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Theorem erclwwlkref 27225
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
erclwwlk.r = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlkref (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 erclwwlkref
StepHypRef Expression
1 anidm 560 . . . 4 ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺)) ↔ 𝑥 ∈ (ClWWalks‘𝐺))
21anbi1i 617 . . 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 1109 . . 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 2764 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
54clwwlkbp 27190 . . . . 5 (𝑥 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅))
6 cshw0 13818 . . . . . . 7 (𝑥 ∈ Word (Vtx‘𝐺) → (𝑥 cyclShift 0) = 𝑥)
7 0nn0 11554 . . . . . . . . . 10 0 ∈ ℕ0
87a1i 11 . . . . . . . . 9 (𝑥 ∈ Word (Vtx‘𝐺) → 0 ∈ ℕ0)
9 lencl 13504 . . . . . . . . 9 (𝑥 ∈ Word (Vtx‘𝐺) → (♯‘𝑥) ∈ ℕ0)
10 hashge0 13377 . . . . . . . . 9 (𝑥 ∈ Word (Vtx‘𝐺) → 0 ≤ (♯‘𝑥))
11 elfz2nn0 12637 . . . . . . . . 9 (0 ∈ (0...(♯‘𝑥)) ↔ (0 ∈ ℕ0 ∧ (♯‘𝑥) ∈ ℕ0 ∧ 0 ≤ (♯‘𝑥)))
128, 9, 10, 11syl3anbrc 1443 . . . . . . . 8 (𝑥 ∈ Word (Vtx‘𝐺) → 0 ∈ (0...(♯‘𝑥)))
13 eqcom 2771 . . . . . . . . 9 ((𝑥 cyclShift 0) = 𝑥𝑥 = (𝑥 cyclShift 0))
1413biimpi 207 . . . . . . . 8 ((𝑥 cyclShift 0) = 𝑥𝑥 = (𝑥 cyclShift 0))
15 oveq2 6849 . . . . . . . . 9 (𝑛 = 0 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 0))
1615rspceeqv 3478 . . . . . . . 8 ((0 ∈ (0...(♯‘𝑥)) ∧ 𝑥 = (𝑥 cyclShift 0)) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
1712, 14, 16syl2an 589 . . . . . . 7 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (𝑥 cyclShift 0) = 𝑥) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
186, 17mpdan 678 . . . . . 6 (𝑥 ∈ Word (Vtx‘𝐺) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
19183ad2ant2 1164 . . . . 5 ((𝐺 ∈ V ∧ 𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
205, 19syl 17 . . . 4 (𝑥 ∈ (ClWWalks‘𝐺) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
2120pm4.71i 555 . . 3 (𝑥 ∈ (ClWWalks‘𝐺) ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
222, 3, 213bitr4ri 295 . 2 (𝑥 ∈ (ClWWalks‘𝐺) ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
23 vex 3352 . . 3 𝑥 ∈ V
24 erclwwlk.r . . . 4 = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
2524erclwwlkeq 27223 . . 3 ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥 𝑥 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))))
2623, 23, 25mp2an 683 . 2 (𝑥 𝑥 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
2722, 26bitr4i 269 1 (𝑥 ∈ (ClWWalks‘𝐺) ↔ 𝑥 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2936  wrex 3055  Vcvv 3349  c0 4078   class class class wbr 4808  {copab 4870  cfv 6067  (class class class)co 6841  0cc0 10188  cle 10328  0cn0 11537  ...cfz 12532  chash 13320  Word cword 13485   cyclShift ccsh 13811  Vtxcvtx 26164  ClWWalkscclwwlk 27186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265  ax-pre-sup 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-oadd 7767  df-er 7946  df-map 8061  df-pm 8062  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-sup 8554  df-inf 8555  df-card 9015  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-div 10938  df-nn 11274  df-n0 11538  df-xnn0 11610  df-z 11624  df-uz 11886  df-rp 12028  df-fz 12533  df-fzo 12673  df-fl 12800  df-mod 12876  df-hash 13321  df-word 13486  df-concat 13541  df-substr 13616  df-pfx 13661  df-csh 13812  df-clwwlk 27187
This theorem is referenced by:  erclwwlk  27228
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