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Theorem List for Metamath Proof Explorer - 15601-15700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremprmgap 15601* 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 15602* Alternate proof of prmgap 15601: in contrast to prmgap 15601, 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 15603 Lemma for prmgapprmo 15604: 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 15604* Alternate proof of prmgap 15601: in contrast to prmgap 15601, 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 15605 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 15606 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ    &   𝐵 < 5        ¬ 5 ∥ 𝐴𝐵

Theoremdec5dvds2 15607 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 15608 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ        ¬ 𝐴5 ∈ ℙ

Theoremdec2nprm 15609 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 15610 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 15611 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 15612 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 15613 Version of mod2xi 15611 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 15614 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 15615 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 15616 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 15617 Calculate a power of two. (Contributed by Mario Carneiro, 19-Feb-2014.)
𝑀 ∈ ℕ0    &   (𝑀 + 2) = 𝑁       ((4 · (2↑𝑀)) + 0) = (2↑𝑁)

Theoremnumexp0 15618 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0       (𝐴↑0) = 1

Theoremnumexp1 15619 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0       (𝐴↑1) = 𝐴

Theoremnumexpp1 15620 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (𝑀 + 1) = 𝑁    &   ((𝐴𝑀) · 𝐴) = 𝐶       (𝐴𝑁) = 𝐶

Theoremnumexp2x 15621 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 15622 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 15623 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 15624 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 15625 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 15626 Obsolete version of decsplit0b 15622 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 15627 Obsolete version of decsplit0 15623 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 15628 Obsolete version of decsplit1 15624 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 15629 Obsolete version of decsplit 15625 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 15630 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 11463. (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 15631 Obsolete version of karatsuba 15630 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 15632 Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
(2↑4) = 16

Theorem2exp6 15633 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 15634 Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
(2↑8) = 256

Theorem2exp16 15635 Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
(2↑16) = 65536

Theorem3exp3 15636 Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.)
(3↑3) = 27

Theorem2expltfac 15637 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 15638 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 15639 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 15640* 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 15641* 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 15642* 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 15643* 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 15644* 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 15645* 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 15646* 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 15647* 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 15648* 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 15649* 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 15650* Lemma for prmlem1 15652 and prmlem2 15665. (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 15651* 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 15652 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 15653 5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
5 ∈ ℙ

Theorem6nprm 15654 6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
¬ 6 ∈ ℙ

Theorem7prm 15655 7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
7 ∈ ℙ

Theorem8nprm 15656 8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
¬ 8 ∈ ℙ

Theorem9nprm 15657 9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
¬ 9 ∈ ℙ

Theorem10nprm 15658 10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
¬ 10 ∈ ℙ

Theorem10nprmOLD 15659 Obsolete version of 10nprm 15658 as of 6-Sep-2021. (Contributed by Mario Carneiro, 18-Feb-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
¬ 10 ∈ ℙ

Theorem11prm 15660 11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
11 ∈ ℙ

Theorem13prm 15661 13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
13 ∈ ℙ

Theorem17prm 15662 17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
17 ∈ ℙ

Theorem19prm 15663 19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
19 ∈ ℙ

Theorem23prm 15664 23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
23 ∈ ℙ

Theoremprmlem2 15665 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 15681).

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 15666 37 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
37 ∈ ℙ

Theorem43prm 15667 43 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
43 ∈ ℙ

Theorem83prm 15668 83 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
83 ∈ ℙ

Theorem139prm 15669 139 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
139 ∈ ℙ

Theorem163prm 15670 163 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
163 ∈ ℙ

Theorem317prm 15671 317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
317 ∈ ℙ

Theorem631prm 15672 631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
631 ∈ ℙ

Theoremprmo4 15673 The primorial of 4. (Contributed by AV, 28-Aug-2020.)
(#p‘4) = 6

Theoremprmo5 15674 The primorial of 5. (Contributed by AV, 28-Aug-2020.)
(#p‘5) = 30

Theoremprmo6 15675 The primorial of 6. (Contributed by AV, 28-Aug-2020.)
(#p‘6) = 30

6.2.20  Very large primes

Theorem1259lem1 15676 Lemma for 1259prm 15681. 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 15677 Lemma for 1259prm 15681. 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 15678 Lemma for 1259prm 15681. 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 15679 Lemma for 1259prm 15681. 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 15680 Lemma for 1259prm 15681. 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 15681 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 15682 Lemma for 2503prm 15685. 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 15683 Lemma for 2503prm 15685. 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 15684 Lemma for 2503prm 15685. 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 15685 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 15686 Lemma for 4001prm 15690. 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 15687 Lemma for 4001prm 15690. 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 15688 Lemma for 4001prm 15690. 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 15689 Lemma for 4001prm 15690. 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 15690 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" 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 15716 and strfv 15735. Using extractors makes it easier to change numeric indices and also makes the components' purpose clearer. For example, as noted in ndxid 15716, 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 15702. 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 19638 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 15931 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 15931 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 16069 and df-acs 16072. 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", homs 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 15691 Extend class notation with the class of structures with components numbered below 𝐴.
class Struct

Syntaxcnx 15692 Extend class notation with the structure component index extractor.
class ndx

Syntaxcsts 15693 Set components of a structure.
class sSet

Syntaxcslot 15694 Extend class notation with the slot function.
class Slot 𝐴

Syntaxcbs 15695 Extend class notation with the class of all base set extractors.
class Base

Syntaxcress 15696 Extend class notation with the extensible structure builder restriction operator.
class s

Definitiondf-struct 15697* 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. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

Definitiondf-ndx 15698 Define the structure component index extractor. See theorem ndxarg 15715 to understand its purpose. The restriction to allows ndx to exist as a set, since I is a proper class. In principle, we could have chosen or (if we revise all structure component definitions such as df-base 15700) another set such as the natural ordinal numbers ω (df-om 6958). (Contributed by NM, 4-Sep-2011.)
ndx = ( I ↾ ℕ)

Definitiondf-slot 15699* Define slot extractor for posets and related structures. Note that the function argument can be any set, although it is meaningful only if it is a member of Poset (df-poset 16769) when used for posets or of Grp (df-grp 17248) when used from groups. (Contributed by Mario Carneiro, 22-Sep-2015.)
Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥𝐴))

Definitiondf-base 15700 Define the base set (also called underlying set or carrier set) extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Base = Slot 1

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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 42301-42360
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