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

Theoremprmgaplem2 15801 Lemma for prmgap 15810: The factorial of a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 13-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 1 < (((!‘𝑁) + 𝐼) gcd 𝐼))

Theoremprmgaplcmlem1 15802 Lemma for prmgaplcm 15811: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number is divisible by that integer. (Contributed by AV, 14-Aug-2020.) (Revised by AV, 27-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ ((lcm‘(1...𝑁)) + 𝐼))

Theoremprmgaplcmlem2 15803 Lemma for prmgaplcm 15811: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 14-Aug-2020.) (Revised by AV, 27-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 1 < (((lcm‘(1...𝑁)) + 𝐼) gcd 𝐼))

Theoremprmgaplem3 15804* Lemma for prmgap 15810. (Contributed by AV, 9-Aug-2020.)
𝐴 = {𝑝 ∈ ℙ ∣ 𝑝 < 𝑁}       (𝑁 ∈ (ℤ‘3) → ∃𝑥𝐴𝑦𝐴 𝑦𝑥)

Theoremprmgaplem4 15805* Lemma for prmgap 15810. (Contributed by AV, 10-Aug-2020.)
𝐴 = {𝑝 ∈ ℙ ∣ (𝑁 < 𝑝𝑝𝑃)}       ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑁 < 𝑃) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)

Theoremprmgaplem5 15806* Lemma for prmgap 15810: for each integer greater than 2 there is a smaller prime closest to this integer, i.e. there is a smaller prime and no other prime is between this prime and the integer. (Contributed by AV, 9-Aug-2020.)
(𝑁 ∈ (ℤ‘3) → ∃𝑝 ∈ ℙ (𝑝 < 𝑁 ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑁)𝑧 ∉ ℙ))

Theoremprmgaplem6 15807* Lemma for prmgap 15810: for each positive integer there is a greater prime closest to this integer, i.e. there is a greater prime and no other prime is between this prime and the integer. (Contributed by AV, 10-Aug-2020.)
(𝑁 ∈ ℕ → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ ∀𝑧 ∈ ((𝑁 + 1)..^𝑝)𝑧 ∉ ℙ))

Theoremprmgaplem7 15808* Lemma for prmgap 15810. (Contributed by AV, 12-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹 ∈ (ℕ ↑𝑚 ℕ))    &   (𝜑 → ∀𝑖 ∈ (2...𝑁)1 < (((𝐹𝑁) + 𝑖) gcd 𝑖))       (𝜑 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 < ((𝐹𝑁) + 2) ∧ ((𝐹𝑁) + 𝑁) < 𝑞 ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ))

Theoremprmgaplem8 15809* Lemma for prmgap 15810. (Contributed by AV, 13-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹 ∈ (ℕ ↑𝑚 ℕ))    &   (𝜑 → ∀𝑖 ∈ (2...𝑁)1 < (((𝐹𝑁) + 𝑖) gcd 𝑖))       (𝜑 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑁 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ))

Theoremprmgap 15810* The prime gap theorem: for each positive integer there are (at least) two successive primes with a difference ("gap") at least as big as the given integer. (Contributed by AV, 13-Aug-2020.)
𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)

Theoremprmgaplcm 15811* Alternate proof of prmgap 15810: in contrast to prmgap 15810, where the gap starts at n! , the factorial of n, the gap starts at the least common multiple of all positive integers less than or equal to n. (Contributed by AV, 13-Aug-2020.) (Revised by AV, 27-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)

Theoremprmgapprmolem 15812 Lemma for prmgapprmo 15813: The primorial of a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 29-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 1 < (((#p𝑁) + 𝐼) gcd 𝐼))

Theoremprmgapprmo 15813* Alternate proof of prmgap 15810: in contrast to prmgap 15810, where the gap starts at n! , the factorial of n, the gap starts at n#, the primorial of n. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 29-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)

6.2.17  Decimal arithmetic (cont.)

Theoremdec2dvds 15814 Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   (𝐵 · 2) = 𝐶    &   𝐷 = (𝐶 + 1)        ¬ 2 ∥ 𝐴𝐷

Theoremdec5dvds 15815 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ    &   𝐵 < 5        ¬ 5 ∥ 𝐴𝐵

Theoremdec5dvds2 15816 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ    &   𝐵 < 5    &   (5 + 𝐵) = 𝐶        ¬ 5 ∥ 𝐴𝐶

Theoremdec5nprm 15817 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ        ¬ 𝐴5 ∈ ℙ

Theoremdec2nprm 15818 Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   (𝐵 · 2) = 𝐶        ¬ 𝐴𝐶 ∈ ℙ

Theoremmodxai 15819 Add exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) (Revised by Mario Carneiro, 5-Feb-2015.)
𝑁 ∈ ℕ    &   𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐷 ∈ ℤ    &   𝐾 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐿 ∈ ℕ0    &   ((𝐴𝐵) mod 𝑁) = (𝐾 mod 𝑁)    &   ((𝐴𝐶) mod 𝑁) = (𝐿 mod 𝑁)    &   (𝐵 + 𝐶) = 𝐸    &   ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐿)       ((𝐴𝐸) mod 𝑁) = (𝑀 mod 𝑁)

Theoremmod2xi 15820 Double exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)
𝑁 ∈ ℕ    &   𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐷 ∈ ℤ    &   𝐾 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   ((𝐴𝐵) mod 𝑁) = (𝐾 mod 𝑁)    &   (2 · 𝐵) = 𝐸    &   ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐾)       ((𝐴𝐸) mod 𝑁) = (𝑀 mod 𝑁)

Theoremmodxp1i 15821 Add one to an exponent in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)
𝑁 ∈ ℕ    &   𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐷 ∈ ℤ    &   𝐾 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   ((𝐴𝐵) mod 𝑁) = (𝐾 mod 𝑁)    &   (𝐵 + 1) = 𝐸    &   ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐴)       ((𝐴𝐸) mod 𝑁) = (𝑀 mod 𝑁)

Theoremmod2xnegi 15822 Version of mod2xi 15820 with a negative mod value. (Contributed by Mario Carneiro, 21-Feb-2014.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐷 ∈ ℤ    &   𝐾 ∈ ℕ    &   𝑀 ∈ ℕ0    &   𝐿 ∈ ℕ0    &   ((𝐴𝐵) mod 𝑁) = (𝐿 mod 𝑁)    &   (2 · 𝐵) = 𝐸    &   (𝐿 + 𝐾) = 𝑁    &   ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐾)       ((𝐴𝐸) mod 𝑁) = (𝑀 mod 𝑁)

Theoremmodsubi 15823 Subtract from within a mod calculation. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁 ∈ ℕ    &   𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (𝐴 mod 𝑁) = (𝐾 mod 𝑁)    &   (𝑀 + 𝐵) = 𝐾       ((𝐴𝐵) mod 𝑁) = (𝑀 mod 𝑁)

Theoremgcdi 15824 Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)
𝐾 ∈ ℕ0    &   𝑅 ∈ ℕ0    &   𝑁 ∈ ℕ0    &   (𝑁 gcd 𝑅) = 𝐺    &   ((𝐾 · 𝑁) + 𝑅) = 𝑀       (𝑀 gcd 𝑁) = 𝐺

Theoremgcdmodi 15825 Calculate a GCD via Euclid's algorithm. Theorem 5.6 in [ApostolNT] p. 109. (Contributed by Mario Carneiro, 19-Feb-2014.)
𝐾 ∈ ℕ0    &   𝑅 ∈ ℕ0    &   𝑁 ∈ ℕ    &   (𝐾 mod 𝑁) = (𝑅 mod 𝑁)    &   (𝑁 gcd 𝑅) = 𝐺       (𝐾 gcd 𝑁) = 𝐺

Theoremdecexp2 15826 Calculate a power of two. (Contributed by Mario Carneiro, 19-Feb-2014.)
𝑀 ∈ ℕ0    &   (𝑀 + 2) = 𝑁       ((4 · (2↑𝑀)) + 0) = (2↑𝑁)

Theoremnumexp0 15827 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0       (𝐴↑0) = 1

Theoremnumexp1 15828 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0       (𝐴↑1) = 𝐴

Theoremnumexpp1 15829 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (𝑀 + 1) = 𝑁    &   ((𝐴𝑀) · 𝐴) = 𝐶       (𝐴𝑁) = 𝐶

Theoremnumexp2x 15830 Double an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (2 · 𝑀) = 𝑁    &   (𝐴𝑀) = 𝐷    &   (𝐷 · 𝐷) = 𝐶       (𝐴𝑁) = 𝐶

Theoremdecsplit0b 15831 Split a decimal number into two parts. Base case: 𝑁 = 0. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
𝐴 ∈ ℕ0       ((𝐴 · (10↑0)) + 𝐵) = (𝐴 + 𝐵)

Theoremdecsplit0 15832 Split a decimal number into two parts. Base case: 𝑁 = 0. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
𝐴 ∈ ℕ0       ((𝐴 · (10↑0)) + 0) = 𝐴

Theoremdecsplit1 15833 Split a decimal number into two parts. Base case: 𝑁 = 1. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
𝐴 ∈ ℕ0       ((𝐴 · (10↑1)) + 𝐵) = 𝐴𝐵

Theoremdecsplit 15834 Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (𝑀 + 1) = 𝑁    &   ((𝐴 · (10↑𝑀)) + 𝐵) = 𝐶       ((𝐴 · (10↑𝑁)) + 𝐵𝐷) = 𝐶𝐷

Theoremdecsplit0bOLD 15835 Obsolete version of decsplit0b 15831 as of 9-Sep-2021. (Contributed by Mario Carneiro, 16-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℕ0       ((𝐴 · (10↑0)) + 𝐵) = (𝐴 + 𝐵)

Theoremdecsplit0OLD 15836 Obsolete version of decsplit0 15832 as of 9-Sep-2021. (Contributed by Mario Carneiro, 16-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℕ0       ((𝐴 · (10↑0)) + 0) = 𝐴

Theoremdecsplit1OLD 15837 Obsolete version of decsplit1 15833 as of 9-Sep-2021. (Contributed by Mario Carneiro, 16-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℕ0       ((𝐴 · (10↑1)) + 𝐵) = 𝐴𝐵

TheoremdecsplitOLD 15838 Obsolete version of decsplit 15834 as of 9-Sep-2021. (Contributed by Mario Carneiro, 16-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (𝑀 + 1) = 𝑁    &   ((𝐴 · (10↑𝑀)) + 𝐵) = 𝐶       ((𝐴 · (10↑𝑁)) + 𝐵𝐷) = 𝐶𝐷

Theoremkaratsuba 15839 The Karatsuba multiplication algorithm. If 𝑋 and 𝑌 are decomposed into two groups of digits of length 𝑀 (only the lower group is known to be this size but the algorithm is most efficient when the partition is chosen near the middle of the digit string), then 𝑋𝑌 can be written in three groups of digits, where each group needs only one multiplication. Thus, we can halve both inputs with only three multiplications on the smaller operands, yielding an asymptotic improvement of n^(log2 3) instead of n^2 for the "naive" algorithm decmul1c 11625. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 9-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑆 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (𝐴 · 𝐶) = 𝑅    &   (𝐵 · 𝐷) = 𝑇    &   ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = ((𝑅 + 𝑆) + 𝑇)    &   ((𝐴 · (10↑𝑀)) + 𝐵) = 𝑋    &   ((𝐶 · (10↑𝑀)) + 𝐷) = 𝑌    &   ((𝑅 · (10↑𝑀)) + 𝑆) = 𝑊    &   ((𝑊 · (10↑𝑀)) + 𝑇) = 𝑍       (𝑋 · 𝑌) = 𝑍

TheoremkaratsubaOLD 15840 Obsolete version of karatsuba 15839 as of 9-Sep-2021. (Contributed by Mario Carneiro, 16-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑆 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (𝐴 · 𝐶) = 𝑅    &   (𝐵 · 𝐷) = 𝑇    &   ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = ((𝑅 + 𝑆) + 𝑇)    &   ((𝐴 · (10↑𝑀)) + 𝐵) = 𝑋    &   ((𝐶 · (10↑𝑀)) + 𝐷) = 𝑌    &   ((𝑅 · (10↑𝑀)) + 𝑆) = 𝑊    &   ((𝑊 · (10↑𝑀)) + 𝑇) = 𝑍       (𝑋 · 𝑌) = 𝑍

Theorem2exp4 15841 Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
(2↑4) = 16

Theorem2exp6 15842 Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
(2↑6) = 64

Theorem2exp8 15843 Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
(2↑8) = 256

Theorem2exp16 15844 Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
(2↑16) = 65536

Theorem3exp3 15845 Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.)
(3↑3) = 27

Theorem2expltfac 15846 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 15847 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 15848 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 15849* 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 15850* 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 15851* 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 15852* 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 15853* 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 15854* 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 15855* 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 15856* 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 15857* 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 15858* 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 15859* Lemma for prmlem1 15861 and prmlem2 15874. (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 15860* 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 15861 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 15862 5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
5 ∈ ℙ

Theorem6nprm 15863 6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
¬ 6 ∈ ℙ

Theorem7prm 15864 7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
7 ∈ ℙ

Theorem8nprm 15865 8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
¬ 8 ∈ ℙ

Theorem9nprm 15866 9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
¬ 9 ∈ ℙ

Theorem10nprm 15867 10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
¬ 10 ∈ ℙ

Theorem10nprmOLD 15868 Obsolete version of 10nprm 15867 as of 6-Sep-2021. (Contributed by Mario Carneiro, 18-Feb-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
¬ 10 ∈ ℙ

Theorem11prm 15869 11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
11 ∈ ℙ

Theorem13prm 15870 13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
13 ∈ ℙ

Theorem17prm 15871 17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
17 ∈ ℙ

Theorem19prm 15872 19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
19 ∈ ℙ

Theorem23prm 15873 23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
23 ∈ ℙ

Theoremprmlem2 15874 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 15890).

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 15875 37 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
37 ∈ ℙ

Theorem43prm 15876 43 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
43 ∈ ℙ

Theorem83prm 15877 83 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
83 ∈ ℙ

Theorem139prm 15878 139 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
139 ∈ ℙ

Theorem163prm 15879 163 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
163 ∈ ℙ

Theorem317prm 15880 317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
317 ∈ ℙ

Theorem631prm 15881 631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
631 ∈ ℙ

Theoremprmo4 15882 The primorial of 4. (Contributed by AV, 28-Aug-2020.)
(#p‘4) = 6

Theoremprmo5 15883 The primorial of 5. (Contributed by AV, 28-Aug-2020.)
(#p‘5) = 30

Theoremprmo6 15884 The primorial of 6. (Contributed by AV, 28-Aug-2020.)
(#p‘6) = 30

6.2.20  Very large primes

Theorem1259lem1 15885 Lemma for 1259prm 15890. 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 15886 Lemma for 1259prm 15890. 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 15887 Lemma for 1259prm 15890. 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 15888 Lemma for 1259prm 15890. 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 15889 Lemma for 1259prm 15890. 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 15890 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 15891 Lemma for 2503prm 15894. 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 15892 Lemma for 2503prm 15894. 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 15893 Lemma for 2503prm 15894. 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 15894 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 15895 Lemma for 4001prm 15899. 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 15896 Lemma for 4001prm 15899. 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 15897 Lemma for 4001prm 15899. 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 15898 Lemma for 4001prm 15899. 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 15899 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 15930 and strfv 15954. Using extractors makes it easier to change numeric indices and also makes the components' purpose clearer. For example, as noted in ndxid 15930, 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 15912. 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 19867 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 16155 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 16155 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 16293 and df-acs 16296. 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 15900 Extend class notation with the class of structures with components numbered below 𝐴.
class Struct

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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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