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Theorem erclwwlkneq 27850
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 2901 . . . 4 (𝑡 = 𝑇 → (𝑡𝑊𝑇𝑊))
21adantr 484 . . 3 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑡𝑊𝑇𝑊))
3 eleq1 2901 . . . 4 (𝑢 = 𝑈 → (𝑢𝑊𝑈𝑊))
43adantl 485 . . 3 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑢𝑊𝑈𝑊))
5 simpl 486 . . . . 5 ((𝑡 = 𝑇𝑢 = 𝑈) → 𝑡 = 𝑇)
6 oveq1 7147 . . . . . 6 (𝑢 = 𝑈 → (𝑢 cyclShift 𝑛) = (𝑈 cyclShift 𝑛))
76adantl 485 . . . . 5 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑢 cyclShift 𝑛) = (𝑈 cyclShift 𝑛))
85, 7eqeq12d 2838 . . . 4 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑡 = (𝑢 cyclShift 𝑛) ↔ 𝑇 = (𝑈 cyclShift 𝑛)))
98rexbidv 3283 . . 3 ((𝑡 = 𝑇𝑢 = 𝑈) → (∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)))
102, 4, 93anbi123d 1433 . 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 5398 1 ((𝑇𝑋𝑈𝑌) → (𝑇 𝑈 ↔ (𝑇𝑊𝑈𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wrex 3131   class class class wbr 5042  {copab 5104  (class class class)co 7140  0cc0 10526  ...cfz 12885   cyclShift ccsh 14141   ClWWalksN cclwwlkn 27807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-iota 6293  df-fv 6342  df-ov 7143
This theorem is referenced by:  erclwwlkneqlen  27851  erclwwlknref  27852  erclwwlknsym  27853  erclwwlkntr  27854  eclclwwlkn1  27858
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