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Theorem erclwwlkseq 41239
 Description: Two classes are equivalent regarding ∼ if both are words and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 29-Apr-2021.)
Hypothesis
Ref Expression
erclwwlks.r = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalkS‘𝐺) ∧ 𝑤 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlkseq ((𝑈𝑋𝑊𝑌) → (𝑈 𝑊 ↔ (𝑈 ∈ (ClWWalkS‘𝐺) ∧ 𝑊 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛))))
Distinct variable groups:   𝑛,𝐺,𝑢,𝑤   𝑈,𝑛,𝑢,𝑤   𝑛,𝑊,𝑢,𝑤
Allowed substitution hints:   (𝑤,𝑢,𝑛)   𝑋(𝑤,𝑢,𝑛)   𝑌(𝑤,𝑢,𝑛)

Proof of Theorem erclwwlkseq
StepHypRef Expression
1 eleq1 2676 . . . 4 (𝑢 = 𝑈 → (𝑢 ∈ (ClWWalkS‘𝐺) ↔ 𝑈 ∈ (ClWWalkS‘𝐺)))
21adantr 480 . . 3 ((𝑢 = 𝑈𝑤 = 𝑊) → (𝑢 ∈ (ClWWalkS‘𝐺) ↔ 𝑈 ∈ (ClWWalkS‘𝐺)))
3 eleq1 2676 . . . 4 (𝑤 = 𝑊 → (𝑤 ∈ (ClWWalkS‘𝐺) ↔ 𝑊 ∈ (ClWWalkS‘𝐺)))
43adantl 481 . . 3 ((𝑢 = 𝑈𝑤 = 𝑊) → (𝑤 ∈ (ClWWalkS‘𝐺) ↔ 𝑊 ∈ (ClWWalkS‘𝐺)))
5 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊))
65oveq2d 6565 . . . . 5 (𝑤 = 𝑊 → (0...(#‘𝑤)) = (0...(#‘𝑊)))
76adantl 481 . . . 4 ((𝑢 = 𝑈𝑤 = 𝑊) → (0...(#‘𝑤)) = (0...(#‘𝑊)))
8 simpl 472 . . . . 5 ((𝑢 = 𝑈𝑤 = 𝑊) → 𝑢 = 𝑈)
9 oveq1 6556 . . . . . 6 (𝑤 = 𝑊 → (𝑤 cyclShift 𝑛) = (𝑊 cyclShift 𝑛))
109adantl 481 . . . . 5 ((𝑢 = 𝑈𝑤 = 𝑊) → (𝑤 cyclShift 𝑛) = (𝑊 cyclShift 𝑛))
118, 10eqeq12d 2625 . . . 4 ((𝑢 = 𝑈𝑤 = 𝑊) → (𝑢 = (𝑤 cyclShift 𝑛) ↔ 𝑈 = (𝑊 cyclShift 𝑛)))
127, 11rexeqbidv 3130 . . 3 ((𝑢 = 𝑈𝑤 = 𝑊) → (∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)))
132, 4, 123anbi123d 1391 . 2 ((𝑢 = 𝑈𝑤 = 𝑊) → ((𝑢 ∈ (ClWWalkS‘𝐺) ∧ 𝑤 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛)) ↔ (𝑈 ∈ (ClWWalkS‘𝐺) ∧ 𝑊 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛))))
14 erclwwlks.r . 2 = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalkS‘𝐺) ∧ 𝑤 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
1513, 14brabga 4914 1 ((𝑈𝑋𝑊𝑌) → (𝑈 𝑊 ↔ (𝑈 ∈ (ClWWalkS‘𝐺) ∧ 𝑊 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   class class class wbr 4583  {copab 4642  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ...cfz 12197  #chash 12979   cyclShift ccsh 13385  ClWWalkScclwwlks 41183 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by:  erclwwlkseqlen  41240  erclwwlksref  41241  erclwwlkssym  41242  erclwwlkstr  41243
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