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Theorem erclwwlkneq 29919
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 2813 . . . 4 (𝑡 = 𝑇 → (𝑡𝑊𝑇𝑊))
21adantr 479 . . 3 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑡𝑊𝑇𝑊))
3 eleq1 2813 . . . 4 (𝑢 = 𝑈 → (𝑢𝑊𝑈𝑊))
43adantl 480 . . 3 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑢𝑊𝑈𝑊))
5 simpl 481 . . . . 5 ((𝑡 = 𝑇𝑢 = 𝑈) → 𝑡 = 𝑇)
6 oveq1 7422 . . . . . 6 (𝑢 = 𝑈 → (𝑢 cyclShift 𝑛) = (𝑈 cyclShift 𝑛))
76adantl 480 . . . . 5 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑢 cyclShift 𝑛) = (𝑈 cyclShift 𝑛))
85, 7eqeq12d 2741 . . . 4 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑡 = (𝑢 cyclShift 𝑛) ↔ 𝑇 = (𝑈 cyclShift 𝑛)))
98rexbidv 3169 . . 3 ((𝑡 = 𝑇𝑢 = 𝑈) → (∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)))
102, 4, 93anbi123d 1432 . 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 5530 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 394  w3a 1084   = wceq 1533  wcel 2098  wrex 3060   class class class wbr 5143  {copab 5205  (class class class)co 7415  0cc0 11136  ...cfz 13514   cyclShift ccsh 14768   ClWWalksN cclwwlkn 29876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-iota 6494  df-fv 6550  df-ov 7418
This theorem is referenced by:  erclwwlkneqlen  29920  erclwwlknref  29921  erclwwlknsym  29922  erclwwlkntr  29923  eclclwwlkn1  29927
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