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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cshwsidrepsw 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.) |
โข ((๐ โ Word ๐ โง (โฏโ๐) โ โ) โ ((๐ฟ โ โค โง (๐ฟ mod (โฏโ๐)) โ 0 โง (๐ cyclShift ๐ฟ) = ๐) โ ๐ = ((๐โ0) repeatS (โฏโ๐)))) | ||
Theorem | cshwsidrepswmod0 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.) |
โข ((๐ โ Word ๐ โง (โฏโ๐) โ โ โง ๐ฟ โ โค) โ ((๐ cyclShift ๐ฟ) = ๐ โ ((๐ฟ mod (โฏโ๐)) = 0 โจ ๐ = ((๐โ0) repeatS (โฏโ๐))))) | ||
Theorem | cshwshashlem1 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.) |
โข (๐ โ (๐ โ Word ๐ โง (โฏโ๐) โ โ)) โ โข ((๐ โง โ๐ โ (0..^(โฏโ๐))(๐โ๐) โ (๐โ0) โง ๐ฟ โ (1..^(โฏโ๐))) โ (๐ cyclShift ๐ฟ) โ ๐) | ||
Theorem | cshwshashlem2 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.) |
โข (๐ โ (๐ โ Word ๐ โง (โฏโ๐) โ โ)) โ โข ((๐ โง โ๐ โ (0..^(โฏโ๐))(๐โ๐) โ (๐โ0)) โ ((๐ฟ โ (0..^(โฏโ๐)) โง ๐พ โ (0..^(โฏโ๐)) โง ๐พ < ๐ฟ) โ (๐ cyclShift ๐ฟ) โ (๐ cyclShift ๐พ))) | ||
Theorem | cshwshashlem3 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.) |
โข (๐ โ (๐ โ Word ๐ โง (โฏโ๐) โ โ)) โ โข ((๐ โง โ๐ โ (0..^(โฏโ๐))(๐โ๐) โ (๐โ0)) โ ((๐ฟ โ (0..^(โฏโ๐)) โง ๐พ โ (0..^(โฏโ๐)) โง ๐พ โ ๐ฟ) โ (๐ cyclShift ๐ฟ) โ (๐ cyclShift ๐พ))) | ||
Theorem | cshwsdisj 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.) |
โข (๐ โ (๐ โ Word ๐ โง (โฏโ๐) โ โ)) โ โข ((๐ โง โ๐ โ (0..^(โฏโ๐))(๐โ๐) โ (๐โ0)) โ Disj ๐ โ (0..^(โฏโ๐)){(๐ cyclShift ๐)}) | ||
Theorem | cshwsiun 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.) |
โข ๐ = {๐ค โ Word ๐ โฃ โ๐ โ (0..^(โฏโ๐))(๐ cyclShift ๐) = ๐ค} โ โข (๐ โ Word ๐ โ ๐ = โช ๐ โ (0..^(โฏโ๐)){(๐ cyclShift ๐)}) | ||
Theorem | cshwsex 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.) |
โข ๐ = {๐ค โ Word ๐ โฃ โ๐ โ (0..^(โฏโ๐))(๐ cyclShift ๐) = ๐ค} โ โข (๐ โ Word ๐ โ ๐ โ V) | ||
Theorem | cshws0 16909* | The size of the set of (different!) words resulting by cyclically shifting an empty word is 0. (Contributed by AV, 8-Nov-2018.) |
โข ๐ = {๐ค โ Word ๐ โฃ โ๐ โ (0..^(โฏโ๐))(๐ cyclShift ๐) = ๐ค} โ โข (๐ = โ โ (โฏโ๐) = 0) | ||
Theorem | cshwrepswhash1 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.) |
โข ๐ = {๐ค โ Word ๐ โฃ โ๐ โ (0..^(โฏโ๐))(๐ cyclShift ๐) = ๐ค} โ โข ((๐ด โ ๐ โง ๐ โ โ โง ๐ = (๐ด repeatS ๐)) โ (โฏโ๐) = 1) | ||
Theorem | cshwshashnsame 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.) |
โข ๐ = {๐ค โ Word ๐ โฃ โ๐ โ (0..^(โฏโ๐))(๐ cyclShift ๐) = ๐ค} โ โข ((๐ โ Word ๐ โง (โฏโ๐) โ โ) โ (โ๐ โ (0..^(โฏโ๐))(๐โ๐) โ (๐โ0) โ (โฏโ๐) = (โฏโ๐))) | ||
Theorem | cshwshash 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.) |
โข ๐ = {๐ค โ Word ๐ โฃ โ๐ โ (0..^(โฏโ๐))(๐ cyclShift ๐) = ๐ค} โ โข ((๐ โ Word ๐ โง (โฏโ๐) โ โ) โ ((โฏโ๐) = (โฏโ๐) โจ (โฏโ๐) = 1)) | ||
Theorem | prmlem0 16913* | Lemma for prmlem1 16915 and prmlem2 16927. (Contributed by Mario Carneiro, 18-Feb-2014.) |
โข ((ยฌ 2 โฅ ๐ โง ๐ฅ โ (โคโฅโ๐)) โ ((๐ฅ โ (โ โ {2}) โง (๐ฅโ2) โค ๐) โ ยฌ ๐ฅ โฅ ๐)) & โข (๐พ โ โ โ ยฌ ๐พ โฅ ๐) & โข (๐พ + 2) = ๐ โ โข ((ยฌ 2 โฅ ๐พ โง ๐ฅ โ (โคโฅโ๐พ)) โ ((๐ฅ โ (โ โ {2}) โง (๐ฅโ2) โค ๐) โ ยฌ ๐ฅ โฅ ๐)) | ||
Theorem | prmlem1a 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.) |
โข ๐ โ โ & โข 1 < ๐ & โข ยฌ 2 โฅ ๐ & โข ยฌ 3 โฅ ๐ & โข ((ยฌ 2 โฅ 5 โง ๐ฅ โ (โคโฅโ5)) โ ((๐ฅ โ (โ โ {2}) โง (๐ฅโ2) โค ๐) โ ยฌ ๐ฅ โฅ ๐)) โ โข ๐ โ โ | ||
Theorem | prmlem1 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.) |
โข ๐ โ โ & โข 1 < ๐ & โข ยฌ 2 โฅ ๐ & โข ยฌ 3 โฅ ๐ & โข ๐ < ;25 โ โข ๐ โ โ | ||
Theorem | 5prm 16916 | 5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
โข 5 โ โ | ||
Theorem | 6nprm 16917 | 6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) |
โข ยฌ 6 โ โ | ||
Theorem | 7prm 16918 | 7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
โข 7 โ โ | ||
Theorem | 8nprm 16919 | 8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) |
โข ยฌ 8 โ โ | ||
Theorem | 9nprm 16920 | 9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) |
โข ยฌ 9 โ โ | ||
Theorem | 10nprm 16921 | 10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
โข ยฌ ;10 โ โ | ||
Theorem | 11prm 16922 | 11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
โข ;11 โ โ | ||
Theorem | 13prm 16923 | 13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
โข ;13 โ โ | ||
Theorem | 17prm 16924 | 17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
โข ;17 โ โ | ||
Theorem | 19prm 16925 | 19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
โข ;19 โ โ | ||
Theorem | 23prm 16926 | 23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
โข ;23 โ โ | ||
Theorem | prmlem2 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
5โ2 = 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 29โ2 = 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.) |
โข ๐ โ โ & โข ๐ < ;;841 & โข 1 < ๐ & โข ยฌ 2 โฅ ๐ & โข ยฌ 3 โฅ ๐ & โข ยฌ 5 โฅ ๐ & โข ยฌ 7 โฅ ๐ & โข ยฌ ;11 โฅ ๐ & โข ยฌ ;13 โฅ ๐ & โข ยฌ ;17 โฅ ๐ & โข ยฌ ;19 โฅ ๐ & โข ยฌ ;23 โฅ ๐ โ โข ๐ โ โ | ||
Theorem | 37prm 16928 | 37 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
โข ;37 โ โ | ||
Theorem | 43prm 16929 | 43 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
โข ;43 โ โ | ||
Theorem | 83prm 16930 | 83 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
โข ;83 โ โ | ||
Theorem | 139prm 16931 | 139 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
โข ;;139 โ โ | ||
Theorem | 163prm 16932 | 163 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
โข ;;163 โ โ | ||
Theorem | 317prm 16933 | 317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
โข ;;317 โ โ | ||
Theorem | 631prm 16934 | 631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
โข ;;631 โ โ | ||
Theorem | prmo4 16935 | The primorial of 4. (Contributed by AV, 28-Aug-2020.) |
โข (#pโ4) = 6 | ||
Theorem | prmo5 16936 | The primorial of 5. (Contributed by AV, 28-Aug-2020.) |
โข (#pโ5) = ;30 | ||
Theorem | prmo6 16937 | The primorial of 6. (Contributed by AV, 28-Aug-2020.) |
โข (#pโ6) = ;30 | ||
Theorem | 1259lem1 16938 | Lemma for 1259prm 16943. Calculate a power mod. In decimal, we calculate 2โ16 = 52๐ + 68โก68 and 2โ17โก68 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.) |
โข ๐ = ;;;1259 โ โข ((2โ;17) mod ๐) = (;;136 mod ๐) | ||
Theorem | 1259lem2 16939 | Lemma for 1259prm 16943. Calculate a power mod. In decimal, we calculate 2โ34 = (2โ17)โ2โก136โ2โก14๐ + 870. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.) |
โข ๐ = ;;;1259 โ โข ((2โ;34) mod ๐) = (;;870 mod ๐) | ||
Theorem | 1259lem3 16940 | Lemma for 1259prm 16943. Calculate a power mod. In decimal, we calculate 2โ38 = 2โ34 ยท 2โ4โก870 ยท 16 = 11๐ + 71 and 2โ76 = (2โ34)โ2โก71โ2 = 4๐ + 5โก5. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
โข ๐ = ;;;1259 โ โข ((2โ;76) mod ๐) = (5 mod ๐) | ||
Theorem | 1259lem4 16941 | Lemma for 1259prm 16943. Calculate a power mod. In decimal, we calculate 2โ306 = (2โ76)โ4 ยท 4โก5โ4 ยท 4 = 2๐ โ 18, 2โ612 = (2โ306)โ2โก18โ2 = 324, 2โ629 = 2โ612 ยท 2โ17โก324 ยท 136 = 35๐ โ 1 and finally 2โ(๐ โ 1) = (2โ629)โ2โก1โ2 = 1. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
โข ๐ = ;;;1259 โ โข ((2โ(๐ โ 1)) mod ๐) = (1 mod ๐) | ||
Theorem | 1259lem5 16942 | Lemma for 1259prm 16943. Calculate the GCD of 2โ34 โ 1โก869 with ๐ = 1259. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) |
โข ๐ = ;;;1259 โ โข (((2โ;34) โ 1) gcd ๐) = 1 | ||
Theorem | 1259prm 16943 | 1259 is a prime number. (Contributed by Mario Carneiro, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
โข ๐ = ;;;1259 โ โข ๐ โ โ | ||
Theorem | 2503lem1 16944 | Lemma for 2503prm 16947. Calculate a power mod. In decimal, we calculate 2โ18 = 512โ2 = 104๐ + 1832โก1832. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
โข ๐ = ;;;2503 โ โข ((2โ;18) mod ๐) = (;;;1832 mod ๐) | ||
Theorem | 2503lem2 16945 | Lemma for 2503prm 16947. Calculate a power mod. We calculate 2โ19 = 2โ18 ยท 2โก1832 ยท 2 = ๐ + 1161, 2โ38 = (2โ19)โ2โก1161โ2 = 538๐ + 1307, 2โ39 = 2โ38 ยท 2โก1307 ยท 2 = ๐ + 111, 2โ78 = (2โ39)โ2โก111โ2 = 5๐ โ 194, 2โ156 = (2โ78)โ2โก194โ2 = 15๐ + 91, 2โ312 = (2โ156)โ2โก91โ2 = 3๐ + 772, 2โ624 = (2โ312)โ2โก772โ2 = 238๐ + 270, 2โ1248 = (2โ624)โ2โก270โ2 = 29๐ + 313, 2โ1251 = 2โ1248 ยท 8โก313 ยท 8 = ๐ + 1 and finally 2โ(๐ โ 1) = (2โ1251)โ2โก1โ2 = 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
โข ๐ = ;;;2503 โ โข ((2โ(๐ โ 1)) mod ๐) = (1 mod ๐) | ||
Theorem | 2503lem3 16946 | Lemma for 2503prm 16947. Calculate the GCD of 2โ18 โ 1โก1831 with ๐ = 2503. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.) |
โข ๐ = ;;;2503 โ โข (((2โ;18) โ 1) gcd ๐) = 1 | ||
Theorem | 2503prm 16947 | 2503 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) |
โข ๐ = ;;;2503 โ โข ๐ โ โ | ||
Theorem | 4001lem1 16948 | Lemma for 4001prm 16952. Calculate a power mod. In decimal, we calculate 2โ12 = 4096 = ๐ + 95, 2โ24 = (2โ12)โ2โก95โ2 = 2๐ + 1023, 2โ25 = 2โ24 ยท 2โก1023 ยท 2 = 2046, 2โ50 = (2โ25)โ2โก2046โ2 = 1046๐ + 1070, 2โ100 = (2โ50)โ2โก1070โ2 = 286๐ + 614 and 2โ200 = (2โ100)โ2โก614โ2 = 94๐ + 902 โก902. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
โข ๐ = ;;;4001 โ โข ((2โ;;200) mod ๐) = (;;902 mod ๐) | ||
Theorem | 4001lem2 16949 | Lemma for 4001prm 16952. Calculate a power mod. In decimal, we calculate 2โ400 = (2โ200)โ2โก902โ2 = 203๐ + 1401 and 2โ800 = (2โ400)โ2โก1401โ2 = 490๐ + 2311 โก2311. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
โข ๐ = ;;;4001 โ โข ((2โ;;800) mod ๐) = (;;;2311 mod ๐) | ||
Theorem | 4001lem3 16950 | Lemma for 4001prm 16952. Calculate a power mod. In decimal, we calculate 2โ1000 = 2โ800 ยท 2โ200โก2311 ยท 902 = 521๐ + 1 and finally 2โ(๐ โ 1) = (2โ1000)โ4โก1โ4 = 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
โข ๐ = ;;;4001 โ โข ((2โ(๐ โ 1)) mod ๐) = (1 mod ๐) | ||
Theorem | 4001lem4 16951 | Lemma for 4001prm 16952. Calculate the GCD of 2โ800 โ 1โก2310 with ๐ = 4001. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
โข ๐ = ;;;4001 โ โข (((2โ;;800) โ 1) gcd ๐) = 1 | ||
Theorem | 4001prm 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.) |
โข ๐ = ;;;4001 โ โข ๐ โ โ | ||
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 ๐ต and order relation ๐ฟ using the extensible structure {โจ(Baseโndx), ๐ตโฉ, โจ(leโndx), ๐ฟโฉ} rather than {โจ1, ๐ตโฉ, โจ;10, ๐ฟโฉ}. 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 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 โพs 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 โคring is defined in df-zring 20793 as simply โคring = (โfld โพs โค). 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 โfld, 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, โพs 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 16953 | Extend class notation with the class of structures with components numbered below ๐ด. |
class Struct | ||
Definition | df-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 โ 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: ๐น Struct ๐ โ Fun (๐น โ {โ }). Allowing an extensible structure to contain the empty set ensures that expressions like {โจ๐ด, ๐ตโฉ, โจ๐ถ, ๐ทโฉ} are structures without asserting or implying that ๐ด, ๐ต, ๐ถ and ๐ท are sets (if ๐ด or ๐ต is a proper class, then โจ๐ด, ๐ตโฉ = โ , 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 โfld does not contain the empty set, and therefore is a function, see cnfldfun 20731. (Contributed by Mario Carneiro, 29-Aug-2015.) |
โข Struct = {โจ๐, ๐ฅโฉ โฃ (๐ฅ โ ( โค โฉ (โ ร โ)) โง Fun (๐ โ {โ }) โง dom ๐ โ (...โ๐ฅ))} | ||
Theorem | brstruct 16955 | The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.) |
โข Rel Struct | ||
Theorem | isstruct2 16956 | The property of being a structure with components in (1st โ๐)...(2nd โ๐). (Contributed by Mario Carneiro, 29-Aug-2015.) |
โข (๐น Struct ๐ โ (๐ โ ( โค โฉ (โ ร โ)) โง Fun (๐น โ {โ }) โง dom ๐น โ (...โ๐))) | ||
Theorem | structex 16957 | A structure is a set. (Contributed by AV, 10-Nov-2021.) |
โข (๐บ Struct ๐ โ ๐บ โ V) | ||
Theorem | structn0fun 16958 | A structure without the empty set is a function. (Contributed by AV, 13-Nov-2021.) |
โข (๐น Struct ๐ โ Fun (๐น โ {โ })) | ||
Theorem | isstruct 16959 | The property of being a structure with components in ๐...๐. (Contributed by Mario Carneiro, 29-Aug-2015.) |
โข (๐น Struct โจ๐, ๐โฉ โ ((๐ โ โ โง ๐ โ โ โง ๐ โค ๐) โง Fun (๐น โ {โ }) โง dom ๐น โ (๐...๐))) | ||
Theorem | structcnvcnv 16960 | Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
โข (๐น Struct ๐ โ โกโก๐น = (๐น โ {โ })) | ||
Theorem | structfung 16961 | The converse of the converse of a structure is a function. Closed form of structfun 16962. (Contributed by AV, 12-Nov-2021.) |
โข (๐น Struct ๐ โ Fun โกโก๐น) | ||
Theorem | structfun 16962 | Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Proof shortened by AV, 12-Nov-2021.) |
โข ๐น Struct ๐ โ โข Fun โกโก๐น | ||
Theorem | structfn 16963 | Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
โข ๐น Struct โจ๐, ๐โฉ โ โข (Fun โกโก๐น โง dom ๐น โ (1...๐)) | ||
Theorem | strleun 16964 | Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) |
โข ๐น Struct โจ๐ด, ๐ตโฉ & โข ๐บ Struct โจ๐ถ, ๐ทโฉ & โข ๐ต < ๐ถ โ โข (๐น โช ๐บ) Struct โจ๐ด, ๐ทโฉ | ||
Theorem | strle1 16965 | Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) |
โข ๐ผ โ โ & โข ๐ด = ๐ผ โ โข {โจ๐ด, ๐โฉ} Struct โจ๐ผ, ๐ผโฉ | ||
Theorem | strle2 16966 | Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) |
โข ๐ผ โ โ & โข ๐ด = ๐ผ & โข ๐ผ < ๐ฝ & โข ๐ฝ โ โ & โข ๐ต = ๐ฝ โ โข {โจ๐ด, ๐โฉ, โจ๐ต, ๐โฉ} Struct โจ๐ผ, ๐ฝโฉ | ||
Theorem | strle3 16967 | Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.) |
โข ๐ผ โ โ & โข ๐ด = ๐ผ & โข ๐ผ < ๐ฝ & โข ๐ฝ โ โ & โข ๐ต = ๐ฝ & โข ๐ฝ < ๐พ & โข ๐พ โ โ & โข ๐ถ = ๐พ โ โข {โจ๐ด, ๐โฉ, โจ๐ต, ๐โฉ, โจ๐ถ, ๐โฉ} Struct โจ๐ผ, ๐พโฉ | ||
Theorem | sbcie2s 16968* | A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.) |
โข ๐ด = (๐ธโ๐) & โข ๐ต = (๐นโ๐) & โข ((๐ = ๐ด โง ๐ = ๐ต) โ (๐ โ ๐)) โ โข (๐ค = ๐ โ ([(๐ธโ๐ค) / ๐][(๐นโ๐ค) / ๐]๐ โ ๐)) | ||
Theorem | sbcie3s 16969* | A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
โข ๐ด = (๐ธโ๐) & โข ๐ต = (๐นโ๐) & โข ๐ถ = (๐บโ๐) & โข ((๐ = ๐ด โง ๐ = ๐ต โง ๐ = ๐ถ) โ (๐ โ ๐)) โ โข (๐ค = ๐ โ ([(๐ธโ๐ค) / ๐][(๐นโ๐ค) / ๐][(๐บโ๐ค) / ๐]๐ โ ๐)) | ||
Syntax | csts 16970 | Set components of a structure. |
class sSet | ||
Definition | df-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.) |
โข sSet = (๐ โ V, ๐ โ V โฆ ((๐ โพ (V โ dom {๐})) โช {๐})) | ||
Theorem | reldmsets 16972 | The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
โข Rel dom sSet | ||
Theorem | setsvalg 16973 | Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.) |
โข ((๐ โ ๐ โง ๐ด โ ๐) โ (๐ sSet ๐ด) = ((๐ โพ (V โ dom {๐ด})) โช {๐ด})) | ||
Theorem | setsval 16974 | Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
โข ((๐ โ ๐ โง ๐ต โ ๐) โ (๐ sSet โจ๐ด, ๐ตโฉ) = ((๐ โพ (V โ {๐ด})) โช {โจ๐ด, ๐ตโฉ})) | ||
Theorem | fvsetsid 16975 | The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.) |
โข ((๐น โ ๐ โง ๐ โ ๐ โง ๐ โ ๐) โ ((๐น sSet โจ๐, ๐โฉ)โ๐) = ๐) | ||
Theorem | fsets 16976 | The structure replacement function is a function. (Contributed by SO, 12-Jul-2018.) |
โข (((๐น โ ๐ โง ๐น:๐ดโถ๐ต) โง ๐ โ ๐ด โง ๐ โ ๐ต) โ (๐น sSet โจ๐, ๐โฉ):๐ดโถ๐ต) | ||
Theorem | setsdm 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.) |
โข ((๐บ โ ๐ โง ๐ธ โ ๐) โ dom (๐บ sSet โจ๐ผ, ๐ธโฉ) = (dom ๐บ โช {๐ผ})) | ||
Theorem | setsfun 16978 | A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.) |
โข (((๐บ โ ๐ โง Fun ๐บ) โง (๐ผ โ ๐ โง ๐ธ โ ๐)) โ Fun (๐บ sSet โจ๐ผ, ๐ธโฉ)) | ||
Theorem | setsfun0 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 (๐น โ {โ }) for ๐น to be an extensible structure. (Contributed by AV, 7-Jun-2021.) |
โข (((๐บ โ ๐ โง Fun (๐บ โ {โ })) โง (๐ผ โ ๐ โง ๐ธ โ ๐)) โ Fun ((๐บ sSet โจ๐ผ, ๐ธโฉ) โ {โ })) | ||
Theorem | setsn0fun 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.) |
โข (๐ โ ๐ Struct ๐) & โข (๐ โ ๐ผ โ ๐) & โข (๐ โ ๐ธ โ ๐) โ โข (๐ โ Fun ((๐ sSet โจ๐ผ, ๐ธโฉ) โ {โ })) | ||
Theorem | setsstruct2 16981 | An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.) |
โข (((๐บ Struct ๐ โง ๐ธ โ ๐ โง ๐ผ โ โ) โง ๐ = โจif(๐ผ โค (1st โ๐), ๐ผ, (1st โ๐)), if(๐ผ โค (2nd โ๐), (2nd โ๐), ๐ผ)โฉ) โ (๐บ sSet โจ๐ผ, ๐ธโฉ) Struct ๐) | ||
Theorem | setsexstruct2 16982* | An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.) |
โข ((๐บ Struct ๐ โง ๐ธ โ ๐ โง ๐ผ โ โ) โ โ๐ฆ(๐บ sSet โจ๐ผ, ๐ธโฉ) Struct ๐ฆ) | ||
Theorem | setsstruct 16983 | An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 9-Jun-2021.) (Revised by AV, 14-Nov-2021.) |
โข ((๐ธ โ ๐ โง ๐ผ โ (โคโฅโ๐) โง ๐บ Struct โจ๐, ๐โฉ) โ (๐บ sSet โจ๐ผ, ๐ธโฉ) Struct โจ๐, if(๐ผ โค ๐, ๐, ๐ผ)โฉ) | ||
Theorem | wunsets 16984 | Closure of structure replacement in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) |
โข (๐ โ ๐ โ WUni) & โข (๐ โ ๐ โ ๐) & โข (๐ โ ๐ด โ ๐) โ โข (๐ โ (๐ sSet ๐ด) โ ๐) | ||
Theorem | setsres 16985 | The structure replacement function does not affect the value of ๐ away from ๐ด. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
โข (๐ โ ๐ โ ((๐ sSet โจ๐ด, ๐ตโฉ) โพ (V โ {๐ด})) = (๐ โพ (V โ {๐ด}))) | ||
Theorem | setsabs 16986 | Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.) |
โข ((๐ โ ๐ โง ๐ถ โ ๐) โ ((๐ sSet โจ๐ด, ๐ตโฉ) sSet โจ๐ด, ๐ถโฉ) = (๐ sSet โจ๐ด, ๐ถโฉ)) | ||
Theorem | setscom 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.) |
โข ๐ด โ V & โข ๐ต โ V โ โข (((๐ โ ๐ โง ๐ด โ ๐ต) โง (๐ถ โ ๐ โง ๐ท โ ๐)) โ ((๐ sSet โจ๐ด, ๐ถโฉ) sSet โจ๐ต, ๐ทโฉ) = ((๐ sSet โจ๐ต, ๐ทโฉ) sSet โจ๐ด, ๐ถโฉ)) | ||
Syntax | cslot 16988 | Extend class notation with the slot function. |
class Slot ๐ด | ||
Definition | df-slot 16989* |
Define the slot extractor for extensible structures. The class
Slot ๐ด 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 ๐ด is implemented as "evaluation at ๐ด". That is, (Slot ๐ดโ๐) is defined to be (๐โ๐ด), where ๐ด 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 ๐ด extracts the structure's component as a function value at a particular slot (with index ๐ด). The special "structure" ndx, defined as the identity function restricted to โ, can be used to extract the number ๐ด from a slot, since (Slot ๐ดโndx) = ๐ด (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 (Baseโndx) 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 ๐ถ = Slot (๐ถโndx)) and *ndx (theorems of the form (๐ถโndx) = 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 ๐บ = {โจ(Baseโndx), ๐ตโฉ, โจ(+gโndx), + โฉ} (see grpstr 17100). The slot extractor +g = Slot 2 (see df-plusg 17081) applied on the structure ๐บ provides the group operation + = (+gโ๐บ). Expanding the defintions, we get + = (Slot 2โ๐บ) = (๐บโ2) = (๐บโ(+gโndx)) (for the last equation, see plusgndx 17094). The class Slot cannot be defined as (๐ฅ โ V โฆ (๐ โ V โฆ (๐โ๐ฅ))) because each Slot ๐ด 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 ๐ด since for instance the class of all (base sets of) groups is proper. (Contributed by Mario Carneiro, 22-Sep-2015.) |
โข Slot ๐ด = (๐ฅ โ V โฆ (๐ฅโ๐ด)) | ||
Theorem | sloteq 16990 | Equality theorem for the Slot construction. The converse holds if ๐ด (or ๐ต) is a set. (Contributed by BJ, 27-Dec-2021.) |
โข (๐ด = ๐ต โ Slot ๐ด = Slot ๐ต) | ||
Theorem | slotfn 16991 | A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.) |
โข ๐ธ = Slot ๐ โ โข ๐ธ Fn V | ||
Theorem | strfvnd 16992 | Deduction version of strfvn 16993. (Contributed by Mario Carneiro, 15-Nov-2014.) |
โข ๐ธ = Slot ๐ & โข (๐ โ ๐ โ ๐) โ โข (๐ โ (๐ธโ๐) = (๐โ๐)) | ||
Theorem | strfvn 16993 |
Value of a structure component extractor ๐ธ. Normally, ๐ธ 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
โ Poset.
Hint: Do not substitute ๐ by a specific (positive) integer to be independent of a hard-coded index value. Often, (๐ธโndx) 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.) |
โข ๐ โ V & โข ๐ธ = Slot ๐ โ โข (๐ธโ๐) = (๐โ๐) | ||
Theorem | strfvss 16994 | A structure component extractor produces a value which is contained in a set dependent on ๐, but not ๐ธ. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.) |
โข ๐ธ = Slot ๐ โ โข (๐ธโ๐) โ โช ran ๐ | ||
Theorem | wunstr 16995 | Closure of a structure index in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) |
โข ๐ธ = Slot ๐ & โข (๐ โ ๐ โ WUni) & โข (๐ โ ๐ โ ๐) โ โข (๐ โ (๐ธโ๐) โ ๐) | ||
Theorem | str0 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.) |
โข ๐น = Slot ๐ผ โ โข โ = (๐นโโ ) | ||
Theorem | strfvi 16997 | Structure slot extractors cannot distinguish between proper classes and โ , so they can be protected using the identity function. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
โข ๐ธ = Slot ๐ & โข ๐ = (๐ธโ๐) โ โข ๐ = (๐ธโ( I โ๐)) | ||
Theorem | fveqprc 16998 | Lemma for showing the equality of values for functions like slot extractors ๐ธ at a proper class. Extracted from several former proofs of lemmas like zlmlem 20840. (Contributed by AV, 31-Oct-2024.) |
โข (๐ธโโ ) = โ & โข ๐ = (๐นโ๐) โ โข (ยฌ ๐ โ V โ (๐ธโ๐) = (๐ธโ๐)) | ||
Theorem | oveqprc 16999 | Lemma for showing the equality of values for functions like slot extractors ๐ธ at a proper class. Extracted from several former proofs of lemmas like resvlem 31903. (Contributed by AV, 31-Oct-2024.) |
โข (๐ธโโ ) = โ & โข ๐ = (๐๐๐) & โข Rel dom ๐ โ โข (ยฌ ๐ โ V โ (๐ธโ๐) = (๐ธโ๐)) | ||
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 (Baseโndx) 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. | ||
Syntax | cnx 17000 | Extend class notation with the structure component index extractor. |
class ndx |
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