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Theorem clwwnisshclwwsn 26796
 Description: Cyclically shifting a closed walk as word of fixed length results in a closed walk as word of the same length (in an undirected graph). (Contributed by Alexander van der Vekens, 10-Jun-2018.) (Revised by AV, 29-Apr-2021.)
Assertion
Ref Expression
clwwnisshclwwsn ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺))

Proof of Theorem clwwnisshclwwsn
StepHypRef Expression
1 eqid 2621 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
21clwwlknbp0 26751 . . 3 (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)))
3 clwwlkclwwlkn 26758 . . . . . . 7 (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ (ClWWalks‘𝐺))
433ad2ant2 1081 . . . . . 6 ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → 𝑊 ∈ (ClWWalks‘𝐺))
5 oveq2 6612 . . . . . . . . . . . 12 (𝑁 = (#‘𝑊) → (0...𝑁) = (0...(#‘𝑊)))
65eleq2d 2684 . . . . . . . . . . 11 (𝑁 = (#‘𝑊) → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...(#‘𝑊))))
76eqcoms 2629 . . . . . . . . . 10 ((#‘𝑊) = 𝑁 → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...(#‘𝑊))))
87ad2antll 764 . . . . . . . . 9 (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...(#‘𝑊))))
98biimpd 219 . . . . . . . 8 (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) → (𝑀 ∈ (0...𝑁) → 𝑀 ∈ (0...(#‘𝑊))))
109a1d 25 . . . . . . 7 (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑀 ∈ (0...𝑁) → 𝑀 ∈ (0...(#‘𝑊)))))
11103imp 1254 . . . . . 6 ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → 𝑀 ∈ (0...(#‘𝑊)))
12 clwwisshclwwsn 26795 . . . . . 6 ((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑀 ∈ (0...(#‘𝑊))) → (𝑊 cyclShift 𝑀) ∈ (ClWWalks‘𝐺))
134, 11, 12syl2anc 692 . . . . 5 ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑀) ∈ (ClWWalks‘𝐺))
14 elfzelz 12284 . . . . . . . . . . 11 (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℤ)
15 cshwlen 13482 . . . . . . . . . . 11 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑀 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))
1614, 15sylan2 491 . . . . . . . . . 10 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))
1716ex 450 . . . . . . . . 9 (𝑊 ∈ Word (Vtx‘𝐺) → (𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊)))
1817ad2antrl 763 . . . . . . . 8 (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) → (𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊)))
1918a1d 25 . . . . . . 7 (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))))
20193imp 1254 . . . . . 6 ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))
21 simp1rr 1125 . . . . . 6 ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (#‘𝑊) = 𝑁)
2220, 21eqtrd 2655 . . . . 5 ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁)
23 simp1lr 1123 . . . . . 6 ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → 𝑁 ∈ ℕ)
24 isclwwlksn 26749 . . . . . 6 (𝑁 ∈ ℕ → ((𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝑊 cyclShift 𝑀) ∈ (ClWWalks‘𝐺) ∧ (#‘(𝑊 cyclShift 𝑀)) = 𝑁)))
2523, 24syl 17 . . . . 5 ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → ((𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝑊 cyclShift 𝑀) ∈ (ClWWalks‘𝐺) ∧ (#‘(𝑊 cyclShift 𝑀)) = 𝑁)))
2613, 22, 25mpbir2and 956 . . . 4 ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺))
27263exp 1261 . . 3 (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = 𝑁)) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑀 ∈ (0...𝑁) → (𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺))))
282, 27mpcom 38 . 2 (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑀 ∈ (0...𝑁) → (𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺)))
2928imp 445 1 ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  Vcvv 3186  ‘cfv 5847  (class class class)co 6604  0cc0 9880  ℕcn 10964  ℤcz 11321  ...cfz 12268  #chash 13057  Word cword 13230   cyclShift ccsh 13471  Vtxcvtx 25774  ClWWalkscclwwlks 26742   ClWWalksN cclwwlksn 26743 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-ico 12123  df-fz 12269  df-fzo 12407  df-fl 12533  df-mod 12609  df-hash 13058  df-word 13238  df-lsw 13239  df-concat 13240  df-substr 13242  df-csh 13472  df-clwwlks 26744  df-clwwlksn 26745 This theorem is referenced by:  clwwlksnscsh  26806
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