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Theorem erclwwlkeq 27801
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
erclwwlk.r = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlkeq ((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 erclwwlkeq
StepHypRef Expression
1 eleq1 2903 . . . 4 (𝑢 = 𝑈 → (𝑢 ∈ (ClWWalks‘𝐺) ↔ 𝑈 ∈ (ClWWalks‘𝐺)))
21adantr 484 . . 3 ((𝑢 = 𝑈𝑤 = 𝑊) → (𝑢 ∈ (ClWWalks‘𝐺) ↔ 𝑈 ∈ (ClWWalks‘𝐺)))
3 eleq1 2903 . . . 4 (𝑤 = 𝑊 → (𝑤 ∈ (ClWWalks‘𝐺) ↔ 𝑊 ∈ (ClWWalks‘𝐺)))
43adantl 485 . . 3 ((𝑢 = 𝑈𝑤 = 𝑊) → (𝑤 ∈ (ClWWalks‘𝐺) ↔ 𝑊 ∈ (ClWWalks‘𝐺)))
5 fveq2 6659 . . . . . 6 (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
65oveq2d 7162 . . . . 5 (𝑤 = 𝑊 → (0...(♯‘𝑤)) = (0...(♯‘𝑊)))
76adantl 485 . . . 4 ((𝑢 = 𝑈𝑤 = 𝑊) → (0...(♯‘𝑤)) = (0...(♯‘𝑊)))
8 simpl 486 . . . . 5 ((𝑢 = 𝑈𝑤 = 𝑊) → 𝑢 = 𝑈)
9 oveq1 7153 . . . . . 6 (𝑤 = 𝑊 → (𝑤 cyclShift 𝑛) = (𝑊 cyclShift 𝑛))
109adantl 485 . . . . 5 ((𝑢 = 𝑈𝑤 = 𝑊) → (𝑤 cyclShift 𝑛) = (𝑊 cyclShift 𝑛))
118, 10eqeq12d 2840 . . . 4 ((𝑢 = 𝑈𝑤 = 𝑊) → (𝑢 = (𝑤 cyclShift 𝑛) ↔ 𝑈 = (𝑊 cyclShift 𝑛)))
127, 11rexeqbidv 3394 . . 3 ((𝑢 = 𝑈𝑤 = 𝑊) → (∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(♯‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)))
132, 4, 123anbi123d 1433 . 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 erclwwlk.r . 2 = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
1513, 14brabga 5409 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wrex 3134   class class class wbr 5053  {copab 5115  cfv 6344  (class class class)co 7146  0cc0 10531  ...cfz 12892  chash 13693   cyclShift ccsh 14148  ClWWalkscclwwlk 27764
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-iota 6303  df-fv 6352  df-ov 7149
This theorem is referenced by:  erclwwlkeqlen  27802  erclwwlkref  27803  erclwwlksym  27804  erclwwlktr  27805
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