P. 151 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 15001-15100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	decexp2 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 )
Theorem	numexp0 15002	Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
|- A e. NN0   =>    |- ( A ^ 0 ) = 1
Theorem	numexp1 15003	Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
|- A e. NN0   =>    |- ( A ^ 1 ) = A
Theorem	numexpp1 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
Theorem	numexp2x 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
Theorem	decsplit0b 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 )
Theorem	decsplit0 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
Theorem	decsplit1 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
Theorem	decsplit 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
Theorem	karatsuba 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
Theorem	2exp4 15011	Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
|- ( 2 ^ 4 ) = ; 1 6
Theorem	2exp6 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
Theorem	2exp6OLD 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
Theorem	2exp8 15014	Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
|- ( 2 ^ 8 ) = ;; 2 5 6
Theorem	2exp16 15015	Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
|- ( 2 ^; 1 6 ) = ;;;; 6 5 5 3 6
Theorem	3exp3 15016	Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.)
|- ( 3 ^ 3 ) = ; 2 7
Theorem	2expltfac 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.)
Theorem	cshwsidrepsw 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 ) ) ) )
Theorem	cshwsidrepswmod0 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 ) ) ) ) )
Theorem	cshwshashlem1 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 )
Theorem	cshwshashlem2 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 ) ) )
Theorem	cshwshashlem3 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 ) ) )
Theorem	cshwsdisj 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 ) } )
Theorem	cshwsiun 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 ) } )
Theorem	cshwsex 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 )
Theorem	cshws0 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 )
Theorem	cshwrepswhash1 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 )
Theorem	cshwshashnsame 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 ) ) )
Theorem	cshwshash 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
Theorem	4nprm 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
Theorem	prmlem0 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 ) )
Theorem	prmlem1a 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
Theorem	prmlem1 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
Theorem	5prm 15034	5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
|- 5 e. Prime
Theorem	6nprm 15035	6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- -. 6 e. Prime
Theorem	7prm 15036	7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
|- 7 e. Prime
Theorem	8nprm 15037	8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- -. 8 e. Prime
Theorem	9nprm 15038	9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- -. 9 e. Prime
Theorem	10nprm 15039	10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- -. 10 e. Prime
Theorem	11prm 15040	11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
|- ; 1 1 e. Prime
Theorem	13prm 15041	13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
|- ; 1 3 e. Prime
Theorem	17prm 15042	17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
|- ; 1 7 e. Prime
Theorem	19prm 15043	19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
|- ; 1 9 e. Prime
Theorem	23prm 15044	23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
|- ; 2 3 e. Prime
Theorem	prmlem2 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
Theorem	37prm 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
Theorem	43prm 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
Theorem	83prm 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
Theorem	139prm 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
Theorem	163prm 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
Theorem	317prm 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
Theorem	631prm 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
Theorem	prmo4 15053	The primorial of 4. (Contributed by AV, 28-Aug-2020.)
|- (#p ` 4 ) = 6
Theorem	prmo5 15054	The primorial of 5. (Contributed by AV, 28-Aug-2020.)
|- (#p ` 5 ) = ; 3 0
Theorem	prmo6 15055	The primorial of 6. (Contributed by AV, 28-Aug-2020.)
|- (#p ` 6 ) = ; 3 0
6.2.20  Very large primes
Theorem	1259lem1 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 )
Theorem	1259lem2 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 )
Theorem	1259lem3 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 )
Theorem	1259lem4 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 )
Theorem	1259lem5 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
Theorem	1259prm 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
Theorem	2503lem1 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 )
Theorem	2503lem2 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 )
Theorem	2503lem3 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
Theorem	2503prm 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
Theorem	4001lem1 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 )
Theorem	4001lem2 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 )
Theorem	4001lem3 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 )
Theorem	4001lem4 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
Theorem	4001prm 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 = (ℂfld ↾s 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.
Syntax	cstr 15071	Extend class notation with the class of structures with components numbered below A.
class Struct
Syntax	cnx 15072	Extend class notation with the structure component index extractor.
class ndx
Syntax	csts 15073	Set components of a structure.
class sSet
Syntax	cslot 15074	Extend class notation with the slot function.
class Slot A
Syntax	cbs 15075	Extend class notation with the class of all base set extractors.
class Base
Syntax	cress 15076	Extend class notation with the extensible structure builder restriction operator.
class ↾s
Definition	df-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 ) ) }
Definition	df-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 )
Definition	df-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 ) )
Definition	df-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
Definition	df-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 } ) )
Definition	df-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 ) ) >. ) ) )
Theorem	brstruct 15083	The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)
|- Rel Struct
Theorem	isstruct2 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 ) ) )
Theorem	isstruct 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 ) ) )
Theorem	structcnvcnv 15086	Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
|- ( F Struct X -> `' `' F = ( F \ { (/) } ) )
Theorem	structfun 15087	Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
|- F Struct X   =>    |- Fun `' `' F
Theorem	structfn 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 ) )
Theorem	slotfn 15089	A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.)
|- E = Slot N   =>    |- E Fn _V
Theorem	strfvnd 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 ) )
Theorem	wunndx 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 )
Theorem	strfvn 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 )
Theorem	strfvss 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
Theorem	wunstr 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 )
Theorem	ndxarg 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
Theorem	ndxid 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 )
Theorem	strndxid 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 ) )
Theorem	reldmsets 15098	The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
|- Rel dom sSet
Theorem	setsvalg 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 } ) )
Theorem	setsval 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 >. } ) )