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Theorem scshwfzeqfzo 14040
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 13684 . . . . . . . . . . . 12 (𝑋 ∈ Word 𝑉 → (♯‘𝑋) ∈ ℕ0)
2 elnn0uz 12090 . . . . . . . . . . . 12 ((♯‘𝑋) ∈ ℕ0 ↔ (♯‘𝑋) ∈ (ℤ‘0))
31, 2sylib 210 . . . . . . . . . . 11 (𝑋 ∈ Word 𝑉 → (♯‘𝑋) ∈ (ℤ‘0))
43adantr 473 . . . . . . . . . 10 ((𝑋 ∈ Word 𝑉𝑁 = (♯‘𝑋)) → (♯‘𝑋) ∈ (ℤ‘0))
5 eleq1 2847 . . . . . . . . . . 11 (𝑁 = (♯‘𝑋) → (𝑁 ∈ (ℤ‘0) ↔ (♯‘𝑋) ∈ (ℤ‘0)))
65adantl 474 . . . . . . . . . 10 ((𝑋 ∈ Word 𝑉𝑁 = (♯‘𝑋)) → (𝑁 ∈ (ℤ‘0) ↔ (♯‘𝑋) ∈ (ℤ‘0)))
74, 6mpbird 249 . . . . . . . . 9 ((𝑋 ∈ Word 𝑉𝑁 = (♯‘𝑋)) → 𝑁 ∈ (ℤ‘0))
873adant2 1111 . . . . . . . 8 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 𝑁 ∈ (ℤ‘0))
98adantr 473 . . . . . . 7 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → 𝑁 ∈ (ℤ‘0))
10 fzisfzounsn 12957 . . . . . . 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 3350 . . . . 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 4050 . . . . 5 (∃𝑛 ∈ ((0..^𝑁) ∪ {𝑁})𝑦 = (𝑋 cyclShift 𝑛) ↔ (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) ∨ ∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛)))
1412, 13syl6bb 279 . . . 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 fvex 6506 . . . . . . . . . . . 12 (♯‘𝑋) ∈ V
16 eleq1 2847 . . . . . . . . . . . 12 (𝑁 = (♯‘𝑋) → (𝑁 ∈ V ↔ (♯‘𝑋) ∈ V))
1715, 16mpbiri 250 . . . . . . . . . . 11 (𝑁 = (♯‘𝑋) → 𝑁 ∈ V)
18 oveq2 6978 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift 𝑁))
1918eqeq2d 2782 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
2019rexsng 4482 . . . . . . . . . . 11 (𝑁 ∈ V → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
2117, 20syl 17 . . . . . . . . . 10 (𝑁 = (♯‘𝑋) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
22213ad2ant3 1115 . . . . . . . . 9 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
2322adantr 473 . . . . . . . 8 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
24 oveq2 6978 . . . . . . . . . . . . 13 (𝑁 = (♯‘𝑋) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (♯‘𝑋)))
25243ad2ant3 1115 . . . . . . . . . . . 12 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (♯‘𝑋)))
26 cshwn 14011 . . . . . . . . . . . . 13 (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift (♯‘𝑋)) = 𝑋)
27263ad2ant1 1113 . . . . . . . . . . . 12 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift (♯‘𝑋)) = 𝑋)
2825, 27eqtrd 2808 . . . . . . . . . . 11 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 𝑁) = 𝑋)
2928eqeq2d 2782 . . . . . . . . . 10 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
3029adantr 473 . . . . . . . . 9 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
31 cshw0 14008 . . . . . . . . . . . . . . 15 (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift 0) = 𝑋)
32313ad2ant1 1113 . . . . . . . . . . . . . 14 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 0) = 𝑋)
33 lennncl 13685 . . . . . . . . . . . . . . . . . 18 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅) → (♯‘𝑋) ∈ ℕ)
34333adant3 1112 . . . . . . . . . . . . . . . . 17 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (♯‘𝑋) ∈ ℕ)
35 eleq1 2847 . . . . . . . . . . . . . . . . . 18 (𝑁 = (♯‘𝑋) → (𝑁 ∈ ℕ ↔ (♯‘𝑋) ∈ ℕ))
36353ad2ant3 1115 . . . . . . . . . . . . . . . . 17 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑁 ∈ ℕ ↔ (♯‘𝑋) ∈ ℕ))
3734, 36mpbird 249 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 𝑁 ∈ ℕ)
38 lbfzo0 12885 . . . . . . . . . . . . . . . 16 (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
3937, 38sylibr 226 . . . . . . . . . . . . . . 15 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 0 ∈ (0..^𝑁))
40 oveq2 6978 . . . . . . . . . . . . . . . . . . . 20 (0 = 𝑛 → (𝑋 cyclShift 0) = (𝑋 cyclShift 𝑛))
4140eqeq1d 2774 . . . . . . . . . . . . . . . . . . 19 (0 = 𝑛 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
4241eqcoms 2780 . . . . . . . . . . . . . . . . . 18 (𝑛 = 0 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
43 eqcom 2779 . . . . . . . . . . . . . . . . . 18 ((𝑋 cyclShift 𝑛) = 𝑋𝑋 = (𝑋 cyclShift 𝑛))
4442, 43syl6bb 279 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → ((𝑋 cyclShift 0) = 𝑋𝑋 = (𝑋 cyclShift 𝑛)))
4544adantl 474 . . . . . . . . . . . . . . . 16 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑛 = 0) → ((𝑋 cyclShift 0) = 𝑋𝑋 = (𝑋 cyclShift 𝑛)))
4645biimpd 221 . . . . . . . . . . . . . . 15 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑛 = 0) → ((𝑋 cyclShift 0) = 𝑋𝑋 = (𝑋 cyclShift 𝑛)))
4739, 46rspcimedv 3531 . . . . . . . . . . . . . 14 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → ((𝑋 cyclShift 0) = 𝑋 → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛)))
4832, 47mpd 15 . . . . . . . . . . . . 13 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
4948adantr 473 . . . . . . . . . . . 12 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
5049adantr 473 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
51 eqeq1 2776 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
5251adantl 474 . . . . . . . . . . . 12 ((((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
5352rexbidv 3236 . . . . . . . . . . 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)))
5450, 53mpbird 249 . . . . . . . . . 10 ((((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛))
5554ex 405 . . . . . . . . 9 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = 𝑋 → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
5630, 55sylbid 232 . . . . . . . 8 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = (𝑋 cyclShift 𝑁) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
5723, 56sylbid 232 . . . . . . 7 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
5857com12 32 . . . . . 6 (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) → (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
5958jao1i 844 . . . . 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)))
6059com12 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)))
6114, 60sylbid 232 . . 3 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
62 fzossfz 12865 . . . 4 (0..^𝑁) ⊆ (0...𝑁)
63 ssrexv 3920 . . . 4 ((0..^𝑁) ⊆ (0...𝑁) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
6462, 63mp1i 13 . . 3 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
6561, 64impbid 204 . 2 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
6665rabbidva 3396 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 198  wa 387  wo 833  w3a 1068   = wceq 1507  wcel 2048  wne 2961  wrex 3083  {crab 3086  Vcvv 3409  cun 3823  wss 3825  c0 4173  {csn 4435  cfv 6182  (class class class)co 6970  0cc0 10327  cn 11431  0cn0 11700  cuz 12051  ...cfz 12701  ..^cfzo 12842  chash 13498  Word cword 13662   cyclShift ccsh 13997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404  ax-pre-sup 10405
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-oadd 7901  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-sup 8693  df-inf 8694  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-div 11091  df-nn 11432  df-n0 11701  df-z 11787  df-uz 12052  df-rp 12198  df-fz 12702  df-fzo 12843  df-fl 12970  df-mod 13046  df-hash 13499  df-word 13663  df-concat 13724  df-substr 13794  df-pfx 13843  df-csh 13999
This theorem is referenced by:  hashecclwwlkn1  27591  umgrhashecclwwlk  27592
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