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Theorem erclwwlkeq 29268
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 ∼ = {βŸ¨π‘’, π‘€βŸ© ∣ (𝑒 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑀 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘€))𝑒 = (𝑀 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlkeq ((π‘ˆ ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (π‘ˆ ∼ π‘Š ↔ (π‘ˆ ∈ (ClWWalksβ€˜πΊ) ∧ π‘Š ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘Š))π‘ˆ = (π‘Š cyclShift 𝑛))))
Distinct variable groups:   𝑛,𝐺,𝑒,𝑀   π‘ˆ,𝑛,𝑒,𝑀   𝑛,π‘Š,𝑒,𝑀
Allowed substitution hints:   ∼ (𝑀,𝑒,𝑛)   𝑋(𝑀,𝑒,𝑛)   π‘Œ(𝑀,𝑒,𝑛)
Proof of Theorem erclwwlkeq
StepHypRef Expression
1 eleq1 2821 . . . 4 (𝑒 = π‘ˆ β†’ (𝑒 ∈ (ClWWalksβ€˜πΊ) ↔ π‘ˆ ∈ (ClWWalksβ€˜πΊ)))
21adantr 481 . . 3 ((𝑒 = π‘ˆ ∧ 𝑀 = π‘Š) β†’ (𝑒 ∈ (ClWWalksβ€˜πΊ) ↔ π‘ˆ ∈ (ClWWalksβ€˜πΊ)))
3 eleq1 2821 . . . 4 (𝑀 = π‘Š β†’ (𝑀 ∈ (ClWWalksβ€˜πΊ) ↔ π‘Š ∈ (ClWWalksβ€˜πΊ)))
43adantl 482 . . 3 ((𝑒 = π‘ˆ ∧ 𝑀 = π‘Š) β†’ (𝑀 ∈ (ClWWalksβ€˜πΊ) ↔ π‘Š ∈ (ClWWalksβ€˜πΊ)))
5 fveq2 6891 . . . . . 6 (𝑀 = π‘Š β†’ (β™―β€˜π‘€) = (β™―β€˜π‘Š))
65oveq2d 7424 . . . . 5 (𝑀 = π‘Š β†’ (0...(β™―β€˜π‘€)) = (0...(β™―β€˜π‘Š)))
76adantl 482 . . . 4 ((𝑒 = π‘ˆ ∧ 𝑀 = π‘Š) β†’ (0...(β™―β€˜π‘€)) = (0...(β™―β€˜π‘Š)))
8 simpl 483 . . . . 5 ((𝑒 = π‘ˆ ∧ 𝑀 = π‘Š) β†’ 𝑒 = π‘ˆ)
9 oveq1 7415 . . . . . 6 (𝑀 = π‘Š β†’ (𝑀 cyclShift 𝑛) = (π‘Š cyclShift 𝑛))
109adantl 482 . . . . 5 ((𝑒 = π‘ˆ ∧ 𝑀 = π‘Š) β†’ (𝑀 cyclShift 𝑛) = (π‘Š cyclShift 𝑛))
118, 10eqeq12d 2748 . . . 4 ((𝑒 = π‘ˆ ∧ 𝑀 = π‘Š) β†’ (𝑒 = (𝑀 cyclShift 𝑛) ↔ π‘ˆ = (π‘Š cyclShift 𝑛)))
127, 11rexeqbidv 3343 . . 3 ((𝑒 = π‘ˆ ∧ 𝑀 = π‘Š) β†’ (βˆƒπ‘› ∈ (0...(β™―β€˜π‘€))𝑒 = (𝑀 cyclShift 𝑛) ↔ βˆƒπ‘› ∈ (0...(β™―β€˜π‘Š))π‘ˆ = (π‘Š cyclShift 𝑛)))
132, 4, 123anbi123d 1436 . 2 ((𝑒 = π‘ˆ ∧ 𝑀 = π‘Š) β†’ ((𝑒 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑀 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘€))𝑒 = (𝑀 cyclShift 𝑛)) ↔ (π‘ˆ ∈ (ClWWalksβ€˜πΊ) ∧ π‘Š ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘Š))π‘ˆ = (π‘Š cyclShift 𝑛))))
14 erclwwlk.r . 2 ∼ = {βŸ¨π‘’, π‘€βŸ© ∣ (𝑒 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑀 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘€))𝑒 = (𝑀 cyclShift 𝑛))}
1513, 14brabga 5534 1 ((π‘ˆ ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (π‘ˆ ∼ π‘Š ↔ (π‘ˆ ∈ (ClWWalksβ€˜πΊ) ∧ π‘Š ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘Š))π‘ˆ = (π‘Š cyclShift 𝑛))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   class class class wbr 5148  {copab 5210  β€˜cfv 6543  (class class class)co 7408  0cc0 11109  ...cfz 13483  β™―chash 14289   cyclShift ccsh 14737  ClWWalkscclwwlk 29231
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-iota 6495  df-fv 6551  df-ov 7411
This theorem is referenced by:  erclwwlkeqlen  29269  erclwwlkref  29270  erclwwlksym  29271  erclwwlktr  29272
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