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Theorem eleclclwwlksnlem2 26805
 Description: Lemma 2 for eleclclwwlksn 26819. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by AV, 30-Apr-2021.)
Hypothesis
Ref Expression
erclwwlksn1.w 𝑊 = (𝑁 ClWWalksN 𝐺)
Assertion
Ref Expression
eleclclwwlksnlem2 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
Distinct variable groups:   𝑚,𝑛,𝐺   𝑚,𝑁,𝑛   𝑚,𝑋,𝑛   𝑚,𝑌,𝑛   𝑘,𝑚,𝑛   𝑥,𝑚,𝑛
Allowed substitution hints:   𝐺(𝑥,𝑘)   𝑁(𝑥,𝑘)   𝑊(𝑥,𝑘,𝑚,𝑛)   𝑋(𝑥,𝑘)   𝑌(𝑥,𝑘)

Proof of Theorem eleclclwwlksnlem2
StepHypRef Expression
1 simpl 473 . . . . 5 ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → 𝑘 ∈ (0...𝑁))
21anim1i 591 . . . 4 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (𝑘 ∈ (0...𝑁) ∧ (𝑋𝑊𝑥𝑊)))
32adantr 481 . . 3 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)) → (𝑘 ∈ (0...𝑁) ∧ (𝑋𝑊𝑥𝑊)))
4 simpr 477 . . . . 5 ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → 𝑋 = (𝑥 cyclShift 𝑘))
54adantr 481 . . . 4 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → 𝑋 = (𝑥 cyclShift 𝑘))
65anim1i 591 . . 3 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)) → (𝑋 = (𝑥 cyclShift 𝑘) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)))
7 erclwwlksn1.w . . . 4 𝑊 = (𝑁 ClWWalksN 𝐺)
87eleclclwwlksnlem1 26804 . . 3 ((𝑘 ∈ (0...𝑁) ∧ (𝑋𝑊𝑥𝑊)) → ((𝑋 = (𝑥 cyclShift 𝑘) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
93, 6, 8sylc 65 . 2 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))
10 eqid 2621 . . . . . . . . . . . 12 (Vtx‘𝐺) = (Vtx‘𝐺)
1110clwwlknbp 26752 . . . . . . . . . . 11 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁))
1211, 7eleq2s 2716 . . . . . . . . . 10 (𝑥𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁))
13 fznn0sub2 12387 . . . . . . . . . . . 12 (𝑘 ∈ (0...𝑁) → (𝑁𝑘) ∈ (0...𝑁))
14 oveq1 6611 . . . . . . . . . . . . 13 ((#‘𝑥) = 𝑁 → ((#‘𝑥) − 𝑘) = (𝑁𝑘))
1514eleq1d 2683 . . . . . . . . . . . 12 ((#‘𝑥) = 𝑁 → (((#‘𝑥) − 𝑘) ∈ (0...𝑁) ↔ (𝑁𝑘) ∈ (0...𝑁)))
1613, 15syl5ibr 236 . . . . . . . . . . 11 ((#‘𝑥) = 𝑁 → (𝑘 ∈ (0...𝑁) → ((#‘𝑥) − 𝑘) ∈ (0...𝑁)))
1716adantl 482 . . . . . . . . . 10 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) → (𝑘 ∈ (0...𝑁) → ((#‘𝑥) − 𝑘) ∈ (0...𝑁)))
1812, 17syl 17 . . . . . . . . 9 (𝑥𝑊 → (𝑘 ∈ (0...𝑁) → ((#‘𝑥) − 𝑘) ∈ (0...𝑁)))
1918adantl 482 . . . . . . . 8 ((𝑋𝑊𝑥𝑊) → (𝑘 ∈ (0...𝑁) → ((#‘𝑥) − 𝑘) ∈ (0...𝑁)))
2019com12 32 . . . . . . 7 (𝑘 ∈ (0...𝑁) → ((𝑋𝑊𝑥𝑊) → ((#‘𝑥) − 𝑘) ∈ (0...𝑁)))
2120adantr 481 . . . . . 6 ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → ((𝑋𝑊𝑥𝑊) → ((#‘𝑥) − 𝑘) ∈ (0...𝑁)))
2221imp 445 . . . . 5 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → ((#‘𝑥) − 𝑘) ∈ (0...𝑁))
2322adantr 481 . . . 4 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → ((#‘𝑥) − 𝑘) ∈ (0...𝑁))
24 simpr 477 . . . . . 6 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (𝑋𝑊𝑥𝑊))
2524ancomd 467 . . . . 5 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (𝑥𝑊𝑋𝑊))
2625adantr 481 . . . 4 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → (𝑥𝑊𝑋𝑊))
2723, 26jca 554 . . 3 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → (((#‘𝑥) − 𝑘) ∈ (0...𝑁) ∧ (𝑥𝑊𝑋𝑊)))
28 simpll 789 . . . . . . . . . . . . 13 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → 𝑥 ∈ Word (Vtx‘𝐺))
29 oveq2 6612 . . . . . . . . . . . . . . . . 17 (𝑁 = (#‘𝑥) → (0...𝑁) = (0...(#‘𝑥)))
3029eleq2d 2684 . . . . . . . . . . . . . . . 16 (𝑁 = (#‘𝑥) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(#‘𝑥))))
3130eqcoms 2629 . . . . . . . . . . . . . . 15 ((#‘𝑥) = 𝑁 → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(#‘𝑥))))
3231adantl 482 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(#‘𝑥))))
3332biimpa 501 . . . . . . . . . . . . 13 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...(#‘𝑥)))
3428, 33jca 554 . . . . . . . . . . . 12 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(#‘𝑥))))
3534ex 450 . . . . . . . . . . 11 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) → (𝑘 ∈ (0...𝑁) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(#‘𝑥)))))
3612, 35syl 17 . . . . . . . . . 10 (𝑥𝑊 → (𝑘 ∈ (0...𝑁) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(#‘𝑥)))))
3736adantl 482 . . . . . . . . 9 ((𝑋𝑊𝑥𝑊) → (𝑘 ∈ (0...𝑁) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(#‘𝑥)))))
3837com12 32 . . . . . . . 8 (𝑘 ∈ (0...𝑁) → ((𝑋𝑊𝑥𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(#‘𝑥)))))
3938adantr 481 . . . . . . 7 ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → ((𝑋𝑊𝑥𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(#‘𝑥)))))
4039imp 445 . . . . . 6 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(#‘𝑥))))
414eqcomd 2627 . . . . . . 7 ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → (𝑥 cyclShift 𝑘) = 𝑋)
4241adantr 481 . . . . . 6 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (𝑥 cyclShift 𝑘) = 𝑋)
43 oveq1 6611 . . . . . . . 8 (𝑋 = (𝑥 cyclShift 𝑘) → (𝑋 cyclShift ((#‘𝑥) − 𝑘)) = ((𝑥 cyclShift 𝑘) cyclShift ((#‘𝑥) − 𝑘)))
4443eqcoms 2629 . . . . . . 7 ((𝑥 cyclShift 𝑘) = 𝑋 → (𝑋 cyclShift ((#‘𝑥) − 𝑘)) = ((𝑥 cyclShift 𝑘) cyclShift ((#‘𝑥) − 𝑘)))
45 elfzelz 12284 . . . . . . . 8 (𝑘 ∈ (0...(#‘𝑥)) → 𝑘 ∈ ℤ)
46 2cshwid 13497 . . . . . . . 8 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ ℤ) → ((𝑥 cyclShift 𝑘) cyclShift ((#‘𝑥) − 𝑘)) = 𝑥)
4745, 46sylan2 491 . . . . . . 7 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(#‘𝑥))) → ((𝑥 cyclShift 𝑘) cyclShift ((#‘𝑥) − 𝑘)) = 𝑥)
4844, 47sylan9eqr 2677 . . . . . 6 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(#‘𝑥))) ∧ (𝑥 cyclShift 𝑘) = 𝑋) → (𝑋 cyclShift ((#‘𝑥) − 𝑘)) = 𝑥)
4940, 42, 48syl2anc 692 . . . . 5 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (𝑋 cyclShift ((#‘𝑥) − 𝑘)) = 𝑥)
5049eqcomd 2627 . . . 4 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → 𝑥 = (𝑋 cyclShift ((#‘𝑥) − 𝑘)))
5150anim1i 591 . . 3 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → (𝑥 = (𝑋 cyclShift ((#‘𝑥) − 𝑘)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
527eleclclwwlksnlem1 26804 . . 3 ((((#‘𝑥) − 𝑘) ∈ (0...𝑁) ∧ (𝑥𝑊𝑋𝑊)) → ((𝑥 = (𝑋 cyclShift ((#‘𝑥) − 𝑘)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)))
5327, 51, 52sylc 65 . 2 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚))
549, 53impbida 876 1 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∃wrex 2908  ‘cfv 5847  (class class class)co 6604  0cc0 9880   − cmin 10210  ℤcz 11321  ...cfz 12268  #chash 13057  Word cword 13230   cyclShift ccsh 13471  Vtxcvtx 25774   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-fz 12269  df-fzo 12407  df-fl 12533  df-mod 12609  df-hash 13058  df-word 13238  df-concat 13240  df-substr 13242  df-csh 13472  df-clwwlks 26744  df-clwwlksn 26745 This theorem is referenced by:  eleclclwwlksn  26819
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