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Theorem clwwlknscsh 26347
 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.)
Assertion
Ref Expression
clwwlknscsh ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)})
Distinct variable groups:   𝑛,𝐸,𝑦   𝑛,𝑁,𝑦   𝑛,𝑉,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 2614 . . . 4 (𝑦 = 𝑥 → (𝑦 = (𝑊 cyclShift 𝑛) ↔ 𝑥 = (𝑊 cyclShift 𝑛)))
21rexbidv 3034 . . 3 (𝑦 = 𝑥 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)))
32cbvrabv 3172 . 2 {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)}
4 clwwlknprop 26300 . . . . . . . 8 (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)))
54simp2d 1067 . . . . . . 7 (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → 𝑤 ∈ Word 𝑉)
65ad2antrl 760 . . . . . 6 (((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → 𝑤 ∈ Word 𝑉)
7 simprr 792 . . . . . 6 (((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))
86, 7jca 553 . . . . 5 (((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
9 simplr 788 . . . . . . . . . . . . 13 ((((𝑤 ∈ Word 𝑉𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → (𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))
10 simpllr 795 . . . . . . . . . . . . 13 ((((𝑤 ∈ Word 𝑉𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → 𝑛 ∈ (0...𝑁))
11 clwwnisshclwwn 26337 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑛 ∈ (0...𝑁) → (𝑊 cyclShift 𝑛) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))
129, 10, 11sylc 63 . . . . . . . . . . . 12 ((((𝑤 ∈ Word 𝑉𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → (𝑊 cyclShift 𝑛) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))
13 eleq1 2676 . . . . . . . . . . . . 13 (𝑤 = (𝑊 cyclShift 𝑛) → (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝑊 cyclShift 𝑛) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))
1413adantl 481 . . . . . . . . . . . 12 ((((𝑤 ∈ Word 𝑉𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝑊 cyclShift 𝑛) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))
1512, 14mpbird 246 . . . . . . . . . . 11 ((((𝑤 ∈ Word 𝑉𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))
1615exp31 628 . . . . . . . . . 10 ((𝑤 ∈ Word 𝑉𝑛 ∈ (0...𝑁)) → ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑤 = (𝑊 cyclShift 𝑛) → 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))))
1716com23 84 . . . . . . . . 9 ((𝑤 ∈ Word 𝑉𝑛 ∈ (0...𝑁)) → (𝑤 = (𝑊 cyclShift 𝑛) → ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))))
1817rexlimdva 3013 . . . . . . . 8 (𝑤 ∈ Word 𝑉 → (∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛) → ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))))
1918imp 444 . . . . . . 7 ((𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)) → ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))
2019impcom 445 . . . . . 6 (((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))
21 simprr 792 . . . . . 6 (((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))
2220, 21jca 553 . . . . 5 (((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
238, 22impbida 873 . . . 4 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → ((𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)) ↔ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))))
24 eqeq1 2614 . . . . . 6 (𝑥 = 𝑤 → (𝑥 = (𝑊 cyclShift 𝑛) ↔ 𝑤 = (𝑊 cyclShift 𝑛)))
2524rexbidv 3034 . . . . 5 (𝑥 = 𝑤 → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
2625elrab 3331 . . . 4 (𝑤 ∈ {𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)} ↔ (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
27 eqeq1 2614 . . . . . 6 (𝑦 = 𝑤 → (𝑦 = (𝑊 cyclShift 𝑛) ↔ 𝑤 = (𝑊 cyclShift 𝑛)))
2827rexbidv 3034 . . . . 5 (𝑦 = 𝑤 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
2928elrab 3331 . . . 4 (𝑤 ∈ {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} ↔ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
3023, 26, 293bitr4g 302 . . 3 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑤 ∈ {𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)} ↔ 𝑤 ∈ {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)}))
3130eqrdv 2608 . 2 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → {𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)})
323, 31syl5eq 2656 1 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℕ0cn0 11169  ...cfz 12197  #chash 12979  Word cword 13146   cyclShift ccsh 13385   ClWWalksN cclwwlkn 26277 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 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 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 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-substr 13158  df-csh 13386  df-clwwlk 26279  df-clwwlkn 26280 This theorem is referenced by:  hashecclwwlkn1  26361  usghashecclwwlk  26362
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