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Theorem scshwfzeqfzo 13423
 Description: For a nonempty word the sets of shifted words, expressd by a finite interval of integers or by a half-open integer range are identical. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
Assertion
Ref Expression
scshwfzeqfzo ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)})
Distinct variable groups:   𝑛,𝑁,𝑦   𝑛,𝑉,𝑦   𝑛,𝑋,𝑦

Proof of Theorem scshwfzeqfzo
StepHypRef Expression
1 lencl 13179 . . . . . . . . . . . 12 (𝑋 ∈ Word 𝑉 → (#‘𝑋) ∈ ℕ0)
2 elnn0uz 11601 . . . . . . . . . . . 12 ((#‘𝑋) ∈ ℕ0 ↔ (#‘𝑋) ∈ (ℤ‘0))
31, 2sylib 207 . . . . . . . . . . 11 (𝑋 ∈ Word 𝑉 → (#‘𝑋) ∈ (ℤ‘0))
43adantr 480 . . . . . . . . . 10 ((𝑋 ∈ Word 𝑉𝑁 = (#‘𝑋)) → (#‘𝑋) ∈ (ℤ‘0))
5 eleq1 2676 . . . . . . . . . . 11 (𝑁 = (#‘𝑋) → (𝑁 ∈ (ℤ‘0) ↔ (#‘𝑋) ∈ (ℤ‘0)))
65adantl 481 . . . . . . . . . 10 ((𝑋 ∈ Word 𝑉𝑁 = (#‘𝑋)) → (𝑁 ∈ (ℤ‘0) ↔ (#‘𝑋) ∈ (ℤ‘0)))
74, 6mpbird 246 . . . . . . . . 9 ((𝑋 ∈ Word 𝑉𝑁 = (#‘𝑋)) → 𝑁 ∈ (ℤ‘0))
873adant2 1073 . . . . . . . 8 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → 𝑁 ∈ (ℤ‘0))
98adantr 480 . . . . . . 7 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → 𝑁 ∈ (ℤ‘0))
10 fzisfzounsn 12445 . . . . . . 7 (𝑁 ∈ (ℤ‘0) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
119, 10syl 17 . . . . . 6 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
1211rexeqdv 3122 . . . . 5 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ ((0..^𝑁) ∪ {𝑁})𝑦 = (𝑋 cyclShift 𝑛)))
13 rexun 3755 . . . . 5 (∃𝑛 ∈ ((0..^𝑁) ∪ {𝑁})𝑦 = (𝑋 cyclShift 𝑛) ↔ (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) ∨ ∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛)))
1412, 13syl6bb 275 . . . 4 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛) ↔ (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) ∨ ∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛))))
15 ax-1 6 . . . . . 6 (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) → (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
16 fvex 6113 . . . . . . . . . . . 12 (#‘𝑋) ∈ V
17 eleq1 2676 . . . . . . . . . . . 12 (𝑁 = (#‘𝑋) → (𝑁 ∈ V ↔ (#‘𝑋) ∈ V))
1816, 17mpbiri 247 . . . . . . . . . . 11 (𝑁 = (#‘𝑋) → 𝑁 ∈ V)
19 oveq2 6557 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift 𝑁))
2019eqeq2d 2620 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
2120rexsng 4166 . . . . . . . . . . 11 (𝑁 ∈ V → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
2218, 21syl 17 . . . . . . . . . 10 (𝑁 = (#‘𝑋) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
23223ad2ant3 1077 . . . . . . . . 9 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
2423adantr 480 . . . . . . . 8 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
25 oveq2 6557 . . . . . . . . . . . . 13 (𝑁 = (#‘𝑋) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (#‘𝑋)))
26253ad2ant3 1077 . . . . . . . . . . . 12 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (#‘𝑋)))
27 cshwn 13394 . . . . . . . . . . . . 13 (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift (#‘𝑋)) = 𝑋)
28273ad2ant1 1075 . . . . . . . . . . . 12 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑋 cyclShift (#‘𝑋)) = 𝑋)
2926, 28eqtrd 2644 . . . . . . . . . . 11 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑋 cyclShift 𝑁) = 𝑋)
3029eqeq2d 2620 . . . . . . . . . 10 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
3130adantr 480 . . . . . . . . 9 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
32 cshw0 13391 . . . . . . . . . . . . . . 15 (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift 0) = 𝑋)
33323ad2ant1 1075 . . . . . . . . . . . . . 14 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑋 cyclShift 0) = 𝑋)
34 lennncl 13180 . . . . . . . . . . . . . . . . . 18 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅) → (#‘𝑋) ∈ ℕ)
35343adant3 1074 . . . . . . . . . . . . . . . . 17 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (#‘𝑋) ∈ ℕ)
36 eleq1 2676 . . . . . . . . . . . . . . . . . 18 (𝑁 = (#‘𝑋) → (𝑁 ∈ ℕ ↔ (#‘𝑋) ∈ ℕ))
37363ad2ant3 1077 . . . . . . . . . . . . . . . . 17 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑁 ∈ ℕ ↔ (#‘𝑋) ∈ ℕ))
3835, 37mpbird 246 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → 𝑁 ∈ ℕ)
39 lbfzo0 12375 . . . . . . . . . . . . . . . 16 (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
4038, 39sylibr 223 . . . . . . . . . . . . . . 15 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → 0 ∈ (0..^𝑁))
41 oveq2 6557 . . . . . . . . . . . . . . . . . . . 20 (0 = 𝑛 → (𝑋 cyclShift 0) = (𝑋 cyclShift 𝑛))
4241eqeq1d 2612 . . . . . . . . . . . . . . . . . . 19 (0 = 𝑛 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
4342eqcoms 2618 . . . . . . . . . . . . . . . . . 18 (𝑛 = 0 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
44 eqcom 2617 . . . . . . . . . . . . . . . . . 18 ((𝑋 cyclShift 𝑛) = 𝑋𝑋 = (𝑋 cyclShift 𝑛))
4543, 44syl6bb 275 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → ((𝑋 cyclShift 0) = 𝑋𝑋 = (𝑋 cyclShift 𝑛)))
4645adantl 481 . . . . . . . . . . . . . . . 16 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑛 = 0) → ((𝑋 cyclShift 0) = 𝑋𝑋 = (𝑋 cyclShift 𝑛)))
4746biimpd 218 . . . . . . . . . . . . . . 15 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑛 = 0) → ((𝑋 cyclShift 0) = 𝑋𝑋 = (𝑋 cyclShift 𝑛)))
4840, 47rspcimedv 3284 . . . . . . . . . . . . . 14 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → ((𝑋 cyclShift 0) = 𝑋 → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛)))
4933, 48mpd 15 . . . . . . . . . . . . 13 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
5049adantr 480 . . . . . . . . . . . 12 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
5150adantr 480 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
52 eqeq1 2614 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
5352adantl 481 . . . . . . . . . . . 12 ((((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
5453rexbidv 3034 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛)))
5551, 54mpbird 246 . . . . . . . . . 10 ((((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛))
5655ex 449 . . . . . . . . 9 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = 𝑋 → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
5731, 56sylbid 229 . . . . . . . 8 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = (𝑋 cyclShift 𝑁) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
5824, 57sylbid 229 . . . . . . 7 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
5958com12 32 . . . . . 6 (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) → (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
6015, 59jaoi 393 . . . . 5 ((∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) ∨ ∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛)) → (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
6160com12 32 . . . 4 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ((∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) ∨ ∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛)) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
6214, 61sylbid 229 . . 3 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
63 fzossfz 12357 . . . 4 (0..^𝑁) ⊆ (0...𝑁)
64 ssrexv 3630 . . . 4 ((0..^𝑁) ⊆ (0...𝑁) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
6563, 64mp1i 13 . . 3 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
6662, 65impbid 201 . 2 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
6766rabbidva 3163 1 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℕcn 10897  ℕ0cn0 11169  ℤ≥cuz 11563  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   cyclShift ccsh 13385 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-csh 13386 This theorem is referenced by:  hashecclwwlkn1  26361  usghashecclwwlk  26362  hashecclwwlksn1  41261  umgrhashecclwwlk  41262
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