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Theorem scshwfzeqfzo 13781
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 13520 . . . . . . . . . . . 12 (𝑋 ∈ Word 𝑉 → (♯‘𝑋) ∈ ℕ0)
2 elnn0uz 11932 . . . . . . . . . . . 12 ((♯‘𝑋) ∈ ℕ0 ↔ (♯‘𝑋) ∈ (ℤ‘0))
31, 2sylib 208 . . . . . . . . . . 11 (𝑋 ∈ Word 𝑉 → (♯‘𝑋) ∈ (ℤ‘0))
43adantr 466 . . . . . . . . . 10 ((𝑋 ∈ Word 𝑉𝑁 = (♯‘𝑋)) → (♯‘𝑋) ∈ (ℤ‘0))
5 eleq1 2838 . . . . . . . . . . 11 (𝑁 = (♯‘𝑋) → (𝑁 ∈ (ℤ‘0) ↔ (♯‘𝑋) ∈ (ℤ‘0)))
65adantl 467 . . . . . . . . . 10 ((𝑋 ∈ Word 𝑉𝑁 = (♯‘𝑋)) → (𝑁 ∈ (ℤ‘0) ↔ (♯‘𝑋) ∈ (ℤ‘0)))
74, 6mpbird 247 . . . . . . . . 9 ((𝑋 ∈ Word 𝑉𝑁 = (♯‘𝑋)) → 𝑁 ∈ (ℤ‘0))
873adant2 1125 . . . . . . . 8 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 𝑁 ∈ (ℤ‘0))
98adantr 466 . . . . . . 7 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → 𝑁 ∈ (ℤ‘0))
10 fzisfzounsn 12788 . . . . . . 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 3294 . . . . 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 3944 . . . . 5 (∃𝑛 ∈ ((0..^𝑁) ∪ {𝑁})𝑦 = (𝑋 cyclShift 𝑛) ↔ (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) ∨ ∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛)))
1412, 13syl6bb 276 . . . 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 6344 . . . . . . . . . . . 12 (♯‘𝑋) ∈ V
17 eleq1 2838 . . . . . . . . . . . 12 (𝑁 = (♯‘𝑋) → (𝑁 ∈ V ↔ (♯‘𝑋) ∈ V))
1816, 17mpbiri 248 . . . . . . . . . . 11 (𝑁 = (♯‘𝑋) → 𝑁 ∈ V)
19 oveq2 6804 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift 𝑁))
2019eqeq2d 2781 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
2120rexsng 4358 . . . . . . . . . . 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 1129 . . . . . . . . 9 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
2423adantr 466 . . . . . . . 8 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
25 oveq2 6804 . . . . . . . . . . . . 13 (𝑁 = (♯‘𝑋) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (♯‘𝑋)))
26253ad2ant3 1129 . . . . . . . . . . . 12 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (♯‘𝑋)))
27 cshwn 13752 . . . . . . . . . . . . 13 (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift (♯‘𝑋)) = 𝑋)
28273ad2ant1 1127 . . . . . . . . . . . 12 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift (♯‘𝑋)) = 𝑋)
2926, 28eqtrd 2805 . . . . . . . . . . 11 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 𝑁) = 𝑋)
3029eqeq2d 2781 . . . . . . . . . 10 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
3130adantr 466 . . . . . . . . 9 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
32 cshw0 13749 . . . . . . . . . . . . . . 15 (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift 0) = 𝑋)
33323ad2ant1 1127 . . . . . . . . . . . . . 14 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 0) = 𝑋)
34 lennncl 13521 . . . . . . . . . . . . . . . . . 18 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅) → (♯‘𝑋) ∈ ℕ)
35343adant3 1126 . . . . . . . . . . . . . . . . 17 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (♯‘𝑋) ∈ ℕ)
36 eleq1 2838 . . . . . . . . . . . . . . . . . 18 (𝑁 = (♯‘𝑋) → (𝑁 ∈ ℕ ↔ (♯‘𝑋) ∈ ℕ))
37363ad2ant3 1129 . . . . . . . . . . . . . . . . 17 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑁 ∈ ℕ ↔ (♯‘𝑋) ∈ ℕ))
3835, 37mpbird 247 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 𝑁 ∈ ℕ)
39 lbfzo0 12716 . . . . . . . . . . . . . . . 16 (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
4038, 39sylibr 224 . . . . . . . . . . . . . . 15 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 0 ∈ (0..^𝑁))
41 oveq2 6804 . . . . . . . . . . . . . . . . . . . 20 (0 = 𝑛 → (𝑋 cyclShift 0) = (𝑋 cyclShift 𝑛))
4241eqeq1d 2773 . . . . . . . . . . . . . . . . . . 19 (0 = 𝑛 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
4342eqcoms 2779 . . . . . . . . . . . . . . . . . 18 (𝑛 = 0 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
44 eqcom 2778 . . . . . . . . . . . . . . . . . 18 ((𝑋 cyclShift 𝑛) = 𝑋𝑋 = (𝑋 cyclShift 𝑛))
4543, 44syl6bb 276 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → ((𝑋 cyclShift 0) = 𝑋𝑋 = (𝑋 cyclShift 𝑛)))
4645adantl 467 . . . . . . . . . . . . . . . 16 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑛 = 0) → ((𝑋 cyclShift 0) = 𝑋𝑋 = (𝑋 cyclShift 𝑛)))
4746biimpd 219 . . . . . . . . . . . . . . 15 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑛 = 0) → ((𝑋 cyclShift 0) = 𝑋𝑋 = (𝑋 cyclShift 𝑛)))
4840, 47rspcimedv 3462 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . 12 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
5150adantr 466 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
52 eqeq1 2775 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
5352adantl 467 . . . . . . . . . . . 12 ((((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
5453rexbidv 3200 . . . . . . . . . . 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 247 . . . . . . . . . 10 ((((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛))
5655ex 397 . . . . . . . . 9 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = 𝑋 → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
5731, 56sylbid 230 . . . . . . . 8 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = (𝑋 cyclShift 𝑁) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
5824, 57sylbid 230 . . . . . . 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 846 . . . . 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 230 . . 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 12696 . . . 4 (0..^𝑁) ⊆ (0...𝑁)
64 ssrexv 3816 . . . 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 202 . 2 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
6766rabbidva 3338 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 196  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wrex 3062  {crab 3065  Vcvv 3351  cun 3721  wss 3723  c0 4063  {csn 4317  cfv 6030  (class class class)co 6796  0cc0 10142  cn 11226  0cn0 11499  cuz 11893  ...cfz 12533  ..^cfzo 12673  chash 13321  Word cword 13487   cyclShift ccsh 13743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8508  df-inf 8509  df-card 8969  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-n0 11500  df-z 11585  df-uz 11894  df-rp 12036  df-fz 12534  df-fzo 12674  df-fl 12801  df-mod 12877  df-hash 13322  df-word 13495  df-concat 13497  df-substr 13499  df-csh 13744
This theorem is referenced by:  hashecclwwlkn1  27235  umgrhashecclwwlk  27236
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