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Theorem scshwfzeqfzo 13371
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 13127 . . . . . . . . . . . 12 (𝑋 ∈ Word 𝑉 → (#‘𝑋) ∈ ℕ0)
2 elnn0uz 11559 . . . . . . . . . . . 12 ((#‘𝑋) ∈ ℕ0 ↔ (#‘𝑋) ∈ (ℤ‘0))
31, 2sylib 206 . . . . . . . . . . 11 (𝑋 ∈ Word 𝑉 → (#‘𝑋) ∈ (ℤ‘0))
43adantr 479 . . . . . . . . . 10 ((𝑋 ∈ Word 𝑉𝑁 = (#‘𝑋)) → (#‘𝑋) ∈ (ℤ‘0))
5 eleq1 2675 . . . . . . . . . . 11 (𝑁 = (#‘𝑋) → (𝑁 ∈ (ℤ‘0) ↔ (#‘𝑋) ∈ (ℤ‘0)))
65adantl 480 . . . . . . . . . 10 ((𝑋 ∈ Word 𝑉𝑁 = (#‘𝑋)) → (𝑁 ∈ (ℤ‘0) ↔ (#‘𝑋) ∈ (ℤ‘0)))
74, 6mpbird 245 . . . . . . . . 9 ((𝑋 ∈ Word 𝑉𝑁 = (#‘𝑋)) → 𝑁 ∈ (ℤ‘0))
873adant2 1072 . . . . . . . 8 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → 𝑁 ∈ (ℤ‘0))
98adantr 479 . . . . . . 7 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → 𝑁 ∈ (ℤ‘0))
10 fzisfzounsn 12402 . . . . . . 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 3121 . . . . 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 3754 . . . . 5 (∃𝑛 ∈ ((0..^𝑁) ∪ {𝑁})𝑦 = (𝑋 cyclShift 𝑛) ↔ (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) ∨ ∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛)))
1412, 13syl6bb 274 . . . 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 6097 . . . . . . . . . . . 12 (#‘𝑋) ∈ V
17 eleq1 2675 . . . . . . . . . . . 12 (𝑁 = (#‘𝑋) → (𝑁 ∈ V ↔ (#‘𝑋) ∈ V))
1816, 17mpbiri 246 . . . . . . . . . . 11 (𝑁 = (#‘𝑋) → 𝑁 ∈ V)
19 oveq2 6534 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift 𝑁))
2019eqeq2d 2619 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
2120rexsng 4165 . . . . . . . . . . 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 1076 . . . . . . . . 9 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
2423adantr 479 . . . . . . . 8 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
25 oveq2 6534 . . . . . . . . . . . . 13 (𝑁 = (#‘𝑋) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (#‘𝑋)))
26253ad2ant3 1076 . . . . . . . . . . . 12 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (#‘𝑋)))
27 cshwn 13342 . . . . . . . . . . . . 13 (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift (#‘𝑋)) = 𝑋)
28273ad2ant1 1074 . . . . . . . . . . . 12 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑋 cyclShift (#‘𝑋)) = 𝑋)
2926, 28eqtrd 2643 . . . . . . . . . . 11 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑋 cyclShift 𝑁) = 𝑋)
3029eqeq2d 2619 . . . . . . . . . 10 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
3130adantr 479 . . . . . . . . 9 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
32 cshw0 13339 . . . . . . . . . . . . . . 15 (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift 0) = 𝑋)
33323ad2ant1 1074 . . . . . . . . . . . . . 14 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑋 cyclShift 0) = 𝑋)
34 lennncl 13128 . . . . . . . . . . . . . . . . . 18 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅) → (#‘𝑋) ∈ ℕ)
35343adant3 1073 . . . . . . . . . . . . . . . . 17 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (#‘𝑋) ∈ ℕ)
36 eleq1 2675 . . . . . . . . . . . . . . . . . 18 (𝑁 = (#‘𝑋) → (𝑁 ∈ ℕ ↔ (#‘𝑋) ∈ ℕ))
37363ad2ant3 1076 . . . . . . . . . . . . . . . . 17 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑁 ∈ ℕ ↔ (#‘𝑋) ∈ ℕ))
3835, 37mpbird 245 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → 𝑁 ∈ ℕ)
39 lbfzo0 12332 . . . . . . . . . . . . . . . 16 (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
4038, 39sylibr 222 . . . . . . . . . . . . . . 15 ((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → 0 ∈ (0..^𝑁))
41 oveq2 6534 . . . . . . . . . . . . . . . . . . . 20 (0 = 𝑛 → (𝑋 cyclShift 0) = (𝑋 cyclShift 𝑛))
4241eqeq1d 2611 . . . . . . . . . . . . . . . . . . 19 (0 = 𝑛 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
4342eqcoms 2617 . . . . . . . . . . . . . . . . . 18 (𝑛 = 0 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
44 eqcom 2616 . . . . . . . . . . . . . . . . . 18 ((𝑋 cyclShift 𝑛) = 𝑋𝑋 = (𝑋 cyclShift 𝑛))
4543, 44syl6bb 274 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → ((𝑋 cyclShift 0) = 𝑋𝑋 = (𝑋 cyclShift 𝑛)))
4645adantl 480 . . . . . . . . . . . . . . . 16 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑛 = 0) → ((𝑋 cyclShift 0) = 𝑋𝑋 = (𝑋 cyclShift 𝑛)))
4746biimpd 217 . . . . . . . . . . . . . . 15 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑛 = 0) → ((𝑋 cyclShift 0) = 𝑋𝑋 = (𝑋 cyclShift 𝑛)))
4840, 47rspcimedv 3283 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . 12 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
5150adantr 479 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
52 eqeq1 2613 . . . . . . . . . . . . 13 (𝑦 = 𝑋 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
5352adantl 480 . . . . . . . . . . . 12 ((((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
5453rexbidv 3033 . . . . . . . . . . 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 245 . . . . . . . . . 10 ((((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛))
5655ex 448 . . . . . . . . 9 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = 𝑋 → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
5731, 56sylbid 228 . . . . . . . 8 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = (𝑋 cyclShift 𝑁) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
5824, 57sylbid 228 . . . . . . 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 392 . . . . 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 228 . . 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 12314 . . . 4 (0..^𝑁) ⊆ (0...𝑁)
64 ssrexv 3629 . . . 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 200 . 2 (((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
6766rabbidva 3162 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 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wrex 2896  {crab 2899  Vcvv 3172  cun 3537  wss 3539  c0 3873  {csn 4124  cfv 5789  (class class class)co 6526  0cc0 9792  cn 10869  0cn0 11141  cuz 11521  ...cfz 12154  ..^cfzo 12291  #chash 12936  Word cword 13094   cyclShift ccsh 13333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
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 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-inf 8209  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10536  df-nn 10870  df-n0 11142  df-z 11213  df-uz 11522  df-rp 11667  df-fz 12155  df-fzo 12292  df-fl 12412  df-mod 12488  df-hash 12937  df-word 13102  df-concat 13104  df-substr 13106  df-csh 13334
This theorem is referenced by:  hashecclwwlkn1  26154  usghashecclwwlk  26155  hashecclwwlksn1  41242  umgrhashecclwwlk  41243
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