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Theorem erclwwlkeq 27165
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 2837 . . . 4 (𝑢 = 𝑈 → (𝑢 ∈ (ClWWalks‘𝐺) ↔ 𝑈 ∈ (ClWWalks‘𝐺)))
21adantr 466 . . 3 ((𝑢 = 𝑈𝑤 = 𝑊) → (𝑢 ∈ (ClWWalks‘𝐺) ↔ 𝑈 ∈ (ClWWalks‘𝐺)))
3 eleq1 2837 . . . 4 (𝑤 = 𝑊 → (𝑤 ∈ (ClWWalks‘𝐺) ↔ 𝑊 ∈ (ClWWalks‘𝐺)))
43adantl 467 . . 3 ((𝑢 = 𝑈𝑤 = 𝑊) → (𝑤 ∈ (ClWWalks‘𝐺) ↔ 𝑊 ∈ (ClWWalks‘𝐺)))
5 fveq2 6332 . . . . . 6 (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
65oveq2d 6808 . . . . 5 (𝑤 = 𝑊 → (0...(♯‘𝑤)) = (0...(♯‘𝑊)))
76adantl 467 . . . 4 ((𝑢 = 𝑈𝑤 = 𝑊) → (0...(♯‘𝑤)) = (0...(♯‘𝑊)))
8 simpl 468 . . . . 5 ((𝑢 = 𝑈𝑤 = 𝑊) → 𝑢 = 𝑈)
9 oveq1 6799 . . . . . 6 (𝑤 = 𝑊 → (𝑤 cyclShift 𝑛) = (𝑊 cyclShift 𝑛))
109adantl 467 . . . . 5 ((𝑢 = 𝑈𝑤 = 𝑊) → (𝑤 cyclShift 𝑛) = (𝑊 cyclShift 𝑛))
118, 10eqeq12d 2785 . . . 4 ((𝑢 = 𝑈𝑤 = 𝑊) → (𝑢 = (𝑤 cyclShift 𝑛) ↔ 𝑈 = (𝑊 cyclShift 𝑛)))
127, 11rexeqbidv 3301 . . 3 ((𝑢 = 𝑈𝑤 = 𝑊) → (∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(♯‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)))
132, 4, 123anbi123d 1546 . 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 5122 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 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wrex 3061   class class class wbr 4784  {copab 4844  cfv 6031  (class class class)co 6792  0cc0 10137  ...cfz 12532  chash 13320   cyclShift ccsh 13742  ClWWalkscclwwlk 27128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-iota 5994  df-fv 6039  df-ov 6795
This theorem is referenced by:  erclwwlkeqlen  27166  erclwwlkref  27167  erclwwlksym  27168  erclwwlktr  27169
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