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Theorem erclwwlkref 27909
 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 568 . . . 4 ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺)) ↔ 𝑥 ∈ (ClWWalks‘𝐺))
21anbi1i 626 . . 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 1086 . . 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 2758 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
54clwwlkbp 27874 . . . . 5 (𝑥 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅))
6 cshw0 14208 . . . . . . 7 (𝑥 ∈ Word (Vtx‘𝐺) → (𝑥 cyclShift 0) = 𝑥)
7 0nn0 11954 . . . . . . . . . 10 0 ∈ ℕ0
87a1i 11 . . . . . . . . 9 (𝑥 ∈ Word (Vtx‘𝐺) → 0 ∈ ℕ0)
9 lencl 13937 . . . . . . . . 9 (𝑥 ∈ Word (Vtx‘𝐺) → (♯‘𝑥) ∈ ℕ0)
10 hashge0 13803 . . . . . . . . 9 (𝑥 ∈ Word (Vtx‘𝐺) → 0 ≤ (♯‘𝑥))
11 elfz2nn0 13052 . . . . . . . . 9 (0 ∈ (0...(♯‘𝑥)) ↔ (0 ∈ ℕ0 ∧ (♯‘𝑥) ∈ ℕ0 ∧ 0 ≤ (♯‘𝑥)))
128, 9, 10, 11syl3anbrc 1340 . . . . . . . 8 (𝑥 ∈ Word (Vtx‘𝐺) → 0 ∈ (0...(♯‘𝑥)))
13 eqcom 2765 . . . . . . . . 9 ((𝑥 cyclShift 0) = 𝑥𝑥 = (𝑥 cyclShift 0))
1413biimpi 219 . . . . . . . 8 ((𝑥 cyclShift 0) = 𝑥𝑥 = (𝑥 cyclShift 0))
15 oveq2 7163 . . . . . . . . 9 (𝑛 = 0 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 0))
1615rspceeqv 3558 . . . . . . . 8 ((0 ∈ (0...(♯‘𝑥)) ∧ 𝑥 = (𝑥 cyclShift 0)) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
1712, 14, 16syl2an 598 . . . . . . 7 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (𝑥 cyclShift 0) = 𝑥) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
186, 17mpdan 686 . . . . . 6 (𝑥 ∈ Word (Vtx‘𝐺) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
19183ad2ant2 1131 . . . . 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 563 . . 3 (𝑥 ∈ (ClWWalks‘𝐺) ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
222, 3, 213bitr4ri 307 . 2 (𝑥 ∈ (ClWWalks‘𝐺) ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
23 erclwwlk.r . . . 4 = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
2423erclwwlkeq 27907 . . 3 ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥 𝑥 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))))
2524el2v 3417 . 2 (𝑥 𝑥 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
2622, 25bitr4i 281 1 (𝑥 ∈ (ClWWalks‘𝐺) ↔ 𝑥 𝑥)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∃wrex 3071  Vcvv 3409  ∅c0 4227   class class class wbr 5035  {copab 5097  ‘cfv 6339  (class class class)co 7155  0cc0 10580   ≤ cle 10719  ℕ0cn0 11939  ...cfz 12944  ♯chash 13745  Word cword 13918   cyclShift ccsh 14202  Vtxcvtx 26893  ClWWalkscclwwlk 27870 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657  ax-pre-sup 10658 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-1o 8117  df-oadd 8121  df-er 8304  df-map 8423  df-en 8533  df-dom 8534  df-sdom 8535  df-fin 8536  df-sup 8944  df-inf 8945  df-card 9406  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-div 11341  df-nn 11680  df-n0 11940  df-xnn0 12012  df-z 12026  df-uz 12288  df-rp 12436  df-fz 12945  df-fzo 13088  df-fl 13216  df-mod 13292  df-hash 13746  df-word 13919  df-concat 13975  df-substr 14055  df-pfx 14085  df-csh 14203  df-clwwlk 27871 This theorem is referenced by:  erclwwlk  27912
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