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Theorem erclwwlksnref 26812
 Description: ∼ is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 26-Mar-2018.) (Revised by AV, 30-Apr-2021.)
Hypotheses
Ref Expression
erclwwlksn.w 𝑊 = (𝑁 ClWWalksN 𝐺)
erclwwlksn.r = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlksnref (𝑥𝑊𝑥 𝑥)
Distinct variable groups:   𝑡,𝑊,𝑢   𝑛,𝑁,𝑢,𝑡,𝑥
Allowed substitution hints:   (𝑥,𝑢,𝑡,𝑛)   𝐺(𝑥,𝑢,𝑡,𝑛)   𝑊(𝑥,𝑛)

Proof of Theorem erclwwlksnref
StepHypRef Expression
1 df-3an 1038 . . 3 ((𝑥𝑊𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)) ↔ ((𝑥𝑊𝑥𝑊) ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)))
2 anidm 675 . . . 4 ((𝑥𝑊𝑥𝑊) ↔ 𝑥𝑊)
32anbi1i 730 . . 3 (((𝑥𝑊𝑥𝑊) ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)) ↔ (𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)))
41, 3bitri 264 . 2 ((𝑥𝑊𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)) ↔ (𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)))
5 vex 3189 . . 3 𝑥 ∈ V
6 erclwwlksn.w . . . 4 𝑊 = (𝑁 ClWWalksN 𝐺)
7 erclwwlksn.r . . . 4 = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
86, 7erclwwlksneq 26810 . . 3 ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥 𝑥 ↔ (𝑥𝑊𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛))))
95, 5, 8mp2an 707 . 2 (𝑥 𝑥 ↔ (𝑥𝑊𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)))
10 eqid 2621 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
1110clwwlknbp0 26751 . . . . 5 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁)))
12 cshw0 13477 . . . . . . . 8 (𝑥 ∈ Word (Vtx‘𝐺) → (𝑥 cyclShift 0) = 𝑥)
1312adantr 481 . . . . . . 7 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) → (𝑥 cyclShift 0) = 𝑥)
1413adantl 482 . . . . . 6 (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁)) → (𝑥 cyclShift 0) = 𝑥)
15 nnnn0 11243 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
16 0elfz 12377 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁))
1715, 16syl 17 . . . . . . . . 9 (𝑁 ∈ ℕ → 0 ∈ (0...𝑁))
1817adantl 482 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) → 0 ∈ (0...𝑁))
1918adantr 481 . . . . . . 7 (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁)) → 0 ∈ (0...𝑁))
20 eqcom 2628 . . . . . . . 8 ((𝑥 cyclShift 0) = 𝑥𝑥 = (𝑥 cyclShift 0))
2120biimpi 206 . . . . . . 7 ((𝑥 cyclShift 0) = 𝑥𝑥 = (𝑥 cyclShift 0))
22 oveq2 6612 . . . . . . . . 9 (𝑛 = 0 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 0))
2322eqeq2d 2631 . . . . . . . 8 (𝑛 = 0 → (𝑥 = (𝑥 cyclShift 𝑛) ↔ 𝑥 = (𝑥 cyclShift 0)))
2423rspcev 3295 . . . . . . 7 ((0 ∈ (0...𝑁) ∧ 𝑥 = (𝑥 cyclShift 0)) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛))
2519, 21, 24syl2an 494 . . . . . 6 ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁)) ∧ (𝑥 cyclShift 0) = 𝑥) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛))
2614, 25mpdan 701 . . . . 5 (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁)) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛))
2711, 26syl 17 . . . 4 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛))
2827, 6eleq2s 2716 . . 3 (𝑥𝑊 → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛))
2928pm4.71i 663 . 2 (𝑥𝑊 ↔ (𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)))
304, 9, 293bitr4ri 293 1 (𝑥𝑊𝑥 𝑥)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∃wrex 2908  Vcvv 3186   class class class wbr 4613  {copab 4672  ‘cfv 5847  (class class class)co 6604  0cc0 9880  ℕcn 10964  ℕ0cn0 11236  ...cfz 12268  #chash 13057  Word cword 13230   cyclShift ccsh 13471  Vtxcvtx 25774   ClWWalksN cclwwlksn 26743 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-fl 12533  df-mod 12609  df-hash 13058  df-word 13238  df-concat 13240  df-substr 13242  df-csh 13472  df-clwwlks 26744  df-clwwlksn 26745 This theorem is referenced by:  erclwwlksn  26815
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