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Theorem erclwwlkref 26103
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 Alexander van der Vekens, 11-Jun-2018.)
Hypothesis
Ref Expression
erclwwlk.r = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlkref (𝑥 ∈ (𝑉 ClWWalks 𝐸) ↔ 𝑥 𝑥)
Distinct variable groups:   𝑛,𝐸,𝑢,𝑤   𝑛,𝑉,𝑢,𝑤   𝑥,𝑛,𝑢,𝑤
Allowed substitution hints:   (𝑥,𝑤,𝑢,𝑛)   𝐸(𝑥)   𝑉(𝑥)

Proof of Theorem erclwwlkref
StepHypRef Expression
1 anidm 673 . . . 4 ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑥 ∈ (𝑉 ClWWalks 𝐸)) ↔ 𝑥 ∈ (𝑉 ClWWalks 𝐸))
21anbi1i 726 . . 3 (((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑥 ∈ (𝑉 ClWWalks 𝐸)) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)) ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
3 df-3an 1032 . . 3 ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)) ↔ ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑥 ∈ (𝑉 ClWWalks 𝐸)) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
4 clwwlkprop 26060 . . . . 5 (𝑥 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑥 ∈ Word 𝑉))
5 cshw0 13333 . . . . . . 7 (𝑥 ∈ Word 𝑉 → (𝑥 cyclShift 0) = 𝑥)
6 0nn0 11150 . . . . . . . . . 10 0 ∈ ℕ0
76a1i 11 . . . . . . . . 9 (𝑥 ∈ Word 𝑉 → 0 ∈ ℕ0)
8 lencl 13121 . . . . . . . . 9 (𝑥 ∈ Word 𝑉 → (#‘𝑥) ∈ ℕ0)
9 hashge0 12985 . . . . . . . . 9 (𝑥 ∈ Word 𝑉 → 0 ≤ (#‘𝑥))
10 elfz2nn0 12251 . . . . . . . . 9 (0 ∈ (0...(#‘𝑥)) ↔ (0 ∈ ℕ0 ∧ (#‘𝑥) ∈ ℕ0 ∧ 0 ≤ (#‘𝑥)))
117, 8, 9, 10syl3anbrc 1238 . . . . . . . 8 (𝑥 ∈ Word 𝑉 → 0 ∈ (0...(#‘𝑥)))
12 eqcom 2612 . . . . . . . . 9 ((𝑥 cyclShift 0) = 𝑥𝑥 = (𝑥 cyclShift 0))
1312biimpi 204 . . . . . . . 8 ((𝑥 cyclShift 0) = 𝑥𝑥 = (𝑥 cyclShift 0))
14 oveq2 6531 . . . . . . . . . 10 (𝑛 = 0 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 0))
1514eqeq2d 2615 . . . . . . . . 9 (𝑛 = 0 → (𝑥 = (𝑥 cyclShift 𝑛) ↔ 𝑥 = (𝑥 cyclShift 0)))
1615rspcev 3277 . . . . . . . 8 ((0 ∈ (0...(#‘𝑥)) ∧ 𝑥 = (𝑥 cyclShift 0)) → ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
1711, 13, 16syl2an 492 . . . . . . 7 ((𝑥 ∈ Word 𝑉 ∧ (𝑥 cyclShift 0) = 𝑥) → ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
185, 17mpdan 698 . . . . . 6 (𝑥 ∈ Word 𝑉 → ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
19183ad2ant3 1076 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑥 ∈ Word 𝑉) → ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
204, 19syl 17 . . . 4 (𝑥 ∈ (𝑉 ClWWalks 𝐸) → ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
2120pm4.71i 661 . . 3 (𝑥 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
222, 3, 213bitr4ri 291 . 2 (𝑥 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
23 vex 3171 . . 3 𝑥 ∈ V
24 erclwwlk.r . . . 4 = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
2524erclwwlkeq 26101 . . 3 ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥 𝑥 ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))))
2623, 23, 25mp2an 703 . 2 (𝑥 𝑥 ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
2722, 26bitr4i 265 1 (𝑥 ∈ (𝑉 ClWWalks 𝐸) ↔ 𝑥 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wrex 2892  Vcvv 3168   class class class wbr 4573  {copab 4632  cfv 5786  (class class class)co 6523  0cc0 9788  cle 9927  0cn0 11135  ...cfz 12148  #chash 12930  Word cword 13088   cyclShift ccsh 13327   ClWWalks cclwwlk 26038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865  ax-pre-sup 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-map 7719  df-pm 7720  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-sup 8204  df-inf 8205  df-card 8621  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-n0 11136  df-z 11207  df-uz 11516  df-rp 11661  df-fz 12149  df-fzo 12286  df-fl 12406  df-mod 12482  df-hash 12931  df-word 13096  df-concat 13098  df-substr 13100  df-csh 13328  df-clwwlk 26041
This theorem is referenced by:  erclwwlk  26106
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