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Theorem clwwlknscsh 27850
Description: The set of cyclical shifts of a word representing a closed walk is the set of closed walks represented by cyclical shifts of a word. (Contributed by Alexander van der Vekens, 15-Jun-2018.) (Revised by AV, 30-Apr-2021.)
Assertion
Ref Expression
clwwlknscsh ((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)})
Distinct variable groups:   𝑛,𝐺,𝑦   𝑛,𝑁,𝑦   𝑛,𝑊,𝑦

Proof of Theorem clwwlknscsh
Dummy variables 𝑤 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2828 . . . 4 (𝑦 = 𝑥 → (𝑦 = (𝑊 cyclShift 𝑛) ↔ 𝑥 = (𝑊 cyclShift 𝑛)))
21rexbidv 3289 . . 3 (𝑦 = 𝑥 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)))
32cbvrabv 3477 . 2 {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑥 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)}
4 eqid 2824 . . . . . . . 8 (Vtx‘𝐺) = (Vtx‘𝐺)
54clwwlknwrd 27822 . . . . . . 7 (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → 𝑤 ∈ Word (Vtx‘𝐺))
65ad2antrl 727 . . . . . 6 (((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → 𝑤 ∈ Word (Vtx‘𝐺))
7 simprr 772 . . . . . 6 (((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))
86, 7jca 515 . . . . 5 (((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
9 simprr 772 . . . . . . . . . . . . 13 (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺))) → 𝑊 ∈ (𝑁 ClWWalksN 𝐺))
10 simpllr 775 . . . . . . . . . . . . 13 ((((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → 𝑛 ∈ (0...𝑁))
11 clwwnisshclwwsn 27847 . . . . . . . . . . . . 13 ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑛 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺))
129, 10, 11syl2an2r 684 . . . . . . . . . . . 12 ((((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺))
13 eleq1 2903 . . . . . . . . . . . . 13 (𝑤 = (𝑊 cyclShift 𝑛) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺)))
1413adantl 485 . . . . . . . . . . . 12 ((((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺)))
1512, 14mpbird 260 . . . . . . . . . . 11 ((((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → 𝑤 ∈ (𝑁 ClWWalksN 𝐺))
1615exp31 423 . . . . . . . . . 10 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) → ((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑤 = (𝑊 cyclShift 𝑛) → 𝑤 ∈ (𝑁 ClWWalksN 𝐺))))
1716com23 86 . . . . . . . . 9 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) → (𝑤 = (𝑊 cyclShift 𝑛) → ((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → 𝑤 ∈ (𝑁 ClWWalksN 𝐺))))
1817rexlimdva 3276 . . . . . . . 8 (𝑤 ∈ Word (Vtx‘𝐺) → (∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛) → ((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → 𝑤 ∈ (𝑁 ClWWalksN 𝐺))))
1918imp 410 . . . . . . 7 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)) → ((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → 𝑤 ∈ (𝑁 ClWWalksN 𝐺)))
2019impcom 411 . . . . . 6 (((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → 𝑤 ∈ (𝑁 ClWWalksN 𝐺))
21 simprr 772 . . . . . 6 (((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))
2220, 21jca 515 . . . . 5 (((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
238, 22impbida 800 . . . 4 ((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → ((𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)) ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))))
24 eqeq1 2828 . . . . . 6 (𝑥 = 𝑤 → (𝑥 = (𝑊 cyclShift 𝑛) ↔ 𝑤 = (𝑊 cyclShift 𝑛)))
2524rexbidv 3289 . . . . 5 (𝑥 = 𝑤 → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
2625elrab 3666 . . . 4 (𝑤 ∈ {𝑥 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)} ↔ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
27 eqeq1 2828 . . . . . 6 (𝑦 = 𝑤 → (𝑦 = (𝑊 cyclShift 𝑛) ↔ 𝑤 = (𝑊 cyclShift 𝑛)))
2827rexbidv 3289 . . . . 5 (𝑦 = 𝑤 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
2928elrab 3666 . . . 4 (𝑤 ∈ {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
3023, 26, 293bitr4g 317 . . 3 ((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑤 ∈ {𝑥 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)} ↔ 𝑤 ∈ {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)}))
3130eqrdv 2822 . 2 ((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → {𝑥 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)})
323, 31syl5eq 2871 1 ((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wrex 3134  {crab 3137  cfv 6343  (class class class)co 7149  0cc0 10535  0cn0 11894  ...cfz 12894  Word cword 13866   cyclShift ccsh 14150  Vtxcvtx 26792   ClWWalksN cclwwlkn 27812
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-inf 8904  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-n0 11895  df-z 11979  df-uz 12241  df-rp 12387  df-ico 12741  df-fz 12895  df-fzo 13038  df-fl 13166  df-mod 13242  df-hash 13696  df-word 13867  df-lsw 13915  df-concat 13923  df-substr 14003  df-pfx 14033  df-csh 14151  df-clwwlk 27770  df-clwwlkn 27813
This theorem is referenced by:  hashecclwwlkn1  27865  umgrhashecclwwlk  27866
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