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Theorem erclwwlksref 41233
Description: is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 29-Apr-2021.)
Hypothesis
Ref Expression
erclwwlks.r = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalkS‘𝐺) ∧ 𝑤 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlksref (𝑥 ∈ (ClWWalkS‘𝐺) ↔ 𝑥 𝑥)
Distinct variable groups:   𝑛,𝐺,𝑢,𝑤   𝑥,𝑛,𝑢,𝑤
Allowed substitution hints:   (𝑥,𝑤,𝑢,𝑛)   𝐺(𝑥)

Proof of Theorem erclwwlksref
StepHypRef Expression
1 anidm 674 . . . 4 ((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑥 ∈ (ClWWalkS‘𝐺)) ↔ 𝑥 ∈ (ClWWalkS‘𝐺))
21anbi1i 727 . . 3 (((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑥 ∈ (ClWWalkS‘𝐺)) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)) ↔ (𝑥 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
3 df-3an 1033 . . 3 ((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑥 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)) ↔ ((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑥 ∈ (ClWWalkS‘𝐺)) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
4 eqid 2610 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
54clwwlkbp 41183 . . . . 5 (𝑥 ∈ (ClWWalkS‘𝐺) → (𝐺 ∈ V ∧ 𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅))
6 cshw0 13340 . . . . . . 7 (𝑥 ∈ Word (Vtx‘𝐺) → (𝑥 cyclShift 0) = 𝑥)
7 0nn0 11157 . . . . . . . . . 10 0 ∈ ℕ0
87a1i 11 . . . . . . . . 9 (𝑥 ∈ Word (Vtx‘𝐺) → 0 ∈ ℕ0)
9 lencl 13128 . . . . . . . . 9 (𝑥 ∈ Word (Vtx‘𝐺) → (#‘𝑥) ∈ ℕ0)
10 hashge0 12992 . . . . . . . . 9 (𝑥 ∈ Word (Vtx‘𝐺) → 0 ≤ (#‘𝑥))
11 elfz2nn0 12258 . . . . . . . . 9 (0 ∈ (0...(#‘𝑥)) ↔ (0 ∈ ℕ0 ∧ (#‘𝑥) ∈ ℕ0 ∧ 0 ≤ (#‘𝑥)))
128, 9, 10, 11syl3anbrc 1239 . . . . . . . 8 (𝑥 ∈ Word (Vtx‘𝐺) → 0 ∈ (0...(#‘𝑥)))
13 eqcom 2617 . . . . . . . . 9 ((𝑥 cyclShift 0) = 𝑥𝑥 = (𝑥 cyclShift 0))
1413biimpi 205 . . . . . . . 8 ((𝑥 cyclShift 0) = 𝑥𝑥 = (𝑥 cyclShift 0))
15 oveq2 6535 . . . . . . . . . 10 (𝑛 = 0 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 0))
1615eqeq2d 2620 . . . . . . . . 9 (𝑛 = 0 → (𝑥 = (𝑥 cyclShift 𝑛) ↔ 𝑥 = (𝑥 cyclShift 0)))
1716rspcev 3282 . . . . . . . 8 ((0 ∈ (0...(#‘𝑥)) ∧ 𝑥 = (𝑥 cyclShift 0)) → ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
1812, 14, 17syl2an 493 . . . . . . 7 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (𝑥 cyclShift 0) = 𝑥) → ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
196, 18mpdan 699 . . . . . 6 (𝑥 ∈ Word (Vtx‘𝐺) → ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
20193ad2ant2 1076 . . . . 5 ((𝐺 ∈ V ∧ 𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅) → ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
215, 20syl 17 . . . 4 (𝑥 ∈ (ClWWalkS‘𝐺) → ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
2221pm4.71i 662 . . 3 (𝑥 ∈ (ClWWalkS‘𝐺) ↔ (𝑥 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
232, 3, 223bitr4ri 292 . 2 (𝑥 ∈ (ClWWalkS‘𝐺) ↔ (𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑥 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
24 vex 3176 . . 3 𝑥 ∈ V
25 erclwwlks.r . . . 4 = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalkS‘𝐺) ∧ 𝑤 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
2625erclwwlkseq 41231 . . 3 ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥 𝑥 ↔ (𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑥 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))))
2724, 24, 26mp2an 704 . 2 (𝑥 𝑥 ↔ (𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑥 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
2823, 27bitr4i 266 1 (𝑥 ∈ (ClWWalkS‘𝐺) ↔ 𝑥 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wrex 2897  Vcvv 3173  c0 3874   class class class wbr 4578  {copab 4637  cfv 5790  (class class class)co 6527  0cc0 9793  cle 9932  0cn0 11142  ...cfz 12155  #chash 12937  Word cword 13095   cyclShift ccsh 13334  Vtxcvtx 40221  ClWWalkScclwwlks 41175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-pre-sup 9871
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4368  df-int 4406  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-map 7724  df-pm 7725  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-sup 8209  df-inf 8210  df-card 8626  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-n0 11143  df-z 11214  df-uz 11523  df-rp 11668  df-fz 12156  df-fzo 12293  df-fl 12413  df-mod 12489  df-hash 12938  df-word 13103  df-concat 13105  df-substr 13107  df-csh 13335  df-clwwlks 41177
This theorem is referenced by:  erclwwlks  41236
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