmod0 15019 If cyclically shifting a word of length being a prime number results in the word itself, the shift must be either by 0 (modulo the length of the word) or the word must be a "repeated symbol word". (Contributed by AV, 18-May-2018.) (Revised by AV, 10-Nov-2018.)
 |-  ( ( W  e. Word  V 
 /\  ( # `  W )  e.  Prime  /\  L  e.  ZZ )  ->  (
 ( W cyclShift  L )  =  W  ->  ( ( L  mod  ( # `  W ) )  =  0  \/  W  =  ( ( W `  0 ) repeatS  ( # `  W ) ) ) ) )
 
Theoremcshwshashlem1 15020* If cyclically shifting a word of length being a prime number not consisting of identical symbols by at least one position (and not by as many positions as the length of the word), the result will not be the word itself. (Contributed by AV, 19-May-2018.) (Revised by AV, 8-Jun-2018.) (Revised by AV, 10-Nov-2018.)
 |-  ( ph  ->  ( W  e. Word  V  /\  ( # `
  W )  e. 
 Prime ) )   =>    |-  ( ( ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W `  0 )  /\  L  e.  ( 1..^ ( # `  W ) ) )  ->  ( W cyclShift  L )  =/=  W )
 
Theoremcshwshashlem2 15021* If cyclically shifting a word of length being a prime number and not of identical symbols by different numbers of positions, the resulting words are different. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
 |-  ( ph  ->  ( W  e. Word  V  /\  ( # `
  W )  e. 
 Prime ) )   =>    |-  ( ( ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W `  0 ) ) 
 ->  ( ( L  e.  ( 0..^ ( # `  W ) )  /\  K  e.  ( 0..^ ( # `  W ) )  /\  K  <  L )  ->  ( W cyclShift  L )  =/=  ( W cyclShift  K ) ) )
 
Theoremcshwshashlem3 15022* If cyclically shifting a word of length being a prime number and not of identical symbols by different numbers of positions, the resulting words are different. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
 |-  ( ph  ->  ( W  e. Word  V  /\  ( # `
  W )  e. 
 Prime ) )   =>    |-  ( ( ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W `  0 ) ) 
 ->  ( ( L  e.  ( 0..^ ( # `  W ) )  /\  K  e.  ( 0..^ ( # `  W ) )  /\  K  =/=  L )  ->  ( W cyclShift  L )  =/=  ( W cyclShift  K ) ) )
 
Theoremcshwsdisj 15023* The singletons resulting by cyclically shifting a given word of length being a prime number and not consisting of identical symbols is a disjoint collection. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
 |-  ( ph  ->  ( W  e. Word  V  /\  ( # `
  W )  e. 
 Prime ) )   =>    |-  ( ( ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W `  0 ) ) 
 -> Disj 
 n  e.  ( 0..^ ( # `  W ) ) { ( W cyclShift  n ) } )
 
Theoremcshwsiun 15024* The set of (different!) words resulting by cyclically shifting a given word is an indexed union. (Contributed by AV, 19-May-2018.) (Revised by AV, 8-Jun-2018.) (Proof shortened by AV, 8-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( W  e. Word  V  ->  M  =  U_ n  e.  ( 0..^ ( # `  W ) ) {
 ( W cyclShift  n ) }
 )
 
Theoremcshwsex 15025* The class of (different!) words resulting by cyclically shifting a given word is a set. (Contributed by AV, 8-Jun-2018.) (Revised by AV, 8-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( W  e. Word  V  ->  M  e.  _V )
 
Theoremcshws0 15026* The size of the set of (different!) words resulting by cyclically shifting an empty word is 0. (Contributed by AV, 8-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( W  =  (/)  ->  ( # `  M )  =  0 )
 
Theoremcshwrepswhash1 15027* The size of the set of (different!) words resulting by cyclically shifting a nonempty "repeated symbol word" is 1. (Contributed by AV, 18-May-2018.) (Revised by AV, 8-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( ( A  e.  V  /\  N  e.  NN  /\  W  =  ( A repeatS  N ) )  ->  ( # `  M )  =  1 )
 
Theoremcshwshashnsame 15028* If a word (not consisting of identical symbols) has a length being a prime number, the size of the set of (different!) words resulting by cyclically shifting the original word equals the length of the original word. (Contributed by AV, 19-May-2018.) (Revised by AV, 10-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( ( W  e. Word  V 
 /\  ( # `  W )  e.  Prime )  ->  ( E. i  e.  (
 0..^ ( # `  W ) ) ( W `
  i )  =/=  ( W `  0
 )  ->  ( # `  M )  =  ( # `  W ) ) )
 
Theoremcshwshash 15029* If a word has a length being a prime number, the size of the set of (different!) words resulting by cyclically shifting the original word equals the length of the original word or 1. (Contributed by AV, 19-May-2018.) (Revised by AV, 10-Nov-2018.)
 |-  M  =  { w  e. Word  V  |  E. n  e.  ( 0..^ ( # `  W ) ) ( W cyclShift  n )  =  w }   =>    |-  ( ( W  e. Word  V 
 /\  ( # `  W )  e.  Prime )  ->  ( ( # `  M )  =  ( # `  W )  \/  ( # `  M )  =  1 )
 )
 
6.2.19  Specific prime numbers
 
Theorem4nprm 15030 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  4  e.  Prime
 
Theoremprmlem0 15031* Lemma for prmlem1 15033 and prmlem2 15045. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  ( ( -.  2  ||  M  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( ( x  e.  ( Prime  \  { 2 } )  /\  ( x ^ 2
 )  <_  N )  ->  -.  x  ||  N ) )   &    |-  ( K  e.  Prime  ->  -.  K  ||  N )   &    |-  ( K  +  2 )  =  M   =>    |-  ( ( -.  2  ||  K  /\  x  e.  ( ZZ>= `  K ) )  ->  ( ( x  e.  ( Prime  \  { 2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N ) )
 
Theoremprmlem1a 15032* A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN   &    |-  1  <  N   &    |-  -.  2  ||  N   &    |- 
 -.  3  ||  N   &    |-  (
 ( -.  2  ||  5  /\  x  e.  ( ZZ>=
 `  5 ) ) 
 ->  ( ( x  e.  ( Prime  \  { 2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N ) )   =>    |-  N  e.  Prime
 
Theoremprmlem1 15033 A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN   &    |-  1  <  N   &    |-  -.  2  ||  N   &    |- 
 -.  3  ||  N   &    |-  N  < ; 2
 5   =>    |-  N  e.  Prime
 
Theorem5prm 15034 5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  5  e.  Prime
 
Theorem6nprm 15035 6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  6  e.  Prime
 
Theorem7prm 15036 7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  7  e.  Prime
 
Theorem8nprm 15037 8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  8  e.  Prime
 
Theorem9nprm 15038 9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  9  e.  Prime
 
Theorem10nprm 15039 10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  10  e.  Prime
 
Theorem11prm 15040 11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |- ; 1
 1  e.  Prime
 
Theorem13prm 15041 13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |- ; 1
 3  e.  Prime
 
Theorem17prm 15042 17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |- ; 1
 7  e.  Prime
 
Theorem19prm 15043 19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |- ; 1
 9  e.  Prime
 
Theorem23prm 15044 23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |- ; 2
 3  e.  Prime
 
Theoremprmlem2 15045 Our last proving session got as far as 25 because we started with the two "bootstrap" primes 2 and 3, and the next prime is 5, so knowing that 2 and 3 are prime and 4 is not allows us to cover the numbers less than  5 ^ 2  =  2 5. Additionally, nonprimes are "easy", so we can extend this range of known prime/nonprimes all the way until 29, which is the first prime larger than 25. Thus, in this lemma we extend another blanket out to  2 9 ^ 2  =  8 4 1, from which we can prove even more primes. If we wanted, we could keep doing this, but the goal is Bertrand's postulate, and for that we only need a few large primes - we don't need to find them all, as we have been doing thus far. So after this blanket runs out, we'll have to switch to another method (see 1259prm 15061).

As a side note, you can see the pattern of the primes in the indentation pattern of this lemma! (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

 |-  N  e.  NN   &    |-  N  < ;; 8 4 1   &    |-  1  <  N   &    |-  -.  2  ||  N   &    |- 
 -.  3  ||  N   &    |-  -.  5  ||  N   &    |-  -.  7  ||  N   &    |- 
 -. ; 1 1  ||  N   &    |-  -. ; 1 3  ||  N   &    |-  -. ; 1 7 
 ||  N   &    |-  -. ; 1 9  ||  N   &    |-  -. ; 2 3 
 ||  N   =>    |-  N  e.  Prime
 
Theorem37prm 15046 37 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ; 3
 7  e.  Prime
 
Theorem43prm 15047 43 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ; 4
 3  e.  Prime
 
Theorem83prm 15048 83 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ; 8
 3  e.  Prime
 
Theorem139prm 15049 139 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ;; 1 3 9  e. 
 Prime
 
Theorem163prm 15050 163 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ;; 1 6 3  e. 
 Prime
 
Theorem317prm 15051 317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ;; 3 1 7  e. 
 Prime
 
Theorem631prm 15052 631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |- ;; 6 3 1  e. 
 Prime
 
Theoremprmo4 15053 The primorial of 4. (Contributed by AV, 28-Aug-2020.)
 |-  (#p `  4 )  =  6
 
Theoremprmo5 15054 The primorial of 5. (Contributed by AV, 28-Aug-2020.)
 |-  (#p `  5 )  = ; 3
 0
 
Theoremprmo6 15055 The primorial of 6. (Contributed by AV, 28-Aug-2020.)
 |-  (#p `  6 )  = ; 3
 0
 
6.2.20  Very large primes
 
Theorem1259lem1 15056 Lemma for 1259prm 15061. Calculate a power mod. In decimal, we calculate  2 ^ 1 6  =  5 2 N  +  6 8  ==  6 8 and  2 ^ 1 7  ==  6 8  x.  2  =  1 3 6 in this lemma. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  ( ( 2 ^; 1 7 )  mod  N )  =  (;; 1 3 6  mod  N )
 
Theorem1259lem2 15057 Lemma for 1259prm 15061. Calculate a power mod. In decimal, we calculate  2 ^ 3 4  =  ( 2 ^ 1 7 ) ^ 2  ==  1
3 6 ^ 2  ==  1 4 N  +  8 7 0. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  ( ( 2 ^; 3 4 )  mod  N )  =  (;; 8 7 0  mod  N )
 
Theorem1259lem3 15058 Lemma for 1259prm 15061. Calculate a power mod. In decimal, we calculate  2 ^ 3 8  =  2 ^ 3 4  x.  2 ^ 4  ==  8
7 0  x.  1 6  =  1 1 N  +  7 1 and  2 ^ 7 6  =  ( 2 ^ 3 4 ) ^ 2  ==  7
1 ^ 2  =  4 N  +  5  ==  5. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  ( ( 2 ^; 7 6 )  mod  N )  =  ( 5  mod 
 N )
 
Theorem1259lem4 15059 Lemma for 1259prm 15061. Calculate a power mod. In decimal, we calculate  2 ^ 3 0 6  =  ( 2 ^ 7 6 ) ^ 4  x.  4  ==  5 ^ 4  x.  4  =  2 N  -  1 8,  2 ^ 6 1 2  =  ( 2 ^ 3 0 6 ) ^ 2  ==  1 8 ^ 2  =  3 2 4,  2 ^ 6 2 9  =  2 ^ 6 1 2  x.  2 ^ 1 7  ==  3 2 4  x.  1 3 6  =  3 5 N  -  1 and finally  2 ^ ( N  -  1 )  =  ( 2 ^ 6 2 9 ) ^ 2  ==  1 ^ 2  =  1. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  ( ( 2 ^
 ( N  -  1
 ) )  mod  N )  =  ( 1  mod  N )
 
Theorem1259lem5 15060 Lemma for 1259prm 15061. Calculate the GCD of  2 ^ 3 4  -  1  ==  8 6 9 with  N  =  1 2 5 9. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  ( ( ( 2 ^; 3 4 )  -  1 )  gcd  N )  =  1
 
Theorem1259prm 15061 1259 is a prime number. (Contributed by Mario Carneiro, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 1 2 5 9   =>    |-  N  e.  Prime
 
Theorem2503lem1 15062 Lemma for 2503prm 15065. Calculate a power mod. In decimal, we calculate  2 ^ 1 8  =  5 1 2 ^ 2  =  1 0 4 N  +  1 8 3 2  ==  1 8 3 2. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 2 5 0 3   =>    |-  ( ( 2 ^; 1 8 )  mod  N )  =  (;;; 1 8 3 2 
 mod  N )
 
Theorem2503lem2 15063 Lemma for 2503prm 15065. Calculate a power mod. We calculate  2 ^ 1 9  =  2 ^ 1 8  x.  2  ==  1 8 3 2  x.  2  =  N  +  1 1 6 1,  2 ^ 3 8  =  ( 2 ^ 1 9 ) ^ 2  ==  1
1 6 1 ^ 2  =  5 3 8 N  +  1 3 0 7,  2 ^ 3 9  =  2 ^ 3 8  x.  2  ==  1 3 0 7  x.  2  =  N  +  1 1 1,  2 ^ 7 8  =  ( 2 ^ 3 9 ) ^ 2  ==  1
1 1 ^ 2  =  5 N  - 
1 9 4,  2 ^ 1 5 6  =  ( 2 ^ 7 8 ) ^ 2  ==  1 9 4 ^ 2  =  1 5 N  +  9 1,  2 ^ 3 1 2  =  ( 2 ^ 1 5 6 ) ^ 2  ==  9 1 ^ 2  =  3 N  +  7 7 2,  2 ^ 6 2 4  =  ( 2 ^ 3 1 2 ) ^ 2  ==  7 7 2 ^ 2  =  2 3 8 N  + 
2 7 0,  2 ^ 1 2 4 8  =  ( 2 ^ 6 2 4 ) ^
2  ==  2 7 0 ^ 2  =  2 9 N  + 
3 1 3,  2 ^ 1 2 5 1  =  2 ^ 1 2 4 8  x.  8  ==  3 1 3  x.  8  =  N  +  1 and finally  2 ^ ( N  -  1 )  =  ( 2 ^ 1 2 5 1 ) ^ 2  ==  1 ^ 2  =  1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 2 5 0 3   =>    |-  ( ( 2 ^
 ( N  -  1
 ) )  mod  N )  =  ( 1  mod  N )
 
Theorem2503lem3 15064 Lemma for 2503prm 15065. Calculate the GCD of  2 ^ 1 8  -  1  ==  1 8 3 1 with  N  =  2 5 0 3. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 2 5 0 3   =>    |-  ( ( ( 2 ^; 1 8 )  -  1 )  gcd  N )  =  1
 
Theorem2503prm 15065 2503 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 2 5 0 3   =>    |-  N  e.  Prime
 
Theorem4001lem1 15066 Lemma for 4001prm 15070. Calculate a power mod. In decimal, we calculate  2 ^ 1 2  =  4 0 9 6  =  N  +  9 5,  2 ^ 2 4  =  ( 2 ^ 1 2 ) ^ 2  ==  9
5 ^ 2  =  2 N  +  1 0 2 3,  2 ^ 2 5  =  2 ^ 2 4  x.  2  ==  1 0 2 3  x.  2  =  2 0 4 6,  2 ^ 5 0  =  ( 2 ^ 2 5 ) ^ 2  ==  2
0 4 6 ^ 2  =  1 0 4 6 N  + 
1 0 7 0,  2 ^ 1 0 0  =  ( 2 ^ 5 0 ) ^ 2  ==  1 0 7 0 ^ 2  =  2 8 6 N  + 
6 1 4 and  2 ^ 2 0 0  =  ( 2 ^ 1 0 0 ) ^ 2  ==  6 1 4 ^ 2  =  9 4 N  +  9 0 2  ==  9 0 2. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 4 0 0 1   =>    |-  ( ( 2 ^;; 2 0 0 ) 
 mod  N )  =  (;; 9 0 2  mod 
 N )
 
Theorem4001lem2 15067 Lemma for 4001prm 15070. Calculate a power mod. In decimal, we calculate  2 ^ 4 0 0  =  ( 2 ^ 2 0 0 ) ^ 2  ==  9 0 2 ^ 2  =  2 0 3 N  + 
1 4 0 1 and  2 ^ 8 0 0  =  ( 2 ^ 4 0 0 ) ^ 2  ==  1 4 0 1 ^ 2  =  4 9 0 N  +  2 3 1 1  ==  2 3 1 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 4 0 0 1   =>    |-  ( ( 2 ^;; 8 0 0 ) 
 mod  N )  =  (;;; 2 3 1 1  mod 
 N )
 
Theorem4001lem3 15068 Lemma for 4001prm 15070. Calculate a power mod. In decimal, we calculate  2 ^ 1 0 0 0  =  2 ^ 8 0 0  x.  2 ^ 2 0 0  ==  2 3 1 1  x.  9 0 2  =  5 2 1 N  +  1 and finally  2 ^ ( N  -  1 )  =  ( 2 ^ 1 0 0 0 ) ^ 4  ==  1 ^ 4  =  1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 4 0 0 1   =>    |-  ( ( 2 ^
 ( N  -  1
 ) )  mod  N )  =  ( 1  mod  N )
 
Theorem4001lem4 15069 Lemma for 4001prm 15070. Calculate the GCD of  2 ^ 8 0 0  -  1  ==  2 3 1 0 with  N  =  4 0 0 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 4 0 0 1   =>    |-  ( ( ( 2 ^;; 8 0 0 )  -  1
 )  gcd  N )  =  1
 
Theorem4001prm 15070 4001 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
 |-  N  = ;;; 4 0 0 1   =>    |-  N  e.  Prime
 
PART 7  BASIC STRUCTURES
 
7.1  Extensible structures
 
7.1.1  Basic definitions

An "extensible structure" is used to define a specific group, ring, poset, and so on. An extensible structure can contain many components. For example, a group will have at least two components (base set and operation), although it can be further specialized by adding other components such as a multiplicative operation for rings (and still remain a group per our definition). Thus, every ring is also a group. This extensible structure approach allows theorems from more general structures (such as groups) to be reused for more specialized structures (such as rings) without having to reprove anything. Structures are common in mathematics, but in informal (natural language) proofs the details are assumed in ways that we must make explicit.

An extensible structure is implemented as a function (set of ordered pairs) on a finite (and not necessarily sequential) subset of  NN. The function's argument is the index of a structure component (such as  1 for the base set of a group), and its value is the component (such as the base set). By convention we normally avoid direct reference to the hard-coded numeric index and instead use structure component extractors such as ndxid 15096 and strfv 15111. Using extractors makes it easier to change numeric indexes and also make the components' purpose clearer. For example, as noted in ndxid 15096, we can refer to a specific poset with base set  B and order relation  L using the extensible structure  { <. ( Base `  ndx ) ,  B >. ,  <. ( le `  ndx ) ,  L >. } rather than  { <. 1 ,  B >. ,  <. 10 ,  L >. }.

There are many other ways to handle structures. We chose this extensible structure approach because this approach (1) results in simpler notation than other approaches we're aware of, and (2) is easier to do proofs with. We cannot use an approach that uses "hidden" arguments; Metamath does not support hidden arguments, and in any case we want nothing hidden. It would be possible to use a categorical approach (e.g., something vaguely similar to Lean's mathlib). However, instances (the chain of proofs that an  X is a  Y via a bunch of forgetful functors) can cause serious performance problems for automated tooling, and the resulting proofs would be painful to look at directly (in the case of Lean, they are long past the level where people would find it acceptable to look at them directly). Metamath is working under much stricter conditions than this, and it has still managed to achieve about the same level of flexibility through this "extensible structure" approach.

To create a substructure of a given extensible structure, you can simply use the multifunction restriction operator for extensible structures ↾s as defined in df-ress 15082. This can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the  Base set while leaving operators alone. Individual kinds of structures will need to handle this behavior by ignoring operators' values outside the range (like  Ring), defining a function using the base set and applying that (like  TopGrp), or explicitly truncating the slot before use (like  MetSp). For example, the unital ring of integers ℤring is defined in df-zring 18965 as simply ℤring  =  (flds  ZZ ). This can be similarly done for all other subsets of  CC, which has all the structure we can show applies to it, and this all comes "for free". Should we come up with some new structure in the future that we wish  CC to inherit, then we change the definition of ℂfld, reprove all the slot extraction theorems, add a new one, and that's it. None of the other downstream theorems have to change.

Note that the construct of df-prds 15296 addresses a different situation. It is not possible to have SubGroup and SubRing be the same thing because they produce different outputs on the same input. The subgroups of an extensible structure treated as a group are not the same as the subrings of that same structure. With df-prds 15296 it can actually reasonably perform the task, that is, being the product group given a family of groups, while also being the product ring given a family of rings. There is no contradiction here because the group part of a product ring is a product group.

There is also a general theory of "substructure algebras", in the form of df-mre 15434 and df-acs 15437. SubGroup is a Moore collection, as is SubRing, SubRng and many other substructure collections. But it's no good for picking out a particular collection of interest; SubRing and SubGroup still need to be defined and they are distinct - nothing is going to select these definitions for us.

Extensible structures only work well when they represent concrete categories, where there is a "base set", homs are functions, and subobjects are subsets with induced operations. In short, they primarily work well for "sets with (some) extra structure". Extensible structures may not suffice for more complicated situations. For example, in manifolds, ↾s would not work. That said, extensible structures are sufficient for many of the structures that set.mm currently considers, and offer a good compromise for a goal-oriented formalization.

 
Syntaxcstr 15071 Extend class notation with the class of structures with components numbered below  A.
 class Struct
 
Syntaxcnx 15072 Extend class notation with the structure component index extractor.
 class  ndx
 
Syntaxcsts 15073 Set components of a structure.
 class sSet
 
Syntaxcslot 15074 Extend class notation with the slot function.
 class Slot  A
 
Syntaxcbs 15075 Extend class notation with the class of all base set extractors.
 class  Base
 
Syntaxcress 15076 Extend class notation with the extensible structure builder restriction operator.
 classs
 
Definitiondf-struct 15077* Define a structure with components in  M ... N. This is not a requirement for groups, posets, etc., but it is a useful assumption for component extraction theorems. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |- Struct  =  { <. f ,  x >.  |  ( x  e.  (  <_  i^i  ( NN 
 X.  NN ) )  /\  Fun  ( f  \  { (/)
 } )  /\  dom  f  C_  ( ... `  x ) ) }
 
Definitiondf-ndx 15078 Define the structure component index extractor. See theorem ndxarg 15095 to understand its purpose. The restriction to  NN allows  ndx to exist as a set, since  _I is a proper class. In principle, we could have chosen  CC or (if we revise all structure component definitions such as df-base 15080) another set such as the natural ordinal numbers  om (df-om 6707). (Contributed by NM, 4-Sep-2011.)
 |- 
 ndx  =  (  _I  |` 
 NN )
 
Definitiondf-slot 15079* Define slot extractor for posets and related structures. Note that the function argument can be any set, although it is meaningful only if it is a member of  Poset (df-poset 16133) when used for posets or of  Grp (df-grp 16615) when used from groups. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |- Slot  A  =  ( x  e.  _V  |->  ( x `  A ) )
 
Definitiondf-base 15080 Define the base set (also called underlying set or carrier set) extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 Base  = Slot  1
 
Definitiondf-sets 15081* Set one or more components of a structure. This function is useful for taking an existing structure and "overriding" one of its components. For example, df-ress 15082 adjusts the base set to match its second argument, which has the effect of making subgroups, subspaces, subrings etc. from the original structures. Or df-mgp 17650, which takes a ring and overrides its addition operation with the multiplicative operation, so that we can consider the "multiplicative group" using group and monoid theorems, which expect the operation to be in the  +g slot instead of the  .r slot. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |- sSet  =  ( s  e.  _V ,  e  e.  _V  |->  ( ( s  |`  ( _V  \  dom  { e } ) )  u. 
 { e } )
 )
 
Definitiondf-ress 15082* Define a multifunction restriction operator for extensible structures, which can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the  Base set while leaving operators alone; individual kinds of structures will need to handle this behavior, by ignoring operators' values outside the range (like  Ring), defining a function using the base set and applying that (like  TopGrp), or explicitly truncating the slot before use (like  MetSp).

(Credit for this operator goes to Mario Carneiro).

See ressbas 15132 for the altered base set, and resslem 15135 (subrg0 17941, ressplusg 15189, subrg1 17944, ressmulr 15200) for the (un)altered other operations. (Contributed by Stefan O'Rear, 29-Nov-2014.)

 |-s  =  ( w  e.  _V ,  x  e.  _V  |->  if ( ( Base `  w )  C_  x ,  w ,  ( w sSet  <. ( Base ` 
 ndx ) ,  ( x  i^i  ( Base `  w ) ) >. ) ) )
 
Theorembrstruct 15083 The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |- 
 Rel Struct
 
Theoremisstruct2 15084 The property of being a structure with components in  ( 1st `  X
) ... ( 2nd `  X
). (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  ( F Struct  X  <->  ( X  e.  (  <_  i^i  ( NN  X. 
 NN ) )  /\  Fun  ( F  \  { (/)
 } )  /\  dom  F 
 C_  ( ... `  X ) ) )
 
Theoremisstruct 15085 The property of being a structure with components in  M ... N. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  ( F Struct  <. M ,  N >. 
 <->  ( ( M  e.  NN  /\  N  e.  NN  /\  M  <_  N )  /\  Fun  ( F  \  { (/) } )  /\  dom 
 F  C_  ( M ... N ) ) )
 
Theoremstructcnvcnv 15086 Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  ( F Struct  X  ->  `' `' F  =  ( F  \  { (/) } )
 )
 
Theoremstructfun 15087 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  F Struct  X   =>    |- 
 Fun  `' `' F
 
Theoremstructfn 15088 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  F Struct  <. M ,  N >.   =>    |-  ( Fun  `' `' F  /\  dom  F  C_  (
 1 ... N ) )
 
Theoremslotfn 15089 A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  E  = Slot  N   =>    |-  E  Fn  _V
 
Theoremstrfvnd 15090 Deduction version of strfvn 15092. (Contributed by Mario Carneiro, 15-Nov-2014.)
 |-  E  = Slot  N   &    |-  ( ph  ->  S  e.  V )   =>    |-  ( ph  ->  ( E `  S )  =  ( S `  N ) )
 
Theoremwunndx 15091 Closure of the index extractor in an infinite weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   =>    |-  ( ph  ->  ndx 
 e.  U )
 
Theoremstrfvn 15092 Value of a structure component extractor  E. Normally,  E is a defined constant symbol such as  Base (df-base 15080) and  N is a fixed integer such as  1.  S is a structure, i.e. a specific member of a class of structures such as  Poset (df-poset 16133) where  S  e.  Poset.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strfv 15111. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2013.) (New usage is discouraged.)

 |-  S  e.  _V   &    |-  E  = Slot  N   =>    |-  ( E `  S )  =  ( S `  N )
 
Theoremstrfvss 15093 A structure component extractor produces a value which is contained in a set dependent on  S, but not  E. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  E  = Slot  N   =>    |-  ( E `  S )  C_  U. ran  S
 
Theoremwunstr 15094 Closure of a structure index in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  E  = Slot  N   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  S  e.  U )   =>    |-  ( ph  ->  ( E `  S )  e.  U )
 
Theoremndxarg 15095 Get the numeric argument from a defined structure component extractor such as df-base 15080. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |-  E  = Slot  N   &    |-  N  e.  NN   =>    |-  ( E `  ndx )  =  N
 
Theoremndxid 15096 A structure component extractor is defined by its own index. This theorem, together with strfv 15111 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the  1 in df-base 15080 and the  10 in df-ple 15163, making it easier to change should the need arise. For example, we can refer to a specific poset with base set  B and order relation  L using  { <. ( Base `  ndx ) ,  B >. ,  <. ( le `  ndx ) ,  L >. } rather than  { <. 1 ,  B >. ,  <. 10 ,  L >. }. The latter, while shorter to state, requires revision if we later change  10 to some other number, and it may also be harder to remember. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.)
 |-  E  = Slot  N   &    |-  N  e.  NN   =>    |-  E  = Slot  ( E `
  ndx )
 
Theoremstrndxid 15097 The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.)
 |-  ( ph  ->  S  e.  V )   &    |-  E  = Slot  N   &    |-  N  e.  NN   =>    |-  ( ph  ->  ( S `  ( E `  ndx ) )  =  ( E `  S ) )
 
Theoremreldmsets 15098 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |- 
 Rel  dom sSet
 
Theoremsetsvalg 15099 Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( S  e.  V  /\  A  e.  W )  ->  ( S sSet  A )  =  ( ( S  |`  ( _V  \  dom  { A } ) )  u.  { A }
 ) )
 
Theoremsetsval 15100 Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( S  e.  V  /\  B  e.  W )  ->  ( S sSet  <. A ,  B >. )  =  ( ( S  |`  ( _V  \  { A } )
 )  u.  { <. A ,  B >. } )
 )
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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39291
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