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Theorem List for Metamath Proof Explorer - 16401-16500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2exp7 16401 Two to the seventh power is 128. (Contributed by AV, 16-Aug-2021.)
(2↑7) = 128
 
Theorem2exp8 16402 Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
(2↑8) = 256
 
Theorem2exp16 16403 Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
(2↑16) = 65536
 
Theorem3exp3 16404 Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.)
(3↑3) = 27
 
Theorem2expltfac 16405 The factorial grows faster than two to the power 𝑁. (Contributed by Mario Carneiro, 15-Sep-2016.)
(𝑁 ∈ (ℤ‘4) → (2↑𝑁) < (!‘𝑁))
 
6.2.18  Cyclical shifts of words (cont.)
 
Theoremcshwsidrepsw 16406 If cyclically shifting a word of length being a prime number by a number of positions which is not divisible by the prime number results in the word itself, the word is a "repeated symbol word". (Contributed by AV, 18-May-2018.) (Revised by AV, 10-Nov-2018.)
((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → ((𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))))
 
Theoremcshwsidrepswmod0 16407 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 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ ∧ 𝐿 ∈ ℤ) → ((𝑊 cyclShift 𝐿) = 𝑊 → ((𝐿 mod (♯‘𝑊)) = 0 ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊)))))
 
Theoremcshwshashlem1 16408* 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.)
(𝜑 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ))       ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) ≠ (𝑊‘0) ∧ 𝐿 ∈ (1..^(♯‘𝑊))) → (𝑊 cyclShift 𝐿) ≠ 𝑊)
 
Theoremcshwshashlem2 16409* 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.)
(𝜑 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ))       ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) ≠ (𝑊‘0)) → ((𝐿 ∈ (0..^(♯‘𝑊)) ∧ 𝐾 ∈ (0..^(♯‘𝑊)) ∧ 𝐾 < 𝐿) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾)))
 
Theoremcshwshashlem3 16410* 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.)
(𝜑 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ))       ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) ≠ (𝑊‘0)) → ((𝐿 ∈ (0..^(♯‘𝑊)) ∧ 𝐾 ∈ (0..^(♯‘𝑊)) ∧ 𝐾𝐿) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾)))
 
Theoremcshwsdisj 16411* 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.)
(𝜑 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ))       ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) ≠ (𝑊‘0)) → Disj 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)})
 
Theoremcshwsiun 16412* 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 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}       (𝑊 ∈ Word 𝑉𝑀 = 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)})
 
Theoremcshwsex 16413* 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 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}       (𝑊 ∈ Word 𝑉𝑀 ∈ V)
 
Theoremcshws0 16414* 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 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}       (𝑊 = ∅ → (♯‘𝑀) = 0)
 
Theoremcshwrepswhash1 16415* 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 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}       ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (♯‘𝑀) = 1)
 
Theoremcshwshashnsame 16416* 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 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}       ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → (∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) ≠ (𝑊‘0) → (♯‘𝑀) = (♯‘𝑊)))
 
Theoremcshwshash 16417* 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 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}       ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → ((♯‘𝑀) = (♯‘𝑊) ∨ (♯‘𝑀) = 1))
 
6.2.19  Specific prime numbers
 
Theoremprmlem0 16418* Lemma for prmlem1 16420 and prmlem2 16432. (Contributed by Mario Carneiro, 18-Feb-2014.)
((¬ 2 ∥ 𝑀𝑥 ∈ (ℤ𝑀)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))    &   (𝐾 ∈ ℙ → ¬ 𝐾𝑁)    &   (𝐾 + 2) = 𝑀       ((¬ 2 ∥ 𝐾𝑥 ∈ (ℤ𝐾)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
 
Theoremprmlem1a 16419* 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 ∈ ℕ    &   1 < 𝑁    &    ¬ 2 ∥ 𝑁    &    ¬ 3 ∥ 𝑁    &   ((¬ 2 ∥ 5 ∧ 𝑥 ∈ (ℤ‘5)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))       𝑁 ∈ ℙ
 
Theoremprmlem1 16420 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 ∈ ℕ    &   1 < 𝑁    &    ¬ 2 ∥ 𝑁    &    ¬ 3 ∥ 𝑁    &   𝑁 < 25       𝑁 ∈ ℙ
 
Theorem5prm 16421 5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
5 ∈ ℙ
 
Theorem6nprm 16422 6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
¬ 6 ∈ ℙ
 
Theorem7prm 16423 7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
7 ∈ ℙ
 
Theorem8nprm 16424 8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
¬ 8 ∈ ℙ
 
Theorem9nprm 16425 9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
¬ 9 ∈ ℙ
 
Theorem10nprm 16426 10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
¬ 10 ∈ ℙ
 
Theorem11prm 16427 11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
11 ∈ ℙ
 
Theorem13prm 16428 13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
13 ∈ ℙ
 
Theorem17prm 16429 17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
17 ∈ ℙ
 
Theorem19prm 16430 19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
19 ∈ ℙ
 
Theorem23prm 16431 23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
23 ∈ ℙ
 
Theoremprmlem2 16432 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 = 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 29↑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 16448).

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 ∈ ℕ    &   𝑁 < 841    &   1 < 𝑁    &    ¬ 2 ∥ 𝑁    &    ¬ 3 ∥ 𝑁    &    ¬ 5 ∥ 𝑁    &    ¬ 7 ∥ 𝑁    &    ¬ 11 ∥ 𝑁    &    ¬ 13 ∥ 𝑁    &    ¬ 17 ∥ 𝑁    &    ¬ 19 ∥ 𝑁    &    ¬ 23 ∥ 𝑁       𝑁 ∈ ℙ
 
Theorem37prm 16433 37 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
37 ∈ ℙ
 
Theorem43prm 16434 43 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
43 ∈ ℙ
 
Theorem83prm 16435 83 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
83 ∈ ℙ
 
Theorem139prm 16436 139 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
139 ∈ ℙ
 
Theorem163prm 16437 163 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
163 ∈ ℙ
 
Theorem317prm 16438 317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
317 ∈ ℙ
 
Theorem631prm 16439 631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
631 ∈ ℙ
 
Theoremprmo4 16440 The primorial of 4. (Contributed by AV, 28-Aug-2020.)
(#p‘4) = 6
 
Theoremprmo5 16441 The primorial of 5. (Contributed by AV, 28-Aug-2020.)
(#p‘5) = 30
 
Theoremprmo6 16442 The primorial of 6. (Contributed by AV, 28-Aug-2020.)
(#p‘6) = 30
 
6.2.20  Very large primes
 
Theorem1259lem1 16443 Lemma for 1259prm 16448. Calculate a power mod. In decimal, we calculate 2↑16 = 52𝑁 + 68≡68 and 2↑17≡68 · 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.)
𝑁 = 1259       ((2↑17) mod 𝑁) = (136 mod 𝑁)
 
Theorem1259lem2 16444 Lemma for 1259prm 16448. Calculate a power mod. In decimal, we calculate 2↑34 = (2↑17)↑2≡136↑2≡14𝑁 + 870. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.)
𝑁 = 1259       ((2↑34) mod 𝑁) = (870 mod 𝑁)
 
Theorem1259lem3 16445 Lemma for 1259prm 16448. Calculate a power mod. In decimal, we calculate 2↑38 = 2↑34 · 2↑4≡870 · 16 = 11𝑁 + 71 and 2↑76 = (2↑34)↑2≡71↑2 = 4𝑁 + 5≡5. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
𝑁 = 1259       ((2↑76) mod 𝑁) = (5 mod 𝑁)
 
Theorem1259lem4 16446 Lemma for 1259prm 16448. Calculate a power mod. In decimal, we calculate 2↑306 = (2↑76)↑4 · 4≡5↑4 · 4 = 2𝑁 − 18, 2↑612 = (2↑306)↑2≡18↑2 = 324, 2↑629 = 2↑612 · 2↑17≡324 · 136 = 35𝑁 − 1 and finally 2↑(𝑁 − 1) = (2↑629)↑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 = 1259       ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
 
Theorem1259lem5 16447 Lemma for 1259prm 16448. Calculate the GCD of 2↑34 − 1≡869 with 𝑁 = 1259. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
𝑁 = 1259       (((2↑34) − 1) gcd 𝑁) = 1
 
Theorem1259prm 16448 1259 is a prime number. (Contributed by Mario Carneiro, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
𝑁 = 1259       𝑁 ∈ ℙ
 
Theorem2503lem1 16449 Lemma for 2503prm 16452. Calculate a power mod. In decimal, we calculate 2↑18 = 512↑2 = 104𝑁 + 1832≡1832. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
𝑁 = 2503       ((2↑18) mod 𝑁) = (1832 mod 𝑁)
 
Theorem2503lem2 16450 Lemma for 2503prm 16452. Calculate a power mod. We calculate 2↑19 = 2↑18 · 2≡1832 · 2 = 𝑁 + 1161, 2↑38 = (2↑19)↑2≡1161↑2 = 538𝑁 + 1307, 2↑39 = 2↑38 · 2≡1307 · 2 = 𝑁 + 111, 2↑78 = (2↑39)↑2≡111↑2 = 5𝑁 − 194, 2↑156 = (2↑78)↑2≡194↑2 = 15𝑁 + 91, 2↑312 = (2↑156)↑2≡91↑2 = 3𝑁 + 772, 2↑624 = (2↑312)↑2≡772↑2 = 238𝑁 + 270, 2↑1248 = (2↑624)↑2≡270↑2 = 29𝑁 + 313, 2↑1251 = 2↑1248 · 8≡313 · 8 = 𝑁 + 1 and finally 2↑(𝑁 − 1) = (2↑1251)↑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 = 2503       ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
 
Theorem2503lem3 16451 Lemma for 2503prm 16452. Calculate the GCD of 2↑18 − 1≡1831 with 𝑁 = 2503. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.)
𝑁 = 2503       (((2↑18) − 1) gcd 𝑁) = 1
 
Theorem2503prm 16452 2503 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
𝑁 = 2503       𝑁 ∈ ℙ
 
Theorem4001lem1 16453 Lemma for 4001prm 16457. Calculate a power mod. In decimal, we calculate 2↑12 = 4096 = 𝑁 + 95, 2↑24 = (2↑12)↑2≡95↑2 = 2𝑁 + 1023, 2↑25 = 2↑24 · 2≡1023 · 2 = 2046, 2↑50 = (2↑25)↑2≡2046↑2 = 1046𝑁 + 1070, 2↑100 = (2↑50)↑2≡1070↑2 = 286𝑁 + 614 and 2↑200 = (2↑100)↑2≡614↑2 = 94𝑁 + 902 ≡902. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
𝑁 = 4001       ((2↑200) mod 𝑁) = (902 mod 𝑁)
 
Theorem4001lem2 16454 Lemma for 4001prm 16457. Calculate a power mod. In decimal, we calculate 2↑400 = (2↑200)↑2≡902↑2 = 203𝑁 + 1401 and 2↑800 = (2↑400)↑2≡1401↑2 = 490𝑁 + 2311 ≡2311. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
𝑁 = 4001       ((2↑800) mod 𝑁) = (2311 mod 𝑁)
 
Theorem4001lem3 16455 Lemma for 4001prm 16457. Calculate a power mod. In decimal, we calculate 2↑1000 = 2↑800 · 2↑200≡2311 · 902 = 521𝑁 + 1 and finally 2↑(𝑁 − 1) = (2↑1000)↑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 = 4001       ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
 
Theorem4001lem4 16456 Lemma for 4001prm 16457. Calculate the GCD of 2↑800 − 1≡2310 with 𝑁 = 4001. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
𝑁 = 4001       (((2↑800) − 1) gcd 𝑁) = 1
 
Theorem4001prm 16457 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 = 4001       𝑁 ∈ ℙ
 
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 . 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 16488 and strfv 16510. Using extractors makes it easier to change numeric indices and also makes the components' purpose clearer. For example, as noted in ndxid 16488, we can refer to a specific poset with base set 𝐵 and order relation 𝐿 using the extensible structure {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), 𝐿⟩} rather than {⟨1, 𝐵⟩, ⟨10, 𝐿⟩}.

There are many other 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 16470. 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 ring is defined in df-zring 20594 as simply ring = (ℂflds ℤ). This can be similarly done for all other subsets of , 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 to inherit, then we change the definition of 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 16700 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 16700 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 16836 and df-acs 16839. 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.

 
Syntaxcstr 16458 Extend class notation with the class of structures with components numbered below 𝐴.
class Struct
 
Syntaxcnx 16459 Extend class notation with the structure component index extractor.
class ndx
 
Syntaxcsts 16460 Set components of a structure.
class sSet
 
Syntaxcslot 16461 Extend class notation with the slot function.
class Slot 𝐴
 
Syntaxcbs 16462 Extend class notation with the class of all base set extractors.
class Base
 
Syntaxcress 16463 Extend class notation with the extensible structure builder restriction operator.
class s
 
Definitiondf-struct 16464* 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 6349, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 16474: 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 4802). This is used critically in strle1 16571, strle2 16572, strle3 16573 and strleun 16570 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 16622 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 16623, which requires that the base set be a set but not any of the other components. Usually, a concrete structure like fld does not contain the empty set, and therefore is a function, see cnfldfun 20533. (Contributed by Mario Carneiro, 29-Aug-2015.)

Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
 
Definitiondf-ndx 16465 Define the structure component index extractor. See theorem ndxarg 16487 to understand its purpose. The restriction to 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 or (if we revise all structure component definitions such as df-base 16468) another set such as the set of finite ordinals ω (df-om 7559). (Contributed by NM, 4-Sep-2011.)
ndx = ( I ↾ ℕ)
 
Definitiondf-slot 16466* 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 17535) or a group (df-grp 18085)).

Note that Slot 𝐴 is implemented as "evaluation at 𝐴". That is, (Slot 𝐴𝑆) is defined to be (𝑆𝐴), 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 , can be used to extract the number 𝐴 from a slot, since (Slot 𝐴‘ndx) = 𝐴 (see ndxarg 16487). 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 ∈ V ↦ (𝑓 ∈ V ↦ (𝑓𝑥))) 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 6659). 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 ∈ V ↦ (𝑥𝐴))
 
Theoremsloteq 16467 Equality theorem for the Slot construction. The converse holds if 𝐴 (or 𝐵) is a set. (Contributed by BJ, 27-Dec-2021.)
(𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)
 
Definitiondf-base 16468 Define the base set (also called underlying set, ground set, carrier set, or carrier) extractor for extensible structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Base = Slot 1
 
Definitiondf-sets 16469* 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 16470 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 19219, 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 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
 
Definitiondf-ress 16470* 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 16533 for the altered base set, and resslem 16536 (subrg0 19518, ressplusg 16591, subrg1 19521, ressmulr 16604) for the (un)altered other operations. (Contributed by Stefan O'Rear, 29-Nov-2014.)

s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
 
Theorembrstruct 16471 The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)
Rel Struct
 
Theoremisstruct2 16472 The property of being a structure with components in (1st𝑋)...(2nd𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.)
(𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))
 
Theoremstructex 16473 A structure is a set. (Contributed by AV, 10-Nov-2021.)
(𝐺 Struct 𝑋𝐺 ∈ V)
 
Theoremstructn0fun 16474 A structure without the empty set is a function. (Contributed by AV, 13-Nov-2021.)
(𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅}))
 
Theoremisstruct 16475 The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.)
(𝐹 Struct ⟨𝑀, 𝑁⟩ ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁)))
 
Theoremstructcnvcnv 16476 Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
(𝐹 Struct 𝑋𝐹 = (𝐹 ∖ {∅}))
 
Theoremstructfung 16477 The converse of the converse of a structure is a function. Closed form of structfun 16478. (Contributed by AV, 12-Nov-2021.)
(𝐹 Struct 𝑋 → Fun 𝐹)
 
Theoremstructfun 16478 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
 
Theoremstructfn 16479 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐹 Struct ⟨𝑀, 𝑁       (Fun 𝐹 ∧ dom 𝐹 ⊆ (1...𝑁))
 
Theoremslotfn 16480 A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝐸 = Slot 𝑁       𝐸 Fn V
 
Theoremstrfvnd 16481 Deduction version of strfvn 16484. (Contributed by Mario Carneiro, 15-Nov-2014.)
𝐸 = Slot 𝑁    &   (𝜑𝑆𝑉)       (𝜑 → (𝐸𝑆) = (𝑆𝑁))
 
Theorembasfn 16482 The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
Base Fn V
 
Theoremwunndx 16483 Closure of the index extractor in an infinite weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)       (𝜑 → ndx ∈ 𝑈)
 
Theoremstrfvn 16484 Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 16468) 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 17535) where 𝑆 ∈ Poset.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strfv 16510. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2013.) (New usage is discouraged.)

𝑆 ∈ V    &   𝐸 = Slot 𝑁       (𝐸𝑆) = (𝑆𝑁)
 
Theoremstrfvss 16485 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       (𝐸𝑆) ⊆ ran 𝑆
 
Theoremwunstr 16486 Closure of a structure index in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐸 = Slot 𝑁    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝑆𝑈)       (𝜑 → (𝐸𝑆) ∈ 𝑈)
 
Theoremndxarg 16487 Get the numeric argument from a defined structure component extractor such as df-base 16468. (Contributed by Mario Carneiro, 6-Oct-2013.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸‘ndx) = 𝑁
 
Theoremndxid 16488 A structure component extractor is defined by its own index. This theorem, together with strfv 16510 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 16468 and the 10 in df-ple 16564, making it easier to change should the need arise.

For example, we can refer to a specific poset with base set 𝐵 and order relation 𝐿 using {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), 𝐿⟩} rather than {⟨1, 𝐵⟩, 10, 𝐿⟩}. The latter, while shorter to state, requires revision if we later change 10 to some other number, and it may also be harder to remember. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.) (Proof shortened by BJ, 27-Dec-2021.)

𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       𝐸 = Slot (𝐸‘ndx)
 
Theoremstrndxid 16489 The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.)
(𝜑𝑆𝑉)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸𝑆))
 
Theoremreldmsets 16490 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Rel dom sSet
 
Theoremsetsvalg 16491 Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝑆𝑉𝐴𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
 
Theoremsetsval 16492 Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
((𝑆𝑉𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
 
Theoremsetsidvald 16493 Value of the structure replacement function, deduction version. (Contributed by AV, 14-Mar-2020.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → (𝐸‘ndx) ∈ dom 𝑆)       (𝜑𝑆 = (𝑆 sSet ⟨(𝐸‘ndx), (𝐸𝑆)⟩))
 
Theoremfvsetsid 16494 The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.)
((𝐹𝑉𝑋𝑊𝑌𝑈) → ((𝐹 sSet ⟨𝑋, 𝑌⟩)‘𝑋) = 𝑌)
 
Theoremfsets 16495 The structure replacement function is a function. (Contributed by SO, 12-Jul-2018.)
(((𝐹𝑉𝐹:𝐴𝐵) ∧ 𝑋𝐴𝑌𝐵) → (𝐹 sSet ⟨𝑋, 𝑌⟩):𝐴𝐵)
 
Theoremsetsdm 16496 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.)
((𝐺𝑉𝐸𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
 
Theoremsetsfun 16497 A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.)
(((𝐺𝑉 ∧ Fun 𝐺) ∧ (𝐼𝑈𝐸𝑊)) → Fun (𝐺 sSet ⟨𝐼, 𝐸⟩))
 
Theoremsetsfun0 16498 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 16497 is useful for proofs based on isstruct2 16472 which requires Fun (𝐹 ∖ {∅}) for 𝐹 to be an extensible structure. (Contributed by AV, 7-Jun-2021.)
(((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼𝑈𝐸𝑊)) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
 
Theoremsetsn0fun 16499 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.)
(𝜑𝑆 Struct 𝑋)    &   (𝜑𝐼𝑈)    &   (𝜑𝐸𝑊)       (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
 
Theoremsetsstruct2 16500 An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)
(((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) ∧ 𝑌 = ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑌)
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45093
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