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Theorem prmlem2 15607
Description: 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 15623).

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.)

Hypotheses
Ref Expression
prmlem2.n 𝑁 ∈ ℕ
prmlem2.lt 𝑁 < 841
prmlem2.gt 1 < 𝑁
prmlem2.2 ¬ 2 ∥ 𝑁
prmlem2.3 ¬ 3 ∥ 𝑁
prmlem2.5 ¬ 5 ∥ 𝑁
prmlem2.7 ¬ 7 ∥ 𝑁
prmlem2.11 ¬ 11 ∥ 𝑁
prmlem2.13 ¬ 13 ∥ 𝑁
prmlem2.17 ¬ 17 ∥ 𝑁
prmlem2.19 ¬ 19 ∥ 𝑁
prmlem2.23 ¬ 23 ∥ 𝑁
Assertion
Ref Expression
prmlem2 𝑁 ∈ ℙ

Proof of Theorem prmlem2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 prmlem2.n . 2 𝑁 ∈ ℕ
2 prmlem2.gt . 2 1 < 𝑁
3 prmlem2.2 . 2 ¬ 2 ∥ 𝑁
4 prmlem2.3 . 2 ¬ 3 ∥ 𝑁
5 eluzelre 11526 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (ℤ29) → 𝑥 ∈ ℝ)
65resqcld 12848 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (ℤ29) → (𝑥↑2) ∈ ℝ)
7 eluzle 11528 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (ℤ29) → 29 ≤ 𝑥)
8 2nn0 11152 . . . . . . . . . . . . . . . . . . . . . . 23 2 ∈ ℕ0
9 9nn0 11159 . . . . . . . . . . . . . . . . . . . . . . 23 9 ∈ ℕ0
108, 9deccl 11340 . . . . . . . . . . . . . . . . . . . . . 22 29 ∈ ℕ0
1110nn0rei 11146 . . . . . . . . . . . . . . . . . . . . 21 29 ∈ ℝ
1210nn0ge0i 11163 . . . . . . . . . . . . . . . . . . . . 21 0 ≤ 29
13 le2sq2 12752 . . . . . . . . . . . . . . . . . . . . 21 (((29 ∈ ℝ ∧ 0 ≤ 29) ∧ (𝑥 ∈ ℝ ∧ 29 ≤ 𝑥)) → (29↑2) ≤ (𝑥↑2))
1411, 12, 13mpanl12 713 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℝ ∧ 29 ≤ 𝑥) → (29↑2) ≤ (𝑥↑2))
155, 7, 14syl2anc 690 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (ℤ29) → (29↑2) ≤ (𝑥↑2))
161nnrei 10872 . . . . . . . . . . . . . . . . . . . 20 𝑁 ∈ ℝ
1711resqcli 12762 . . . . . . . . . . . . . . . . . . . 20 (29↑2) ∈ ℝ
18 prmlem2.lt . . . . . . . . . . . . . . . . . . . . . 22 𝑁 < 841
1910nn0cni 11147 . . . . . . . . . . . . . . . . . . . . . . . 24 29 ∈ ℂ
2019sqvali 12756 . . . . . . . . . . . . . . . . . . . . . . 23 (29↑2) = (29 · 29)
21 eqid 2605 . . . . . . . . . . . . . . . . . . . . . . . 24 29 = 29
22 1nn0 11151 . . . . . . . . . . . . . . . . . . . . . . . 24 1 ∈ ℕ0
23 6nn0 11156 . . . . . . . . . . . . . . . . . . . . . . . . 25 6 ∈ ℕ0
248, 23deccl 11340 . . . . . . . . . . . . . . . . . . . . . . . 24 26 ∈ ℕ0
25 5nn0 11155 . . . . . . . . . . . . . . . . . . . . . . . . 25 5 ∈ ℕ0
26 8nn0 11158 . . . . . . . . . . . . . . . . . . . . . . . . 25 8 ∈ ℕ0
27192timesi 10990 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (2 · 29) = (29 + 29)
28 2p2e4 10987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (2 + 2) = 4
2928oveq1i 6533 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((2 + 2) + 1) = (4 + 1)
30 4p1e5 10997 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (4 + 1) = 5
3129, 30eqtri 2627 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((2 + 2) + 1) = 5
32 9p9e18 11455 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (9 + 9) = 18
338, 9, 8, 9, 21, 21, 31, 26, 32decaddc 11400 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (29 + 29) = 58
3427, 33eqtri 2627 . . . . . . . . . . . . . . . . . . . . . . . . 25 (2 · 29) = 58
35 eqid 2605 . . . . . . . . . . . . . . . . . . . . . . . . 25 26 = 26
36 5p2e7 11008 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (5 + 2) = 7
3736oveq1i 6533 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((5 + 2) + 1) = (7 + 1)
38 7p1e8 11000 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (7 + 1) = 8
3937, 38eqtri 2627 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((5 + 2) + 1) = 8
40 4nn0 11154 . . . . . . . . . . . . . . . . . . . . . . . . 25 4 ∈ ℕ0
41 8p6e14 11444 . . . . . . . . . . . . . . . . . . . . . . . . 25 (8 + 6) = 14
4225, 26, 8, 23, 34, 35, 39, 40, 41decaddc 11400 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2 · 29) + 26) = 84
43 9t2e18 11491 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (9 · 2) = 18
44 1p1e2 10977 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (1 + 1) = 2
45 8p8e16 11446 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (8 + 8) = 16
4622, 26, 26, 43, 44, 23, 45decaddci 11408 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((9 · 2) + 8) = 26
47 9t9e81 11498 . . . . . . . . . . . . . . . . . . . . . . . . 25 (9 · 9) = 81
489, 8, 9, 21, 22, 26, 46, 47decmul2c 11417 . . . . . . . . . . . . . . . . . . . . . . . 24 (9 · 29) = 261
4910, 8, 9, 21, 22, 24, 42, 48decmul1c 11415 . . . . . . . . . . . . . . . . . . . . . . 23 (29 · 29) = 841
5020, 49eqtri 2627 . . . . . . . . . . . . . . . . . . . . . 22 (29↑2) = 841
5118, 50breqtrri 4600 . . . . . . . . . . . . . . . . . . . . 21 𝑁 < (29↑2)
52 ltletr 9976 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℝ ∧ (29↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → ((𝑁 < (29↑2) ∧ (29↑2) ≤ (𝑥↑2)) → 𝑁 < (𝑥↑2)))
5351, 52mpani 707 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℝ ∧ (29↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → ((29↑2) ≤ (𝑥↑2) → 𝑁 < (𝑥↑2)))
5416, 17, 53mp3an12 1405 . . . . . . . . . . . . . . . . . . 19 ((𝑥↑2) ∈ ℝ → ((29↑2) ≤ (𝑥↑2) → 𝑁 < (𝑥↑2)))
556, 15, 54sylc 62 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (ℤ29) → 𝑁 < (𝑥↑2))
56 ltnle 9964 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → (𝑁 < (𝑥↑2) ↔ ¬ (𝑥↑2) ≤ 𝑁))
5716, 6, 56sylancr 693 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (ℤ29) → (𝑁 < (𝑥↑2) ↔ ¬ (𝑥↑2) ≤ 𝑁))
5855, 57mpbid 220 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℤ29) → ¬ (𝑥↑2) ≤ 𝑁)
5958pm2.21d 116 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (ℤ29) → ((𝑥↑2) ≤ 𝑁 → ¬ 𝑥𝑁))
6059adantld 481 . . . . . . . . . . . . . . 15 (𝑥 ∈ (ℤ29) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
6160adantl 480 . . . . . . . . . . . . . 14 ((¬ 2 ∥ 29 ∧ 𝑥 ∈ (ℤ29)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
62 9nn 11035 . . . . . . . . . . . . . . . 16 9 ∈ ℕ
63 3nn 11029 . . . . . . . . . . . . . . . 16 3 ∈ ℕ
64 1lt9 11072 . . . . . . . . . . . . . . . 16 1 < 9
65 1lt3 11039 . . . . . . . . . . . . . . . 16 1 < 3
66 9t3e27 11492 . . . . . . . . . . . . . . . 16 (9 · 3) = 27
6762, 63, 64, 65, 66nprmi 15182 . . . . . . . . . . . . . . 15 ¬ 27 ∈ ℙ
6867pm2.21i 114 . . . . . . . . . . . . . 14 (27 ∈ ℙ → ¬ 27 ∥ 𝑁)
69 7nn0 11157 . . . . . . . . . . . . . . 15 7 ∈ ℕ0
70 eqid 2605 . . . . . . . . . . . . . . 15 27 = 27
71 7p2e9 11015 . . . . . . . . . . . . . . 15 (7 + 2) = 9
728, 69, 8, 70, 71decaddi 11407 . . . . . . . . . . . . . 14 (27 + 2) = 29
7361, 68, 72prmlem0 15592 . . . . . . . . . . . . 13 ((¬ 2 ∥ 27 ∧ 𝑥 ∈ (ℤ27)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
74 5nn 11031 . . . . . . . . . . . . . . 15 5 ∈ ℕ
75 1lt5 11046 . . . . . . . . . . . . . . 15 1 < 5
76 5t5e25 11467 . . . . . . . . . . . . . . 15 (5 · 5) = 25
7774, 74, 75, 75, 76nprmi 15182 . . . . . . . . . . . . . 14 ¬ 25 ∈ ℙ
7877pm2.21i 114 . . . . . . . . . . . . 13 (25 ∈ ℙ → ¬ 25 ∥ 𝑁)
79 eqid 2605 . . . . . . . . . . . . . 14 25 = 25
808, 25, 8, 79, 36decaddi 11407 . . . . . . . . . . . . 13 (25 + 2) = 27
8173, 78, 80prmlem0 15592 . . . . . . . . . . . 12 ((¬ 2 ∥ 25 ∧ 𝑥 ∈ (ℤ25)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
82 prmlem2.23 . . . . . . . . . . . . 13 ¬ 23 ∥ 𝑁
8382a1i 11 . . . . . . . . . . . 12 (23 ∈ ℙ → ¬ 23 ∥ 𝑁)
84 3nn0 11153 . . . . . . . . . . . . 13 3 ∈ ℕ0
85 eqid 2605 . . . . . . . . . . . . 13 23 = 23
86 3p2e5 11003 . . . . . . . . . . . . 13 (3 + 2) = 5
878, 84, 8, 85, 86decaddi 11407 . . . . . . . . . . . 12 (23 + 2) = 25
8881, 83, 87prmlem0 15592 . . . . . . . . . . 11 ((¬ 2 ∥ 23 ∧ 𝑥 ∈ (ℤ23)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
89 7nn 11033 . . . . . . . . . . . . 13 7 ∈ ℕ
90 1lt7 11057 . . . . . . . . . . . . 13 1 < 7
91 7t3e21 11477 . . . . . . . . . . . . 13 (7 · 3) = 21
9289, 63, 90, 65, 91nprmi 15182 . . . . . . . . . . . 12 ¬ 21 ∈ ℙ
9392pm2.21i 114 . . . . . . . . . . 11 (21 ∈ ℙ → ¬ 21 ∥ 𝑁)
94 eqid 2605 . . . . . . . . . . . 12 21 = 21
95 1p2e3 10995 . . . . . . . . . . . 12 (1 + 2) = 3
968, 22, 8, 94, 95decaddi 11407 . . . . . . . . . . 11 (21 + 2) = 23
9788, 93, 96prmlem0 15592 . . . . . . . . . 10 ((¬ 2 ∥ 21 ∧ 𝑥 ∈ (ℤ21)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
98 prmlem2.19 . . . . . . . . . . 11 ¬ 19 ∥ 𝑁
9998a1i 11 . . . . . . . . . 10 (19 ∈ ℙ → ¬ 19 ∥ 𝑁)
100 eqid 2605 . . . . . . . . . . 11 19 = 19
101 9p2e11 11447 . . . . . . . . . . 11 (9 + 2) = 11
10222, 9, 8, 100, 44, 22, 101decaddci 11408 . . . . . . . . . 10 (19 + 2) = 21
10397, 99, 102prmlem0 15592 . . . . . . . . 9 ((¬ 2 ∥ 19 ∧ 𝑥 ∈ (ℤ19)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
104 prmlem2.17 . . . . . . . . . 10 ¬ 17 ∥ 𝑁
105104a1i 11 . . . . . . . . 9 (17 ∈ ℙ → ¬ 17 ∥ 𝑁)
106 eqid 2605 . . . . . . . . . 10 17 = 17
10722, 69, 8, 106, 71decaddi 11407 . . . . . . . . 9 (17 + 2) = 19
108103, 105, 107prmlem0 15592 . . . . . . . 8 ((¬ 2 ∥ 17 ∧ 𝑥 ∈ (ℤ17)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
109 5t3e15 11463 . . . . . . . . . 10 (5 · 3) = 15
11074, 63, 75, 65, 109nprmi 15182 . . . . . . . . 9 ¬ 15 ∈ ℙ
111110pm2.21i 114 . . . . . . . 8 (15 ∈ ℙ → ¬ 15 ∥ 𝑁)
112 eqid 2605 . . . . . . . . 9 15 = 15
11322, 25, 8, 112, 36decaddi 11407 . . . . . . . 8 (15 + 2) = 17
114108, 111, 113prmlem0 15592 . . . . . . 7 ((¬ 2 ∥ 15 ∧ 𝑥 ∈ (ℤ15)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
115 prmlem2.13 . . . . . . . 8 ¬ 13 ∥ 𝑁
116115a1i 11 . . . . . . 7 (13 ∈ ℙ → ¬ 13 ∥ 𝑁)
117 eqid 2605 . . . . . . . 8 13 = 13
11822, 84, 8, 117, 86decaddi 11407 . . . . . . 7 (13 + 2) = 15
119114, 116, 118prmlem0 15592 . . . . . 6 ((¬ 2 ∥ 13 ∧ 𝑥 ∈ (ℤ13)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
120 prmlem2.11 . . . . . . 7 ¬ 11 ∥ 𝑁
121120a1i 11 . . . . . 6 (11 ∈ ℙ → ¬ 11 ∥ 𝑁)
122 eqid 2605 . . . . . . 7 11 = 11
12322, 22, 8, 122, 95decaddi 11407 . . . . . 6 (11 + 2) = 13
124119, 121, 123prmlem0 15592 . . . . 5 ((¬ 2 ∥ 11 ∧ 𝑥 ∈ (ℤ11)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
125 9nprm 15599 . . . . . 6 ¬ 9 ∈ ℙ
126125pm2.21i 114 . . . . 5 (9 ∈ ℙ → ¬ 9 ∥ 𝑁)
127124, 126, 101prmlem0 15592 . . . 4 ((¬ 2 ∥ 9 ∧ 𝑥 ∈ (ℤ‘9)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
128 prmlem2.7 . . . . 5 ¬ 7 ∥ 𝑁
129128a1i 11 . . . 4 (7 ∈ ℙ → ¬ 7 ∥ 𝑁)
130127, 129, 71prmlem0 15592 . . 3 ((¬ 2 ∥ 7 ∧ 𝑥 ∈ (ℤ‘7)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
131 prmlem2.5 . . . 4 ¬ 5 ∥ 𝑁
132131a1i 11 . . 3 (5 ∈ ℙ → ¬ 5 ∥ 𝑁)
133130, 132, 36prmlem0 15592 . 2 ((¬ 2 ∥ 5 ∧ 𝑥 ∈ (ℤ‘5)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
1341, 2, 3, 4, 133prmlem1a 15593 1 𝑁 ∈ ℙ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030  wcel 1975  cdif 3532  {csn 4120   class class class wbr 4573  cfv 5786  (class class class)co 6523  cr 9787  0cc0 9788  1c1 9789   + caddc 9791   · cmul 9793   < clt 9926  cle 9927  cn 10863  2c2 10913  3c3 10914  4c4 10915  5c5 10916  6c6 10917  7c7 10918  8c8 10919  9c9 10920  cdc 11321  cuz 11515  cexp 12673  cdvds 14763  cprime 15165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865  ax-pre-sup 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-sup 8204  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-2 10922  df-3 10923  df-4 10924  df-5 10925  df-6 10926  df-7 10927  df-8 10928  df-9 10929  df-n0 11136  df-z 11207  df-dec 11322  df-uz 11516  df-rp 11661  df-fz 12149  df-seq 12615  df-exp 12674  df-cj 13629  df-re 13630  df-im 13631  df-sqrt 13765  df-abs 13766  df-dvds 14764  df-prm 15166
This theorem is referenced by:  37prm  15608  43prm  15609  83prm  15610  139prm  15611  163prm  15612  317prm  15613  631prm  15614  257prm  39812  139prmALT  39850  127prm  39854
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