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Theorem List for Metamath Proof Explorer - 13101-13200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmodxp1i 13101 Add one to an exponent in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)
 |-  N  e.  NN   &    |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  D  e.  ZZ   &    |-  K  e.  NN0   &    |-  M  e.  NN0   &    |-  ( ( A ^ B )  mod  N )  =  ( K  mod  N )   &    |-  ( B  +  1 )  =  E   &    |-  (
 ( D  x.  N )  +  M )  =  ( K  x.  A )   =>    |-  ( ( A ^ E )  mod  N )  =  ( M  mod  N )
 
Theoremmod2xnegi 13102 Version of mod2xi 13100 with a negaive mod value. (Contributed by Mario Carneiro, 21-Feb-2014.)
 |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  D  e.  ZZ   &    |-  K  e.  NN   &    |-  M  e.  NN0   &    |-  L  e.  NN0   &    |-  ( ( A ^ B )  mod  N )  =  ( L  mod  N )   &    |-  ( 2  x.  B )  =  E   &    |-  ( L  +  K )  =  N   &    |-  ( ( D  x.  N )  +  M )  =  ( K  x.  K )   =>    |-  ( ( A ^ E )  mod  N )  =  ( M 
 mod  N )
 
Theoremmodsubi 13103 Subtract from within a mod calculation. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN   &    |-  A  e.  NN   &    |-  B  e.  NN0   &    |-  M  e.  NN0   &    |-  ( A  mod  N )  =  ( K  mod  N )   &    |-  ( M  +  B )  =  K   =>    |-  (
 ( A  -  B )  mod  N )  =  ( M  mod  N )
 
Theoremgcdi 13104 Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  K  e.  NN0   &    |-  R  e.  NN0   &    |-  N  e.  NN0   &    |-  ( N  gcd  R )  =  G   &    |-  ( ( K  x.  N )  +  R )  =  M   =>    |-  ( M  gcd  N )  =  G
 
Theoremgcdmodi 13105 Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  K  e.  NN0   &    |-  R  e.  NN0   &    |-  N  e.  NN   &    |-  ( K  mod  N )  =  ( R  mod  N )   &    |-  ( N  gcd  R )  =  G   =>    |-  ( K  gcd  N )  =  G
 
Theoremdecexp2 13106 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 13107 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   =>    |-  ( A ^ 0
 )  =  1
 
Theoremnumexp1 13108 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   =>    |-  ( A ^ 1
 )  =  A
 
Theoremnumexpp1 13109 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 13110 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 13111 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 13112 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 13113 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 13114 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 13115 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 10187. (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 13116 Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 2 ^ 4
 )  = ; 1 6
 
Theorem2exp6 13117 Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 2 ^ 6
 )  = ; 6 4
 
Theorem2exp8 13118 Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 2 ^ 8
 )  = ;; 2 5 6
 
Theorem2exp16 13119 Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 2 ^; 1 6 )  = ;;;; 6 5 5 3 6
 
Theorem3exp3 13120 Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.)
 |-  ( 3 ^ 3
 )  = ; 2 7
 
Theorem2expltfac 13121 The factorial grows faster than two to the power  N. (Contributed by Mario Carneiro, 15-Sep-2016.)
 |-  ( N  e.  ( ZZ>=
 `  4 )  ->  ( 2 ^ N )  <  ( ! `  N ) )
 
6.2.14  Specific prime numbers
 
Theorem4nprm 13122 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 13123* Lemma for prmlem1 13125 and prmlem2 13137. (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 13124* 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 13125 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 13126 5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  5  e.  Prime
 
Theorem6nprm 13127 6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  6  e.  Prime
 
Theorem7prm 13128 7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |-  7  e.  Prime
 
Theorem8nprm 13129 8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  8  e.  Prime
 
Theorem9nprm 13130 9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  9  e.  Prime
 
Theorem10nprm 13131 10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |- 
 -.  10  e.  Prime
 
Theorem11prm 13132 11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |- ; 1
 1  e.  Prime
 
Theorem13prm 13133 13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |- ; 1
 3  e.  Prime
 
Theorem17prm 13134 17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |- ; 1
 7  e.  Prime
 
Theorem19prm 13135 19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |- ; 1
 9  e.  Prime
 
Theorem23prm 13136 23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
 |- ; 2
 3  e.  Prime
 
Theoremprmlem2 13137 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 13150).

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 13138 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 13139 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 13140 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 13141 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 13142 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 13143 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 13144 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
 
6.2.15  Very large primes
 
Theorem1259lem1 13145 Lemma for 1259prm 13150. 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 13146 Lemma for 1259prm 13150. 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 13147 Lemma for 1259prm 13150. 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 13148 Lemma for 1259prm 13150. 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 13149 Lemma for 1259prm 13150. 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 13150 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 13151 Lemma for 2503prm 13154. 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 13152 Lemma for 2503prm 13154. 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 13153 Lemma for 2503prm 13154. 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 13154 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 13155 Lemma for 4001prm 13159. 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 13156 Lemma for 2503prm 13154. 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 13157 Lemma for 4001prm 13159. 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 13158 Lemma for 4001prm 13159. 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 13159 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 a function (set of ordered pairs) on a finite (and not necessarily sequential) subset of  NN, used to define a specific group, ring, poset, etc. 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). 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 allows theorems from more general structures (groups) to be reused for more specialized structures (rings) without having to reprove them.

 
Syntaxcstr 13160 Extend class notation with the class of structures with components numbered below  A.
 class Struct
 
Syntaxcnx 13161 Extend class notation with the structure component index extractor.
 class  ndx
 
Syntaxcsts 13162 Set components of a structure.
 class sSet
 
Syntaxcslot 13163 Extend class notation with the slot function.
 class Slot  A
 
Syntaxcbs 13164 Extend class notation with the class of all base set extractors.
 class  Base
 
Syntaxcress 13165 Extend class notation with the extensible structure builder restriction operator.
 classs
 
Definitiondf-struct 13166* 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 13167 Define the structure component index extractor. See theorem ndxarg 13184 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 13169) another set such as the natural ordinal numbers  om (df-om 4673). (Contributed by NM, 4-Sep-2011.)
 |- 
 ndx  =  (  _I  |` 
 NN )
 
Definitiondf-slot 13168* 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 14096) when used for posets or of  Grp (df-grp 14505) when used from groups. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |- Slot  A  =  ( x  e.  _V  |->  ( x `  A ) )
 
Definitiondf-base 13169 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 13170* 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 13171 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 15342, 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 13171* 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 13214 for the altered base set, and resslem 13217 (subrg0 15568, ressplusg 13266, subrg1 15571, ressmulr 13277) 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 13172 The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |- 
 Rel Struct
 
Theoremisstruct2 13173 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 13174 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 13175 Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  ( F Struct  X  ->  `' `' F  =  ( F  \  { (/) } )
 )
 
Theoremstructfun 13176 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  F Struct  X   =>    |- 
 Fun  `' `' F
 
Theoremstructfn 13177 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 13178 A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  E  = Slot  N   =>    |-  E  Fn  _V
 
Theoremstrfvnd 13179 Deduction version of strfvn 13181. (Contributed by Mario Carneiro, 15-Nov-2014.)
 |-  E  = Slot  N   &    |-  ( ph  ->  S  e.  V )   =>    |-  ( ph  ->  ( E `  S )  =  ( S `  N ) )
 
Theoremwunndx 13180 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 13181 Value of a structure component extractor  E. Normally,  E is a defined constant symbol such as  Base (df-base 13169) 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 14096) 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 13196. (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 13182 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 13183 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 13184 Get the numeric argument from a defined structure component extractor such as df-base 13169. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |-  E  = Slot  N   &    |-  N  e.  NN   =>    |-  ( E `  ndx )  =  N
 
Theoremndxid 13185 A structure component extractor is defined by its own index. This theorem, together with strfv 13196 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the  1 in df-base 13169 and the  10 in df-ple 13244, 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 )
 
Theoremreldmsets 13186 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |- 
 Rel  dom sSet
 
Theoremsetsvalg 13187 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 13188 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 >. } )
 )
 
Theoremwunsets 13189 Closure of structure replacement in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  S  e.  U )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  ( S sSet  A )  e.  U )
 
Theoremsetsres 13190 The structure replacement function does not affect the value of  S away from  A. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( S  e.  V  ->  ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } )
 )  =  ( S  |`  ( _V  \  { A } ) ) )
 
Theoremsetsabs 13191 Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( ( S sSet  <. A ,  B >. ) sSet  <. A ,  C >. )  =  ( S sSet  <. A ,  C >. ) )
 
Theoremsetscom 13192 Component-setting is commutative when the x-values are different. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( ( S  e.  V  /\  A  =/=  B )  /\  ( C  e.  W  /\  D  e.  X )
 )  ->  ( ( S sSet  <. A ,  C >. ) sSet  <. B ,  D >. )  =  ( ( S sSet  <. B ,  D >. ) sSet  <. A ,  C >. ) )
 
Theoremstrfvd 13193 Deduction version of strfv 13196. (Contributed by Mario Carneiro, 15-Nov-2014.)
 |-  E  = Slot  ( E `
  ndx )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  Fun  S )   &    |-  ( ph  ->  <. ( E `  ndx ) ,  C >.  e.  S )   =>    |-  ( ph  ->  C  =  ( E `  S ) )
 
Theoremstrfv2d 13194 Deduction version of strfv 13196. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  E  = Slot  ( E `
  ndx )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  Fun  `' `' S )   &    |-  ( ph  ->  <.
 ( E `  ndx ) ,  C >.  e.  S )   &    |-  ( ph  ->  C  e.  W )   =>    |-  ( ph  ->  C  =  ( E `  S ) )
 
Theoremstrfv2 13195 A variation on strfv 13196 to avoid asserting that  S itself is a function, which involves sethood of all the ordered pair components of  S. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  S  e.  _V   &    |-  Fun  `' `' S   &    |-  E  = Slot  ( E `  ndx )   &    |-  <. ( E `
  ndx ) ,  C >.  e.  S   =>    |-  ( C  e.  V  ->  C  =  ( E `
  S ) )
 
Theoremstrfv 13196 Extract a structure component  C (such as the base set) from a structure  S (such as a member of  Poset, df-poset 14096) with a component extractor  E (such as the base set extractor df-base 13169). By virtue of ndxid 13185, this can be done without having to refer to the hard-coded numeric index of 
E. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  S Struct  X   &    |-  E  = Slot  ( E `  ndx )   &    |-  { <. ( E `  ndx ) ,  C >. }  C_  S   =>    |-  ( C  e.  V  ->  C  =  ( E `  S ) )
 
Theoremstrfv3 13197 Variant on strfv 13196 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.)
 |-  ( ph  ->  U  =  S )   &    |-  S Struct  X   &    |-  E  = Slot  ( E `  ndx )   &    |-  { <. ( E `  ndx ) ,  C >. }  C_  S   &    |-  ( ph  ->  C  e.  V )   &    |-  A  =  ( E `
  U )   =>    |-  ( ph  ->  A  =  C )
 
Theoremstrssd 13198 Deduction version of strss 13199. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  E  = Slot  ( E `
  ndx )   &    |-  ( ph  ->  T  e.  V )   &    |-  ( ph  ->  Fun  T )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ph  ->  <. ( E `
  ndx ) ,  C >.  e.  S )   =>    |-  ( ph  ->  ( E `  T )  =  ( E `  S ) )
 
Theoremstrss 13199 Propagate component extraction to a structure  T from a subset structure  S. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 15-Jan-2014.)
 |-  T  e.  _V   &    |-  Fun  T   &    |-  S  C_  T   &    |-  E  = Slot  ( E `  ndx )   &    |-  <. ( E `
  ndx ) ,  C >.  e.  S   =>    |-  ( E `  T )  =  ( E `  S )
 
Theoremstr0 13200 All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
 |-  F  = Slot  I   =>    |-  (/)  =  ( F `
  (/) )
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