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Theorem erclwwlkneq 28428
Description: Two classes are equivalent regarding if both are words of the same fixed length and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 30-Apr-2021.)
Hypotheses
Ref Expression
erclwwlkn.w 𝑊 = (𝑁 ClWWalksN 𝐺)
erclwwlkn.r = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlkneq ((𝑇𝑋𝑈𝑌) → (𝑇 𝑈 ↔ (𝑇𝑊𝑈𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛))))
Distinct variable groups:   𝑡,𝑊,𝑢   𝑡,𝑁,𝑢   𝑇,𝑛,𝑡,𝑢   𝑈,𝑛,𝑡,𝑢
Allowed substitution hints:   (𝑢,𝑡,𝑛)   𝐺(𝑢,𝑡,𝑛)   𝑁(𝑛)   𝑊(𝑛)   𝑋(𝑢,𝑡,𝑛)   𝑌(𝑢,𝑡,𝑛)

Proof of Theorem erclwwlkneq
StepHypRef Expression
1 eleq1 2826 . . . 4 (𝑡 = 𝑇 → (𝑡𝑊𝑇𝑊))
21adantr 481 . . 3 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑡𝑊𝑇𝑊))
3 eleq1 2826 . . . 4 (𝑢 = 𝑈 → (𝑢𝑊𝑈𝑊))
43adantl 482 . . 3 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑢𝑊𝑈𝑊))
5 simpl 483 . . . . 5 ((𝑡 = 𝑇𝑢 = 𝑈) → 𝑡 = 𝑇)
6 oveq1 7284 . . . . . 6 (𝑢 = 𝑈 → (𝑢 cyclShift 𝑛) = (𝑈 cyclShift 𝑛))
76adantl 482 . . . . 5 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑢 cyclShift 𝑛) = (𝑈 cyclShift 𝑛))
85, 7eqeq12d 2754 . . . 4 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑡 = (𝑢 cyclShift 𝑛) ↔ 𝑇 = (𝑈 cyclShift 𝑛)))
98rexbidv 3225 . . 3 ((𝑡 = 𝑇𝑢 = 𝑈) → (∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)))
102, 4, 93anbi123d 1435 . 2 ((𝑡 = 𝑇𝑢 = 𝑈) → ((𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛)) ↔ (𝑇𝑊𝑈𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛))))
11 erclwwlkn.r . 2 = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
1210, 11brabga 5449 1 ((𝑇𝑋𝑈𝑌) → (𝑇 𝑈 ↔ (𝑇𝑊𝑈𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wrex 3065   class class class wbr 5076  {copab 5138  (class class class)co 7277  0cc0 10869  ...cfz 13237   cyclShift ccsh 14499   ClWWalksN cclwwlkn 28385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5225  ax-nul 5232  ax-pr 5354
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rex 3070  df-rab 3073  df-v 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5077  df-opab 5139  df-iota 6393  df-fv 6443  df-ov 7280
This theorem is referenced by:  erclwwlkneqlen  28429  erclwwlknref  28430  erclwwlknsym  28431  erclwwlkntr  28432  eclclwwlkn1  28436
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