P. 159 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 15801-15900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cshwsidrepswmod0 15801	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 15802*	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 15803*	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 15804*	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 15805*	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 15806*	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 15807*	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 15808*	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 15809*	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 15810*	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 15811*	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	prmlem0 15812*	Lemma for prmlem1 15814 and prmlem2 15827. (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 15813*	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 15814	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 15815	5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
|- 5 e. Prime
Theorem	6nprm 15816	6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- -. 6 e. Prime
Theorem	7prm 15817	7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
|- 7 e. Prime
Theorem	8nprm 15818	8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- -. 8 e. Prime
Theorem	9nprm 15819	9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- -. 9 e. Prime
Theorem	10nprm 15820	10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
|- -. ; 1 0 e. Prime
Theorem	10nprmOLD 15821	Obsolete version of 10nprm 15820 as of 6-Sep-2021. (Contributed by Mario Carneiro, 18-Feb-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
|- -. 10 e. Prime
Theorem	11prm 15822	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 15823	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 15824	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 15825	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 15826	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 15827	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 15843). 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 15828	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 15829	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 15830	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 15831	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 15832	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 15833	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 15834	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 15835	The primorial of 4. (Contributed by AV, 28-Aug-2020.)
|- (#p ` 4 ) = 6
Theorem	prmo5 15836	The primorial of 5. (Contributed by AV, 28-Aug-2020.)
|- (#p ` 5 ) = ; 3 0
Theorem	prmo6 15837	The primorial of 6. (Contributed by AV, 28-Aug-2020.)
|- (#p ` 6 ) = ; 3 0
6.2.20  Very large primes
Theorem	1259lem1 15838	Lemma for 1259prm 15843. 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.) (Proof shortened by AV, 16-Sep-2021.)
|- N = ;;; 1 2 5 9   =>    |- ( ( 2 ^; 1 7 ) mod N ) = (;; 1 3 6 mod N )
Theorem	1259lem2 15839	Lemma for 1259prm 15843. 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.) (Proof shortened by AV, 15-Sep-2021.)
|- N = ;;; 1 2 5 9   =>    |- ( ( 2 ^; 3 4 ) mod N ) = (;; 8 7 0 mod N )
Theorem	1259lem3 15840	Lemma for 1259prm 15843. 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.) (Proof shortened by AV, 16-Sep-2021.)
|- N = ;;; 1 2 5 9   =>    |- ( ( 2 ^; 7 6 ) mod N ) = ( 5 mod N )
Theorem	1259lem4 15841	Lemma for 1259prm 15843. 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.) (Proof shortened by AV, 16-Sep-2021.)
|- N = ;;; 1 2 5 9   =>    |- ( ( 2 ^ ( N - 1 ) ) mod N ) = ( 1 mod N )
Theorem	1259lem5 15842	Lemma for 1259prm 15843. 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 15843	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 15844	Lemma for 2503prm 15847. 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.) (Proof shortened by AV, 16-Sep-2021.)
|- N = ;;; 2 5 0 3   =>    |- ( ( 2 ^; 1 8 ) mod N ) = (;;; 1 8 3 2 mod N )
Theorem	2503lem2 15845	Lemma for 2503prm 15847. 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.) (Proof shortened by AV, 16-Sep-2021.)
|- N = ;;; 2 5 0 3   =>    |- ( ( 2 ^ ( N - 1 ) ) mod N ) = ( 1 mod N )
Theorem	2503lem3 15846	Lemma for 2503prm 15847. 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.) (Proof shortened by AV, 15-Sep-2021.)
|- N = ;;; 2 5 0 3   =>    |- ( ( ( 2 ^; 1 8 ) - 1 ) gcd N ) = 1
Theorem	2503prm 15847	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 15848	Lemma for 4001prm 15852. 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.) (Proof shortened by AV, 16-Sep-2021.)
|- N = ;;; 4 0 0 1   =>    |- ( ( 2 ^;; 2 0 0 ) mod N ) = (;; 9 0 2 mod N )
Theorem	4001lem2 15849	Lemma for 4001prm 15852. 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.) (Proof shortened by AV, 16-Sep-2021.)
|- N = ;;; 4 0 0 1   =>    |- ( ( 2 ^;; 8 0 0 ) mod N ) = (;;; 2 3 1 1 mod N )
Theorem	4001lem3 15850	Lemma for 4001prm 15852. 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.) (Proof shortened by AV, 16-Sep-2021.)
|- N = ;;; 4 0 0 1   =>    |- ( ( 2 ^ ( N - 1 ) ) mod N ) = ( 1 mod N )
Theorem	4001lem4 15851	Lemma for 4001prm 15852. 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.) (Proof shortened by AV, 16-Sep-2021.)
|- N = ;;; 4 0 0 1   =>    |- ( ( ( 2 ^;; 8 0 0 ) - 1 ) gcd N ) = 1
Theorem	4001prm 15852	4001 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
|- N = ;;; 4 0 0 1   =>    |- N e. Prime
PART 7  BASIC STRUCTURES
7.1  Extensible structures
7.1.1  Basic definitions An "extensible structure" (or "structure" in short, at least in this section) 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 (a 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 15883 and strfv 15907. Using extractors makes it easier to change numeric indices and also makes the components' purpose clearer. For example, as noted in ndxid 15883, 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 >. , <.; 1 0 , L >. }. There are many other possible ways to handle structures. We chose this extensible structure approach because this approach (1) results in simpler notation than other approaches we are 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 15865. 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 19819 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 16108 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 16108 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 16246 and df-acs 16249. SubGroup is a Moore collection, as is SubRing, SubRng and many other substructure collections. But it is not useful 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", morphisms 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 15853	Extend class notation with the class of structures with components numbered below A.
class Struct
Syntax	cnx 15854	Extend class notation with the structure component index extractor.
class ndx
Syntax	csts 15855	Set components of a structure.
class sSet
Syntax	cslot 15856	Extend class notation with the slot function.
class Slot A
Syntax	cbs 15857	Extend class notation with the class of all base set extractors.
class Base
Syntax	cress 15858	Extend class notation with the extensible structure builder restriction operator.
class ↾s
Definition	df-struct 15859*	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. As mentioned in the section header, an "extensible structure should be implemented as a function (a set of ordered pairs)". The current definition, however, is less restrictive: it allows for classes which contain the empty set (/) to be extensible structures. Because of 0nelfun 5906, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 15869: F Struct X -> Fun ( F \ { (/) } ). Allowing an extensible structure to contain the empty set ensures that expressions like { <. A , B >. , <. C , D >. } are structures without asserting or implying that A, B, C and D are sets (if A or B is a proper class, then <. A , B >. = (/), see opprc 4424). This is used critically in strle1 15973, strle2 15974, strle3 15975 and strleun 15972 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 16024 and theorems like it will have many sethood assumptions, and may not even be usable in the empty context. Instead, the sethood assumption is deferred until it is actually needed, e.g. ipsbase 16025, which requires that the base set is a set but not any of the other components. Usually, a concrete structure like ℂfld does not contain the empty set, and therefore is a function, see cnfldfun 19758. (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 15860	Define the structure component index extractor. See theorem ndxarg 15882 to understand its purpose. The restriction to NN ensures that ndx is a set. The restriction to some set is necessary 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 15863) another set such as the set of finite ordinals om (df-om 7066). (Contributed by NM, 4-Sep-2011.)
|- ndx = ( _I |` NN )
Definition	df-slot 15861*	Define the slot extractor for extensible structures. The class Slot A is a function whose argument can be any set, although it is meaningful only if that set is a member of an extensible structure (such as a partially ordered set (df-poset 16946) or a group (df-grp 17425)). Note that Slot A is implemented as "evaluation at A". That is, (Slot A ` S ) is defined to be ( S ` A ), where A will typically be a small nonzero natural number. Each extensible structure S is a function defined on specific natural number "slots", and this function extracts the value at a particular slot. The special "structure" ndx, defined as the identity function restricted to NN, can be used to extract the number A from a slot, since (Slot A ` ndx ) = A (see ndxarg 15882). This is typically used to refer to the number of a slot when defining structures without having to expose the detail of what that number is (for instance, we use the expression ( Base ` ndx ) in theorems and proofs instead of its value 1). The class Slot cannot be defined as ( x e. V |-> ( f e. _V |-> ( f ` x ) ) ) because each Slot A is a function on the proper class _V so is itself a proper class, and the values of functions are sets (fvex 6201). It is necessary to allow proper classes as values of Slot A since for instance the class of all (base sets of) groups is proper. (Contributed by Mario Carneiro, 22-Sep-2015.)
|- Slot A = ( x e. _V |-> ( x ` A ) )
Theorem	sloteq 15862	Equality theorem for the Slot construction. (Contributed by BJ, 27-Dec-2021.)
|- ( A = B -> Slot A = Slot B )
Definition	df-base 15863	Define the base set (also called underlying set or carrier set) extractor for extensible structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- Base = Slot 1
Definition	df-sets 15864*	Set a component of an extensible structure. This function is useful for taking an existing structure and "overriding" one of its components. For example, df-ress 15865 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 18490, 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 15865*	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 15930 for the altered base set, and resslem 15933 (subrg0 18787, ressplusg 15993, subrg1 18790, ressmulr 16006) 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 15866	The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)
|- Rel Struct
Theorem	isstruct2 15867	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	structex 15868	A structure is a set. (Contributed by AV, 10-Nov-2021.)
|- ( G Struct X -> G e. _V )
Theorem	structn0fun 15869	A structure witout the empty set is a function. (Contributed by AV, 13-Nov-2021.)
|- ( F Struct X -> Fun ( F \ { (/) } ) )
Theorem	isstruct 15870	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 15871	Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
|- ( F Struct X -> `' `' F = ( F \ { (/) } ) )
Theorem	structfung 15872	The converse of the converse of a structure is a function. Closed form of structfun 15873. (Contributed by AV, 12-Nov-2021.)
|- ( F Struct X -> Fun `' `' F )
Theorem	structfun 15873	Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Proof shortened by AV, 12-Nov-2021.)
|- F Struct X   =>    |- Fun `' `' F
Theorem	structfn 15874	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 15875	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 15876	Deduction version of strfvn 15879. (Contributed by Mario Carneiro, 15-Nov-2014.)
|- E = Slot N   &    |- ( ph -> S e. V )   =>    |- ( ph -> ( E ` S ) = ( S ` N ) )
Theorem	basfn 15877	The base set extractor is a function on _V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
|- Base Fn _V
Theorem	wunndx 15878	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 15879	Value of a structure component extractor E. Normally, E is a defined constant symbol such as Base (df-base 15863) 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 16946) 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 15907. (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 15880	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 15881	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 15882	Get the numeric argument from a defined structure component extractor such as df-base 15863. (Contributed by Mario Carneiro, 6-Oct-2013.)
|- E = Slot N   &    |- N e. NN   =>    |- ( E ` ndx ) = N
Theorem	ndxid 15883	A structure component extractor is defined by its own index. This theorem, together with strfv 15907 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 15863 and the ; 1 0 in df-ple 15961, 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 >. , <.; 1 0 , L >. }. The latter, while shorter to state, requires revision if we later change ; 1 0 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.) (Proof shortened by BJ, 27-Dec-2021.)
|- E = Slot N   &    |- N e. NN   =>    |- E = Slot ( E ` ndx )
Theorem	ndxidOLD 15884	Obsolete proof of ndxid 15883 as of 28-Dec-2021. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
|- E = Slot N   &    |- N e. NN   =>    |- E = Slot ( E ` ndx )
Theorem	strndxid 15885	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 15886	The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
|- Rel dom sSet
Theorem	setsvalg 15887	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 15888	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 >. } ) )
Theorem	setsidvald 15889	Value of the structure replacement function, deduction version. (Contributed by AV, 14-Mar-2020.)
|- E = Slot N   &    |- N e. NN   &    |- ( ph -> S e. V )   &    |- ( ph -> Fun S )   &    |- ( ph -> ( E ` ndx ) e. dom S )   =>    |- ( ph -> S = ( S sSet <. ( E ` ndx ) , ( E ` S ) >. ) )
Theorem	fvsetsid 15890	The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.)
|- ( ( F e. V /\ X e. W /\ Y e. U ) -> ( ( F sSet <. X , Y >. ) ` X ) = Y )
Theorem	fsets 15891	The structure replacement function is a function. (Contributed by SO, 12-Jul-2018.)
|- ( ( ( F e. V /\ F : A --> B ) /\ X e. A /\ Y e. B ) -> ( F sSet <. X , Y >. ) : A --> B )
Theorem	setsdm 15892	The domain of a structure with replacement is the domain of the original structure extended by the index of the replacement. (Contributed by AV, 7-Jun-2021.)
|- ( ( G e. V /\ E e. W ) -> dom ( G sSet <. I , E >. ) = ( dom G u. { I } ) )
Theorem	setsfun 15893	A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.)
|- ( ( ( G e. V /\ Fun G ) /\ ( I e. U /\ E e. W ) ) -> Fun ( G sSet <. I , E >. ) )
Theorem	setsfun0 15894	A structure with replacement without the empty set is a function if the original structure without the empty set is a function. This variant of setsfun 15893 is useful for proofs based on isstruct2 15867 which requires Fun ( F \ { (/) } ) for F to be an extensible structure. (Contributed by AV, 7-Jun-2021.)
|- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ ( I e. U /\ E e. W ) ) -> Fun ( ( G sSet <. I , E >. ) \ { (/) } ) )
Theorem	setsn0fun 15895	The value of the structure replacement function (without the empty set) is a function if the structure (without the empty set)is a function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
|- ( ph -> S Struct X )   &    |- ( ph -> I e. U )   &    |- ( ph -> E e. W )   =>    |- ( ph -> Fun ( ( S sSet <. I , E >. ) \ { (/) } ) )
Theorem	setsstruct2 15896	An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)
|- ( ( ( G Struct X /\ E e. V /\ I e. NN ) /\ Y = <. if ( I <_ ( 1st ` X ) , I , ( 1st ` X ) ) , if ( I <_ ( 2nd ` X ) , ( 2nd ` X ) , I ) >. ) -> ( G sSet <. I , E >. ) Struct Y )
Theorem	setsexstruct2 15897*	An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)
|- ( ( G Struct X /\ E e. V /\ I e. NN ) -> E. y ( G sSet <. I , E >. ) Struct y )
Theorem	setsstruct 15898	An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 9-Jun-2021.) (Revised by AV, 14-Nov-2021.)
|- ( ( E e. V /\ I e. ( ZZ>= ` M ) /\ G Struct <. M , N >. ) -> ( G sSet <. I , E >. ) Struct <. M , if ( I <_ N , N , I ) >. )
Theorem	setsstructOLD 15899	Obsolete version of setsstruct 15898 as of 14-Nov-2021. (Contributed by AV, 9-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ( G e. U /\ E e. V /\ I e. ( ZZ>= ` M ) ) -> ( G Struct <. M , N >. -> ( G sSet <. I , E >. ) Struct <. M , if ( I <_ N , N , I ) >. ) )
Theorem	wunsets 15900	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 )