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Theorem List for Metamath Proof Explorer - 16901-17000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
6.2.18  Cyclical shifts of words (cont.)
 
Theoremcshwsidrepsw 16901 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.)
((๐‘Š โˆˆ Word ๐‘‰ โˆง (โ™ฏโ€˜๐‘Š) โˆˆ โ„™) โ†’ ((๐ฟ โˆˆ โ„ค โˆง (๐ฟ mod (โ™ฏโ€˜๐‘Š)) โ‰  0 โˆง (๐‘Š cyclShift ๐ฟ) = ๐‘Š) โ†’ ๐‘Š = ((๐‘Šโ€˜0) repeatS (โ™ฏโ€˜๐‘Š))))
 
Theoremcshwsidrepswmod0 16902 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.)
((๐‘Š โˆˆ Word ๐‘‰ โˆง (โ™ฏโ€˜๐‘Š) โˆˆ โ„™ โˆง ๐ฟ โˆˆ โ„ค) โ†’ ((๐‘Š cyclShift ๐ฟ) = ๐‘Š โ†’ ((๐ฟ mod (โ™ฏโ€˜๐‘Š)) = 0 โˆจ ๐‘Š = ((๐‘Šโ€˜0) repeatS (โ™ฏโ€˜๐‘Š)))))
 
Theoremcshwshashlem1 16903* 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.)
(๐œ‘ โ†’ (๐‘Š โˆˆ Word ๐‘‰ โˆง (โ™ฏโ€˜๐‘Š) โˆˆ โ„™))    โ‡’   ((๐œ‘ โˆง โˆƒ๐‘– โˆˆ (0..^(โ™ฏโ€˜๐‘Š))(๐‘Šโ€˜๐‘–) โ‰  (๐‘Šโ€˜0) โˆง ๐ฟ โˆˆ (1..^(โ™ฏโ€˜๐‘Š))) โ†’ (๐‘Š cyclShift ๐ฟ) โ‰  ๐‘Š)
 
Theoremcshwshashlem2 16904* 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.)
(๐œ‘ โ†’ (๐‘Š โˆˆ Word ๐‘‰ โˆง (โ™ฏโ€˜๐‘Š) โˆˆ โ„™))    โ‡’   ((๐œ‘ โˆง โˆƒ๐‘– โˆˆ (0..^(โ™ฏโ€˜๐‘Š))(๐‘Šโ€˜๐‘–) โ‰  (๐‘Šโ€˜0)) โ†’ ((๐ฟ โˆˆ (0..^(โ™ฏโ€˜๐‘Š)) โˆง ๐พ โˆˆ (0..^(โ™ฏโ€˜๐‘Š)) โˆง ๐พ < ๐ฟ) โ†’ (๐‘Š cyclShift ๐ฟ) โ‰  (๐‘Š cyclShift ๐พ)))
 
Theoremcshwshashlem3 16905* 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.)
(๐œ‘ โ†’ (๐‘Š โˆˆ Word ๐‘‰ โˆง (โ™ฏโ€˜๐‘Š) โˆˆ โ„™))    โ‡’   ((๐œ‘ โˆง โˆƒ๐‘– โˆˆ (0..^(โ™ฏโ€˜๐‘Š))(๐‘Šโ€˜๐‘–) โ‰  (๐‘Šโ€˜0)) โ†’ ((๐ฟ โˆˆ (0..^(โ™ฏโ€˜๐‘Š)) โˆง ๐พ โˆˆ (0..^(โ™ฏโ€˜๐‘Š)) โˆง ๐พ โ‰  ๐ฟ) โ†’ (๐‘Š cyclShift ๐ฟ) โ‰  (๐‘Š cyclShift ๐พ)))
 
Theoremcshwsdisj 16906* 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.)
(๐œ‘ โ†’ (๐‘Š โˆˆ Word ๐‘‰ โˆง (โ™ฏโ€˜๐‘Š) โˆˆ โ„™))    โ‡’   ((๐œ‘ โˆง โˆƒ๐‘– โˆˆ (0..^(โ™ฏโ€˜๐‘Š))(๐‘Šโ€˜๐‘–) โ‰  (๐‘Šโ€˜0)) โ†’ Disj ๐‘› โˆˆ (0..^(โ™ฏโ€˜๐‘Š)){(๐‘Š cyclShift ๐‘›)})
 
Theoremcshwsiun 16907* 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.)
๐‘€ = {๐‘ค โˆˆ Word ๐‘‰ โˆฃ โˆƒ๐‘› โˆˆ (0..^(โ™ฏโ€˜๐‘Š))(๐‘Š cyclShift ๐‘›) = ๐‘ค}    โ‡’   (๐‘Š โˆˆ Word ๐‘‰ โ†’ ๐‘€ = โˆช ๐‘› โˆˆ (0..^(โ™ฏโ€˜๐‘Š)){(๐‘Š cyclShift ๐‘›)})
 
Theoremcshwsex 16908* 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.)
๐‘€ = {๐‘ค โˆˆ Word ๐‘‰ โˆฃ โˆƒ๐‘› โˆˆ (0..^(โ™ฏโ€˜๐‘Š))(๐‘Š cyclShift ๐‘›) = ๐‘ค}    โ‡’   (๐‘Š โˆˆ Word ๐‘‰ โ†’ ๐‘€ โˆˆ V)
 
Theoremcshws0 16909* The size of the set of (different!) words resulting by cyclically shifting an empty word is 0. (Contributed by AV, 8-Nov-2018.)
๐‘€ = {๐‘ค โˆˆ Word ๐‘‰ โˆฃ โˆƒ๐‘› โˆˆ (0..^(โ™ฏโ€˜๐‘Š))(๐‘Š cyclShift ๐‘›) = ๐‘ค}    โ‡’   (๐‘Š = โˆ… โ†’ (โ™ฏโ€˜๐‘€) = 0)
 
Theoremcshwrepswhash1 16910* 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.)
๐‘€ = {๐‘ค โˆˆ Word ๐‘‰ โˆฃ โˆƒ๐‘› โˆˆ (0..^(โ™ฏโ€˜๐‘Š))(๐‘Š cyclShift ๐‘›) = ๐‘ค}    โ‡’   ((๐ด โˆˆ ๐‘‰ โˆง ๐‘ โˆˆ โ„• โˆง ๐‘Š = (๐ด repeatS ๐‘)) โ†’ (โ™ฏโ€˜๐‘€) = 1)
 
Theoremcshwshashnsame 16911* 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.)
๐‘€ = {๐‘ค โˆˆ Word ๐‘‰ โˆฃ โˆƒ๐‘› โˆˆ (0..^(โ™ฏโ€˜๐‘Š))(๐‘Š cyclShift ๐‘›) = ๐‘ค}    โ‡’   ((๐‘Š โˆˆ Word ๐‘‰ โˆง (โ™ฏโ€˜๐‘Š) โˆˆ โ„™) โ†’ (โˆƒ๐‘– โˆˆ (0..^(โ™ฏโ€˜๐‘Š))(๐‘Šโ€˜๐‘–) โ‰  (๐‘Šโ€˜0) โ†’ (โ™ฏโ€˜๐‘€) = (โ™ฏโ€˜๐‘Š)))
 
Theoremcshwshash 16912* 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.)
๐‘€ = {๐‘ค โˆˆ Word ๐‘‰ โˆฃ โˆƒ๐‘› โˆˆ (0..^(โ™ฏโ€˜๐‘Š))(๐‘Š cyclShift ๐‘›) = ๐‘ค}    โ‡’   ((๐‘Š โˆˆ Word ๐‘‰ โˆง (โ™ฏโ€˜๐‘Š) โˆˆ โ„™) โ†’ ((โ™ฏโ€˜๐‘€) = (โ™ฏโ€˜๐‘Š) โˆจ (โ™ฏโ€˜๐‘€) = 1))
 
6.2.19  Specific prime numbers
 
Theoremprmlem0 16913* Lemma for prmlem1 16915 and prmlem2 16927. (Contributed by Mario Carneiro, 18-Feb-2014.)
((ยฌ 2 โˆฅ ๐‘€ โˆง ๐‘ฅ โˆˆ (โ„คโ‰ฅโ€˜๐‘€)) โ†’ ((๐‘ฅ โˆˆ (โ„™ โˆ– {2}) โˆง (๐‘ฅโ†‘2) โ‰ค ๐‘) โ†’ ยฌ ๐‘ฅ โˆฅ ๐‘))    &   (๐พ โˆˆ โ„™ โ†’ ยฌ ๐พ โˆฅ ๐‘)    &   (๐พ + 2) = ๐‘€    โ‡’   ((ยฌ 2 โˆฅ ๐พ โˆง ๐‘ฅ โˆˆ (โ„คโ‰ฅโ€˜๐พ)) โ†’ ((๐‘ฅ โˆˆ (โ„™ โˆ– {2}) โˆง (๐‘ฅโ†‘2) โ‰ค ๐‘) โ†’ ยฌ ๐‘ฅ โˆฅ ๐‘))
 
Theoremprmlem1a 16914* A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.)
๐‘ โˆˆ โ„•    &   1 < ๐‘    &    ยฌ 2 โˆฅ ๐‘    &    ยฌ 3 โˆฅ ๐‘    &   ((ยฌ 2 โˆฅ 5 โˆง ๐‘ฅ โˆˆ (โ„คโ‰ฅโ€˜5)) โ†’ ((๐‘ฅ โˆˆ (โ„™ โˆ– {2}) โˆง (๐‘ฅโ†‘2) โ‰ค ๐‘) โ†’ ยฌ ๐‘ฅ โˆฅ ๐‘))    โ‡’   ๐‘ โˆˆ โ„™
 
Theoremprmlem1 16915 A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.)
๐‘ โˆˆ โ„•    &   1 < ๐‘    &    ยฌ 2 โˆฅ ๐‘    &    ยฌ 3 โˆฅ ๐‘    &   ๐‘ < 25    โ‡’   ๐‘ โˆˆ โ„™
 
Theorem5prm 16916 5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
5 โˆˆ โ„™
 
Theorem6nprm 16917 6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
ยฌ 6 โˆˆ โ„™
 
Theorem7prm 16918 7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
7 โˆˆ โ„™
 
Theorem8nprm 16919 8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
ยฌ 8 โˆˆ โ„™
 
Theorem9nprm 16920 9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
ยฌ 9 โˆˆ โ„™
 
Theorem10nprm 16921 10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
ยฌ 10 โˆˆ โ„™
 
Theorem11prm 16922 11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
11 โˆˆ โ„™
 
Theorem13prm 16923 13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
13 โˆˆ โ„™
 
Theorem17prm 16924 17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
17 โˆˆ โ„™
 
Theorem19prm 16925 19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
19 โˆˆ โ„™
 
Theorem23prm 16926 23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
23 โˆˆ โ„™
 
Theoremprmlem2 16927 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 to cover the numbers less than 5โ†‘2 = 25. 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 29โ†‘2 = 841, 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 16943).

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.)

๐‘ โˆˆ โ„•    &   ๐‘ < 841    &   1 < ๐‘    &    ยฌ 2 โˆฅ ๐‘    &    ยฌ 3 โˆฅ ๐‘    &    ยฌ 5 โˆฅ ๐‘    &    ยฌ 7 โˆฅ ๐‘    &    ยฌ 11 โˆฅ ๐‘    &    ยฌ 13 โˆฅ ๐‘    &    ยฌ 17 โˆฅ ๐‘    &    ยฌ 19 โˆฅ ๐‘    &    ยฌ 23 โˆฅ ๐‘    โ‡’   ๐‘ โˆˆ โ„™
 
Theorem37prm 16928 37 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
37 โˆˆ โ„™
 
Theorem43prm 16929 43 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
43 โˆˆ โ„™
 
Theorem83prm 16930 83 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
83 โˆˆ โ„™
 
Theorem139prm 16931 139 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
139 โˆˆ โ„™
 
Theorem163prm 16932 163 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
163 โˆˆ โ„™
 
Theorem317prm 16933 317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
317 โˆˆ โ„™
 
Theorem631prm 16934 631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
631 โˆˆ โ„™
 
Theoremprmo4 16935 The primorial of 4. (Contributed by AV, 28-Aug-2020.)
(#pโ€˜4) = 6
 
Theoremprmo5 16936 The primorial of 5. (Contributed by AV, 28-Aug-2020.)
(#pโ€˜5) = 30
 
Theoremprmo6 16937 The primorial of 6. (Contributed by AV, 28-Aug-2020.)
(#pโ€˜6) = 30
 
6.2.20  Very large primes
 
Theorem1259lem1 16938 Lemma for 1259prm 16943. Calculate a power mod. In decimal, we calculate 2โ†‘16 = 52๐‘ + 68โ‰ก68 and 2โ†‘17โ‰ก68 ยท 2 = 136 in this lemma. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
๐‘ = 1259    โ‡’   ((2โ†‘17) mod ๐‘) = (136 mod ๐‘)
 
Theorem1259lem2 16939 Lemma for 1259prm 16943. Calculate a power mod. In decimal, we calculate 2โ†‘34 = (2โ†‘17)โ†‘2โ‰ก136โ†‘2โ‰ก14๐‘ + 870. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.)
๐‘ = 1259    โ‡’   ((2โ†‘34) mod ๐‘) = (870 mod ๐‘)
 
Theorem1259lem3 16940 Lemma for 1259prm 16943. Calculate a power mod. In decimal, we calculate 2โ†‘38 = 2โ†‘34 ยท 2โ†‘4โ‰ก870 ยท 16 = 11๐‘ + 71 and 2โ†‘76 = (2โ†‘34)โ†‘2โ‰ก71โ†‘2 = 4๐‘ + 5โ‰ก5. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
๐‘ = 1259    โ‡’   ((2โ†‘76) mod ๐‘) = (5 mod ๐‘)
 
Theorem1259lem4 16941 Lemma for 1259prm 16943. Calculate a power mod. In decimal, we calculate 2โ†‘306 = (2โ†‘76)โ†‘4 ยท 4โ‰ก5โ†‘4 ยท 4 = 2๐‘ โˆ’ 18, 2โ†‘612 = (2โ†‘306)โ†‘2โ‰ก18โ†‘2 = 324, 2โ†‘629 = 2โ†‘612 ยท 2โ†‘17โ‰ก324 ยท 136 = 35๐‘ โˆ’ 1 and finally 2โ†‘(๐‘ โˆ’ 1) = (2โ†‘629)โ†‘2โ‰ก1โ†‘2 = 1. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
๐‘ = 1259    โ‡’   ((2โ†‘(๐‘ โˆ’ 1)) mod ๐‘) = (1 mod ๐‘)
 
Theorem1259lem5 16942 Lemma for 1259prm 16943. Calculate the GCD of 2โ†‘34 โˆ’ 1โ‰ก869 with ๐‘ = 1259. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
๐‘ = 1259    โ‡’   (((2โ†‘34) โˆ’ 1) gcd ๐‘) = 1
 
Theorem1259prm 16943 1259 is a prime number. (Contributed by Mario Carneiro, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
๐‘ = 1259    โ‡’   ๐‘ โˆˆ โ„™
 
Theorem2503lem1 16944 Lemma for 2503prm 16947. Calculate a power mod. In decimal, we calculate 2โ†‘18 = 512โ†‘2 = 104๐‘ + 1832โ‰ก1832. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
๐‘ = 2503    โ‡’   ((2โ†‘18) mod ๐‘) = (1832 mod ๐‘)
 
Theorem2503lem2 16945 Lemma for 2503prm 16947. Calculate a power mod. We calculate 2โ†‘19 = 2โ†‘18 ยท 2โ‰ก1832 ยท 2 = ๐‘ + 1161, 2โ†‘38 = (2โ†‘19)โ†‘2โ‰ก1161โ†‘2 = 538๐‘ + 1307, 2โ†‘39 = 2โ†‘38 ยท 2โ‰ก1307 ยท 2 = ๐‘ + 111, 2โ†‘78 = (2โ†‘39)โ†‘2โ‰ก111โ†‘2 = 5๐‘ โˆ’ 194, 2โ†‘156 = (2โ†‘78)โ†‘2โ‰ก194โ†‘2 = 15๐‘ + 91, 2โ†‘312 = (2โ†‘156)โ†‘2โ‰ก91โ†‘2 = 3๐‘ + 772, 2โ†‘624 = (2โ†‘312)โ†‘2โ‰ก772โ†‘2 = 238๐‘ + 270, 2โ†‘1248 = (2โ†‘624)โ†‘2โ‰ก270โ†‘2 = 29๐‘ + 313, 2โ†‘1251 = 2โ†‘1248 ยท 8โ‰ก313 ยท 8 = ๐‘ + 1 and finally 2โ†‘(๐‘ โˆ’ 1) = (2โ†‘1251)โ†‘2โ‰ก1โ†‘2 = 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
๐‘ = 2503    โ‡’   ((2โ†‘(๐‘ โˆ’ 1)) mod ๐‘) = (1 mod ๐‘)
 
Theorem2503lem3 16946 Lemma for 2503prm 16947. Calculate the GCD of 2โ†‘18 โˆ’ 1โ‰ก1831 with ๐‘ = 2503. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.)
๐‘ = 2503    โ‡’   (((2โ†‘18) โˆ’ 1) gcd ๐‘) = 1
 
Theorem2503prm 16947 2503 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
๐‘ = 2503    โ‡’   ๐‘ โˆˆ โ„™
 
Theorem4001lem1 16948 Lemma for 4001prm 16952. Calculate a power mod. In decimal, we calculate 2โ†‘12 = 4096 = ๐‘ + 95, 2โ†‘24 = (2โ†‘12)โ†‘2โ‰ก95โ†‘2 = 2๐‘ + 1023, 2โ†‘25 = 2โ†‘24 ยท 2โ‰ก1023 ยท 2 = 2046, 2โ†‘50 = (2โ†‘25)โ†‘2โ‰ก2046โ†‘2 = 1046๐‘ + 1070, 2โ†‘100 = (2โ†‘50)โ†‘2โ‰ก1070โ†‘2 = 286๐‘ + 614 and 2โ†‘200 = (2โ†‘100)โ†‘2โ‰ก614โ†‘2 = 94๐‘ + 902 โ‰ก902. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
๐‘ = 4001    โ‡’   ((2โ†‘200) mod ๐‘) = (902 mod ๐‘)
 
Theorem4001lem2 16949 Lemma for 4001prm 16952. Calculate a power mod. In decimal, we calculate 2โ†‘400 = (2โ†‘200)โ†‘2โ‰ก902โ†‘2 = 203๐‘ + 1401 and 2โ†‘800 = (2โ†‘400)โ†‘2โ‰ก1401โ†‘2 = 490๐‘ + 2311 โ‰ก2311. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
๐‘ = 4001    โ‡’   ((2โ†‘800) mod ๐‘) = (2311 mod ๐‘)
 
Theorem4001lem3 16950 Lemma for 4001prm 16952. Calculate a power mod. In decimal, we calculate 2โ†‘1000 = 2โ†‘800 ยท 2โ†‘200โ‰ก2311 ยท 902 = 521๐‘ + 1 and finally 2โ†‘(๐‘ โˆ’ 1) = (2โ†‘1000)โ†‘4โ‰ก1โ†‘4 = 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
๐‘ = 4001    โ‡’   ((2โ†‘(๐‘ โˆ’ 1)) mod ๐‘) = (1 mod ๐‘)
 
Theorem4001lem4 16951 Lemma for 4001prm 16952. Calculate the GCD of 2โ†‘800 โˆ’ 1โ‰ก2310 with ๐‘ = 4001. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
๐‘ = 4001    โ‡’   (((2โ†‘800) โˆ’ 1) gcd ๐‘) = 1
 
Theorem4001prm 16952 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.)
๐‘ = 4001    โ‡’   ๐‘ โˆˆ โ„™
 
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 โ„•. 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 17004 and strfv 17011. Using extractors makes it easier to change numeric indices and also makes the components' purpose clearer. For example, as noted in ndxid 17004, we can refer to a specific poset with base set ๐ต and order relation ๐ฟ using the extensible structure {โŸจ(Baseโ€˜ndx), ๐ตโŸฉ, โŸจ(leโ€˜ndx), ๐ฟโŸฉ} rather than {โŸจ1, ๐ตโŸฉ, โŸจ10, ๐ฟโŸฉ}. See section header comment mmtheorems.html#cnx 17004 for more details on numeric indices versus the structure component extractors.

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 ๐‘‹ is a ๐‘Œ 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 โ†พs as defined in df-ress 17048. 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 โ„คring is defined in df-zring 20793 as simply โ„คring = (โ„‚fld โ†พs โ„ค). This can be similarly done for all other subsets of โ„‚, 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 โ„‚ to inherit, then we change the definition of โ„‚fld, 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 17264 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 17264 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 17401 and df-acs 17404. 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, โ†พs 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.

 
7.1.1.1  Extensible structures as structures with components
 
Syntaxcstr 16953 Extend class notation with the class of structures with components numbered below ๐ด.
class Struct
 
Definitiondf-struct 16954* Define a structure with components in ๐‘€...๐‘. 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 โˆ… to be extensible structures. Because of 0nelfun 6515, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 16958: ๐น Struct ๐‘‹ โ†’ Fun (๐น โˆ– {โˆ…}).

Allowing an extensible structure to contain the empty set ensures that expressions like {โŸจ๐ด, ๐ตโŸฉ, โŸจ๐ถ, ๐ทโŸฉ} are structures without asserting or implying that ๐ด, ๐ต, ๐ถ and ๐ท are sets (if ๐ด or ๐ต is a proper class, then โŸจ๐ด, ๐ตโŸฉ = โˆ…, see opprc 4852). This is used critically in strle1 16965, strle2 16966, strle3 16967 and strleun 16964 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17152 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 17153, which requires that the base set be a set but not any of the other components. Usually, a concrete structure like โ„‚fld does not contain the empty set, and therefore is a function, see cnfldfun 20731. (Contributed by Mario Carneiro, 29-Aug-2015.)

Struct = {โŸจ๐‘“, ๐‘ฅโŸฉ โˆฃ (๐‘ฅ โˆˆ ( โ‰ค โˆฉ (โ„• ร— โ„•)) โˆง Fun (๐‘“ โˆ– {โˆ…}) โˆง dom ๐‘“ โŠ† (...โ€˜๐‘ฅ))}
 
Theorembrstruct 16955 The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)
Rel Struct
 
Theoremisstruct2 16956 The property of being a structure with components in (1st โ€˜๐‘‹)...(2nd โ€˜๐‘‹). (Contributed by Mario Carneiro, 29-Aug-2015.)
(๐น Struct ๐‘‹ โ†” (๐‘‹ โˆˆ ( โ‰ค โˆฉ (โ„• ร— โ„•)) โˆง Fun (๐น โˆ– {โˆ…}) โˆง dom ๐น โŠ† (...โ€˜๐‘‹)))
 
Theoremstructex 16957 A structure is a set. (Contributed by AV, 10-Nov-2021.)
(๐บ Struct ๐‘‹ โ†’ ๐บ โˆˆ V)
 
Theoremstructn0fun 16958 A structure without the empty set is a function. (Contributed by AV, 13-Nov-2021.)
(๐น Struct ๐‘‹ โ†’ Fun (๐น โˆ– {โˆ…}))
 
Theoremisstruct 16959 The property of being a structure with components in ๐‘€...๐‘. (Contributed by Mario Carneiro, 29-Aug-2015.)
(๐น Struct โŸจ๐‘€, ๐‘โŸฉ โ†” ((๐‘€ โˆˆ โ„• โˆง ๐‘ โˆˆ โ„• โˆง ๐‘€ โ‰ค ๐‘) โˆง Fun (๐น โˆ– {โˆ…}) โˆง dom ๐น โŠ† (๐‘€...๐‘)))
 
Theoremstructcnvcnv 16960 Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
(๐น Struct ๐‘‹ โ†’ โ—กโ—ก๐น = (๐น โˆ– {โˆ…}))
 
Theoremstructfung 16961 The converse of the converse of a structure is a function. Closed form of structfun 16962. (Contributed by AV, 12-Nov-2021.)
(๐น Struct ๐‘‹ โ†’ Fun โ—กโ—ก๐น)
 
Theoremstructfun 16962 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Proof shortened by AV, 12-Nov-2021.)
๐น Struct ๐‘‹    โ‡’   Fun โ—กโ—ก๐น
 
Theoremstructfn 16963 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
๐น Struct โŸจ๐‘€, ๐‘โŸฉ    โ‡’   (Fun โ—กโ—ก๐น โˆง dom ๐น โŠ† (1...๐‘))
 
Theoremstrleun 16964 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)
๐น Struct โŸจ๐ด, ๐ตโŸฉ    &   ๐บ Struct โŸจ๐ถ, ๐ทโŸฉ    &   ๐ต < ๐ถ    โ‡’   (๐น โˆช ๐บ) Struct โŸจ๐ด, ๐ทโŸฉ
 
Theoremstrle1 16965 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.)
๐ผ โˆˆ โ„•    &   ๐ด = ๐ผ    โ‡’   {โŸจ๐ด, ๐‘‹โŸฉ} Struct โŸจ๐ผ, ๐ผโŸฉ
 
Theoremstrle2 16966 Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.)
๐ผ โˆˆ โ„•    &   ๐ด = ๐ผ    &   ๐ผ < ๐ฝ    &   ๐ฝ โˆˆ โ„•    &   ๐ต = ๐ฝ    โ‡’   {โŸจ๐ด, ๐‘‹โŸฉ, โŸจ๐ต, ๐‘ŒโŸฉ} Struct โŸจ๐ผ, ๐ฝโŸฉ
 
Theoremstrle3 16967 Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.)
๐ผ โˆˆ โ„•    &   ๐ด = ๐ผ    &   ๐ผ < ๐ฝ    &   ๐ฝ โˆˆ โ„•    &   ๐ต = ๐ฝ    &   ๐ฝ < ๐พ    &   ๐พ โˆˆ โ„•    &   ๐ถ = ๐พ    โ‡’   {โŸจ๐ด, ๐‘‹โŸฉ, โŸจ๐ต, ๐‘ŒโŸฉ, โŸจ๐ถ, ๐‘โŸฉ} Struct โŸจ๐ผ, ๐พโŸฉ
 
Theoremsbcie2s 16968* A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.)
๐ด = (๐ธโ€˜๐‘Š)    &   ๐ต = (๐นโ€˜๐‘Š)    &   ((๐‘Ž = ๐ด โˆง ๐‘ = ๐ต) โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   (๐‘ค = ๐‘Š โ†’ ([(๐ธโ€˜๐‘ค) / ๐‘Ž][(๐นโ€˜๐‘ค) / ๐‘]๐œ“ โ†” ๐œ‘))
 
Theoremsbcie3s 16969* A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 15-Mar-2019.)
๐ด = (๐ธโ€˜๐‘Š)    &   ๐ต = (๐นโ€˜๐‘Š)    &   ๐ถ = (๐บโ€˜๐‘Š)    &   ((๐‘Ž = ๐ด โˆง ๐‘ = ๐ต โˆง ๐‘ = ๐ถ) โ†’ (๐œ‘ โ†” ๐œ“))    โ‡’   (๐‘ค = ๐‘Š โ†’ ([(๐ธโ€˜๐‘ค) / ๐‘Ž][(๐นโ€˜๐‘ค) / ๐‘][(๐บโ€˜๐‘ค) / ๐‘]๐œ“ โ†” ๐œ‘))
 
7.1.1.2  Substitution of components
 
Syntaxcsts 16970 Set components of a structure.
class sSet
 
Definitiondf-sets 16971* 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 17048 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 19826, 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 = (๐‘  โˆˆ V, ๐‘’ โˆˆ V โ†ฆ ((๐‘  โ†พ (V โˆ– dom {๐‘’})) โˆช {๐‘’}))
 
Theoremreldmsets 16972 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Rel dom sSet
 
Theoremsetsvalg 16973 Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
((๐‘† โˆˆ ๐‘‰ โˆง ๐ด โˆˆ ๐‘Š) โ†’ (๐‘† sSet ๐ด) = ((๐‘† โ†พ (V โˆ– dom {๐ด})) โˆช {๐ด}))
 
Theoremsetsval 16974 Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
((๐‘† โˆˆ ๐‘‰ โˆง ๐ต โˆˆ ๐‘Š) โ†’ (๐‘† sSet โŸจ๐ด, ๐ตโŸฉ) = ((๐‘† โ†พ (V โˆ– {๐ด})) โˆช {โŸจ๐ด, ๐ตโŸฉ}))
 
Theoremfvsetsid 16975 The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.)
((๐น โˆˆ ๐‘‰ โˆง ๐‘‹ โˆˆ ๐‘Š โˆง ๐‘Œ โˆˆ ๐‘ˆ) โ†’ ((๐น sSet โŸจ๐‘‹, ๐‘ŒโŸฉ)โ€˜๐‘‹) = ๐‘Œ)
 
Theoremfsets 16976 The structure replacement function is a function. (Contributed by SO, 12-Jul-2018.)
(((๐น โˆˆ ๐‘‰ โˆง ๐น:๐ดโŸถ๐ต) โˆง ๐‘‹ โˆˆ ๐ด โˆง ๐‘Œ โˆˆ ๐ต) โ†’ (๐น sSet โŸจ๐‘‹, ๐‘ŒโŸฉ):๐ดโŸถ๐ต)
 
Theoremsetsdm 16977 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.)
((๐บ โˆˆ ๐‘‰ โˆง ๐ธ โˆˆ ๐‘Š) โ†’ dom (๐บ sSet โŸจ๐ผ, ๐ธโŸฉ) = (dom ๐บ โˆช {๐ผ}))
 
Theoremsetsfun 16978 A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.)
(((๐บ โˆˆ ๐‘‰ โˆง Fun ๐บ) โˆง (๐ผ โˆˆ ๐‘ˆ โˆง ๐ธ โˆˆ ๐‘Š)) โ†’ Fun (๐บ sSet โŸจ๐ผ, ๐ธโŸฉ))
 
Theoremsetsfun0 16979 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 16978 is useful for proofs based on isstruct2 16956 which requires Fun (๐น โˆ– {โˆ…}) for ๐น to be an extensible structure. (Contributed by AV, 7-Jun-2021.)
(((๐บ โˆˆ ๐‘‰ โˆง Fun (๐บ โˆ– {โˆ…})) โˆง (๐ผ โˆˆ ๐‘ˆ โˆง ๐ธ โˆˆ ๐‘Š)) โ†’ Fun ((๐บ sSet โŸจ๐ผ, ๐ธโŸฉ) โˆ– {โˆ…}))
 
Theoremsetsn0fun 16980 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.)
(๐œ‘ โ†’ ๐‘† Struct ๐‘‹)    &   (๐œ‘ โ†’ ๐ผ โˆˆ ๐‘ˆ)    &   (๐œ‘ โ†’ ๐ธ โˆˆ ๐‘Š)    โ‡’   (๐œ‘ โ†’ Fun ((๐‘† sSet โŸจ๐ผ, ๐ธโŸฉ) โˆ– {โˆ…}))
 
Theoremsetsstruct2 16981 An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)
(((๐บ Struct ๐‘‹ โˆง ๐ธ โˆˆ ๐‘‰ โˆง ๐ผ โˆˆ โ„•) โˆง ๐‘Œ = โŸจif(๐ผ โ‰ค (1st โ€˜๐‘‹), ๐ผ, (1st โ€˜๐‘‹)), if(๐ผ โ‰ค (2nd โ€˜๐‘‹), (2nd โ€˜๐‘‹), ๐ผ)โŸฉ) โ†’ (๐บ sSet โŸจ๐ผ, ๐ธโŸฉ) Struct ๐‘Œ)
 
Theoremsetsexstruct2 16982* An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)
((๐บ Struct ๐‘‹ โˆง ๐ธ โˆˆ ๐‘‰ โˆง ๐ผ โˆˆ โ„•) โ†’ โˆƒ๐‘ฆ(๐บ sSet โŸจ๐ผ, ๐ธโŸฉ) Struct ๐‘ฆ)
 
Theoremsetsstruct 16983 An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 9-Jun-2021.) (Revised by AV, 14-Nov-2021.)
((๐ธ โˆˆ ๐‘‰ โˆง ๐ผ โˆˆ (โ„คโ‰ฅโ€˜๐‘€) โˆง ๐บ Struct โŸจ๐‘€, ๐‘โŸฉ) โ†’ (๐บ sSet โŸจ๐ผ, ๐ธโŸฉ) Struct โŸจ๐‘€, if(๐ผ โ‰ค ๐‘, ๐‘, ๐ผ)โŸฉ)
 
Theoremwunsets 16984 Closure of structure replacement in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
(๐œ‘ โ†’ ๐‘ˆ โˆˆ WUni)    &   (๐œ‘ โ†’ ๐‘† โˆˆ ๐‘ˆ)    &   (๐œ‘ โ†’ ๐ด โˆˆ ๐‘ˆ)    โ‡’   (๐œ‘ โ†’ (๐‘† sSet ๐ด) โˆˆ ๐‘ˆ)
 
Theoremsetsres 16985 The structure replacement function does not affect the value of ๐‘† away from ๐ด. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
(๐‘† โˆˆ ๐‘‰ โ†’ ((๐‘† sSet โŸจ๐ด, ๐ตโŸฉ) โ†พ (V โˆ– {๐ด})) = (๐‘† โ†พ (V โˆ– {๐ด})))
 
Theoremsetsabs 16986 Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.)
((๐‘† โˆˆ ๐‘‰ โˆง ๐ถ โˆˆ ๐‘Š) โ†’ ((๐‘† sSet โŸจ๐ด, ๐ตโŸฉ) sSet โŸจ๐ด, ๐ถโŸฉ) = (๐‘† sSet โŸจ๐ด, ๐ถโŸฉ))
 
Theoremsetscom 16987 Component-setting is commutative when the x-values are different. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
๐ด โˆˆ V    &   ๐ต โˆˆ V    โ‡’   (((๐‘† โˆˆ ๐‘‰ โˆง ๐ด โ‰  ๐ต) โˆง (๐ถ โˆˆ ๐‘Š โˆง ๐ท โˆˆ ๐‘‹)) โ†’ ((๐‘† sSet โŸจ๐ด, ๐ถโŸฉ) sSet โŸจ๐ต, ๐ทโŸฉ) = ((๐‘† sSet โŸจ๐ต, ๐ทโŸฉ) sSet โŸจ๐ด, ๐ถโŸฉ))
 
7.1.1.3  Slots
 
Syntaxcslot 16988 Extend class notation with the slot function.
class Slot ๐ด
 
Definitiondf-slot 16989* Define the slot extractor for extensible structures. The class Slot ๐ด 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 18137) or a group (df-grp 18686)).

Note that Slot ๐ด is implemented as "evaluation at ๐ด". That is, (Slot ๐ดโ€˜๐‘†) is defined to be (๐‘†โ€˜๐ด), where ๐ด will typically be an index (which is implemented as a small natural number) of a component of an extensible structure ๐‘†. Each extensible structure is a function defined on specific (natural number) "slots", and the function Slot ๐ด extracts the structure's component as a function value at a particular slot (with index ๐ด).

The special "structure" ndx, defined as the identity function restricted to โ„•, can be used to extract the number ๐ด from a slot, since (Slot ๐ดโ€˜ndx) = ๐ด (see ndxarg 17003). 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 (Baseโ€˜ndx) in theorems and proofs instead of its hard-coded, numeric value 1), and discourage using the specific definition of slot extractors like Base = Slot 1 (see df-base 17019). Actually, these definitions are used in two basic theorems named *id (theorems of the form ๐ถ = Slot (๐ถโ€˜ndx)) and *ndx (theorems of the form (๐ถโ€˜ndx) = ๐‘) only (see, for example, baseid 17021 and basendx 17027), except additionally in the discouraged theorem baseval 17020 to demonstrate the representations of the value of the base set extractor. The *id theorems are implementation independent equivalents of the definitions by the means of ndxid 17004, but the *ndx theorems still depend on the hard-coded values of the indices. Therefore, the usage of these *ndx theorems is also discouraged (for more details see the section header comment mmtheorems.html#cnx 17004).

Example: The group operation is the second component, i.e., the component in the second slot, of a group-like structure ๐บ = {โŸจ(Baseโ€˜ndx), ๐ตโŸฉ, โŸจ(+gโ€˜ndx), + โŸฉ} (see grpstr 17100). The slot extractor +g = Slot 2 (see df-plusg 17081) applied on the structure ๐บ provides the group operation + = (+gโ€˜๐บ). Expanding the defintions, we get + = (Slot 2โ€˜๐บ) = (๐บโ€˜2) = (๐บโ€˜(+gโ€˜ndx)) (for the last equation, see plusgndx 17094).

The class Slot cannot be defined as (๐‘ฅ โˆˆ V โ†ฆ (๐‘“ โˆˆ V โ†ฆ (๐‘“โ€˜๐‘ฅ))) because each Slot ๐ด is a function on the proper class V so is itself a proper class, and the values of functions are sets (fvex 6851). It is necessary to allow proper classes as values of Slot ๐ด since for instance the class of all (base sets of) groups is proper. (Contributed by Mario Carneiro, 22-Sep-2015.)

Slot ๐ด = (๐‘ฅ โˆˆ V โ†ฆ (๐‘ฅโ€˜๐ด))
 
Theoremsloteq 16990 Equality theorem for the Slot construction. The converse holds if ๐ด (or ๐ต) is a set. (Contributed by BJ, 27-Dec-2021.)
(๐ด = ๐ต โ†’ Slot ๐ด = Slot ๐ต)
 
Theoremslotfn 16991 A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.)
๐ธ = Slot ๐‘    โ‡’   ๐ธ Fn V
 
Theoremstrfvnd 16992 Deduction version of strfvn 16993. (Contributed by Mario Carneiro, 15-Nov-2014.)
๐ธ = Slot ๐‘    &   (๐œ‘ โ†’ ๐‘† โˆˆ ๐‘‰)    โ‡’   (๐œ‘ โ†’ (๐ธโ€˜๐‘†) = (๐‘†โ€˜๐‘))
 
Theoremstrfvn 16993 Value of a structure component extractor ๐ธ. Normally, ๐ธ is a defined constant symbol such as Base (df-base 17019) and ๐‘ is the index of the component. ๐‘† is a structure, i.e. a specific member of a class of structures such as Poset (df-poset 18137) where ๐‘† โˆˆ Poset.

Hint: Do not substitute ๐‘ by a specific (positive) integer to be independent of a hard-coded index value. Often, (๐ธโ€˜ndx) can be used instead of ๐‘. Alternatively, use strfv 17011 instead of strfvn 16993. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2013.) (New usage is discouraged.)

๐‘† โˆˆ V    &   ๐ธ = Slot ๐‘    โ‡’   (๐ธโ€˜๐‘†) = (๐‘†โ€˜๐‘)
 
Theoremstrfvss 16994 A structure component extractor produces a value which is contained in a set dependent on ๐‘†, but not ๐ธ. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.)
๐ธ = Slot ๐‘    โ‡’   (๐ธโ€˜๐‘†) โŠ† โˆช ran ๐‘†
 
Theoremwunstr 16995 Closure of a structure index in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
๐ธ = Slot ๐‘    &   (๐œ‘ โ†’ ๐‘ˆ โˆˆ WUni)    &   (๐œ‘ โ†’ ๐‘† โˆˆ ๐‘ˆ)    โ‡’   (๐œ‘ โ†’ (๐ธโ€˜๐‘†) โˆˆ ๐‘ˆ)
 
Theoremstr0 16996 All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
๐น = Slot ๐ผ    โ‡’   โˆ… = (๐นโ€˜โˆ…)
 
Theoremstrfvi 16997 Structure slot extractors cannot distinguish between proper classes and โˆ…, so they can be protected using the identity function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
๐ธ = Slot ๐‘    &   ๐‘‹ = (๐ธโ€˜๐‘†)    โ‡’   ๐‘‹ = (๐ธโ€˜( I โ€˜๐‘†))
 
Theoremfveqprc 16998 Lemma for showing the equality of values for functions like slot extractors ๐ธ at a proper class. Extracted from several former proofs of lemmas like zlmlem 20840. (Contributed by AV, 31-Oct-2024.)
(๐ธโ€˜โˆ…) = โˆ…    &   ๐‘Œ = (๐นโ€˜๐‘‹)    โ‡’   (ยฌ ๐‘‹ โˆˆ V โ†’ (๐ธโ€˜๐‘‹) = (๐ธโ€˜๐‘Œ))
 
Theoremoveqprc 16999 Lemma for showing the equality of values for functions like slot extractors ๐ธ at a proper class. Extracted from several former proofs of lemmas like resvlem 31903. (Contributed by AV, 31-Oct-2024.)
(๐ธโ€˜โˆ…) = โˆ…    &   ๐‘ = (๐‘‹๐‘‚๐‘Œ)    &   Rel dom ๐‘‚    โ‡’   (ยฌ ๐‘‹ โˆˆ V โ†’ (๐ธโ€˜๐‘‹) = (๐ธโ€˜๐‘))
 
7.1.1.4  Structure component indices

The structure component index extractor ndx, defined in this subsection, is used to get the numeric argument from a defined structure component extractor such as df-base 17019 (see ndxarg 17003). For each defined structure component extractor, there should be a corresponding specific theorem providing its index, like basendx 17027. The usage of these theorems, however, is discouraged since the particular value for the index is an implementation detail. It is generally sufficient to work with (Baseโ€˜ndx) instead of the hard-coded index value, and use theorems such as baseid 17021 and basendxnplusgndx 17098.

The main circumstance in which it is necessary to look at indices directly is when showing that a set of indices are disjoint (for example in proofs such as cznabel 45970, based on setsnid 17016) or even ordered (in proofs such as lmodstr 17141). The requirement that the indices are distinct is necessary for sets of ordered pairs to be extensible structures, whereas the ordering allows for proofs avoiding the usage of quadradically many inequalities (compare cnfldfun 20731 with cnfldfunALT 20732).

As for the inequalities, it is recommended to provide them explicitly as theorems like basendxnplusgndx 17098, whenever they are required. Since these theorems use discouraged slot theorems, they should be placed near the definition of a slot (within the same subsection), so that the range of usages of discouraged theorems is tightly limited. Although there could be quadradically many of them in the total number of indices, much less are actually available (and not much more are expected).

As for the ordering, there are some theorems like basendxltplusgndx 17097 providing the less-than relationship between two indices. These theorems are also proved by discouraged theorems, so they should be placed near the definition of a slot (within the same subsection), too. However, since such theorems are rarely used (in structure building theorems *str like rngstr 17114), it is not recommended to provide explicit theorems for all of them, but to use the (discouraged) *ndx theorems as in lmodstr 17141. Therefore, *str theorems generally depend on the hard-coded values of the indices.

 
Syntaxcnx 17000 Extend class notation with the structure component index extractor.
class ndx
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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