Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eleclclwwlksnlem2 Structured version   Visualization version   GIF version

Theorem eleclclwwlksnlem2 41246
 Description: Lemma 2 for eleclclwwlksn 41260. (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 472 . . . . 5 ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → 𝑘 ∈ (0...𝑁))
21anim1i 590 . . . 4 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (𝑘 ∈ (0...𝑁) ∧ (𝑋𝑊𝑥𝑊)))
32adantr 480 . . 3 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)) → (𝑘 ∈ (0...𝑁) ∧ (𝑋𝑊𝑥𝑊)))
4 simpr 476 . . . . 5 ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → 𝑋 = (𝑥 cyclShift 𝑘))
54adantr 480 . . . 4 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → 𝑋 = (𝑥 cyclShift 𝑘))
65anim1i 590 . . 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 41245 . . 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 63 . 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 2610 . . . . . . . . . . . 12 (Vtx‘𝐺) = (Vtx‘𝐺)
1110clwwlknbp 41193 . . . . . . . . . . 11 (𝑥 ∈ (𝑁 ClWWalkSN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁))
1211, 7eleq2s 2706 . . . . . . . . . 10 (𝑥𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁))
13 fznn0sub2 12315 . . . . . . . . . . . 12 (𝑘 ∈ (0...𝑁) → (𝑁𝑘) ∈ (0...𝑁))
14 oveq1 6556 . . . . . . . . . . . . 13 ((#‘𝑥) = 𝑁 → ((#‘𝑥) − 𝑘) = (𝑁𝑘))
1514eleq1d 2672 . . . . . . . . . . . 12 ((#‘𝑥) = 𝑁 → (((#‘𝑥) − 𝑘) ∈ (0...𝑁) ↔ (𝑁𝑘) ∈ (0...𝑁)))
1613, 15syl5ibr 235 . . . . . . . . . . 11 ((#‘𝑥) = 𝑁 → (𝑘 ∈ (0...𝑁) → ((#‘𝑥) − 𝑘) ∈ (0...𝑁)))
1716adantl 481 . . . . . . . . . 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 481 . . . . . . . 8 ((𝑋𝑊𝑥𝑊) → (𝑘 ∈ (0...𝑁) → ((#‘𝑥) − 𝑘) ∈ (0...𝑁)))
2019com12 32 . . . . . . 7 (𝑘 ∈ (0...𝑁) → ((𝑋𝑊𝑥𝑊) → ((#‘𝑥) − 𝑘) ∈ (0...𝑁)))
2120adantr 480 . . . . . 6 ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → ((𝑋𝑊𝑥𝑊) → ((#‘𝑥) − 𝑘) ∈ (0...𝑁)))
2221imp 444 . . . . 5 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → ((#‘𝑥) − 𝑘) ∈ (0...𝑁))
2322adantr 480 . . . 4 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → ((#‘𝑥) − 𝑘) ∈ (0...𝑁))
24 simpr 476 . . . . . 6 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (𝑋𝑊𝑥𝑊))
2524ancomd 466 . . . . 5 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (𝑥𝑊𝑋𝑊))
2625adantr 480 . . . 4 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → (𝑥𝑊𝑋𝑊))
2723, 26jca 553 . . 3 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → (((#‘𝑥) − 𝑘) ∈ (0...𝑁) ∧ (𝑥𝑊𝑋𝑊)))
28 simpll 786 . . . . . . . . . . . . 13 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → 𝑥 ∈ Word (Vtx‘𝐺))
29 oveq2 6557 . . . . . . . . . . . . . . . . 17 (𝑁 = (#‘𝑥) → (0...𝑁) = (0...(#‘𝑥)))
3029eleq2d 2673 . . . . . . . . . . . . . . . 16 (𝑁 = (#‘𝑥) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(#‘𝑥))))
3130eqcoms 2618 . . . . . . . . . . . . . . 15 ((#‘𝑥) = 𝑁 → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(#‘𝑥))))
3231adantl 481 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(#‘𝑥))))
3332biimpa 500 . . . . . . . . . . . . 13 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...(#‘𝑥)))
3428, 33jca 553 . . . . . . . . . . . 12 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(#‘𝑥))))
3534ex 449 . . . . . . . . . . 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 481 . . . . . . . . 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 480 . . . . . . 7 ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → ((𝑋𝑊𝑥𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(#‘𝑥)))))
4039imp 444 . . . . . 6 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(#‘𝑥))))
414eqcomd 2616 . . . . . . 7 ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → (𝑥 cyclShift 𝑘) = 𝑋)
4241adantr 480 . . . . . 6 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (𝑥 cyclShift 𝑘) = 𝑋)
43 oveq1 6556 . . . . . . . 8 (𝑋 = (𝑥 cyclShift 𝑘) → (𝑋 cyclShift ((#‘𝑥) − 𝑘)) = ((𝑥 cyclShift 𝑘) cyclShift ((#‘𝑥) − 𝑘)))
4443eqcoms 2618 . . . . . . 7 ((𝑥 cyclShift 𝑘) = 𝑋 → (𝑋 cyclShift ((#‘𝑥) − 𝑘)) = ((𝑥 cyclShift 𝑘) cyclShift ((#‘𝑥) − 𝑘)))
45 elfzelz 12213 . . . . . . . 8 (𝑘 ∈ (0...(#‘𝑥)) → 𝑘 ∈ ℤ)
46 2cshwid 13411 . . . . . . . 8 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ ℤ) → ((𝑥 cyclShift 𝑘) cyclShift ((#‘𝑥) − 𝑘)) = 𝑥)
4745, 46sylan2 490 . . . . . . 7 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(#‘𝑥))) → ((𝑥 cyclShift 𝑘) cyclShift ((#‘𝑥) − 𝑘)) = 𝑥)
4844, 47sylan9eqr 2666 . . . . . 6 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(#‘𝑥))) ∧ (𝑥 cyclShift 𝑘) = 𝑋) → (𝑋 cyclShift ((#‘𝑥) − 𝑘)) = 𝑥)
4940, 42, 48syl2anc 691 . . . . 5 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (𝑋 cyclShift ((#‘𝑥) − 𝑘)) = 𝑥)
5049eqcomd 2616 . . . 4 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → 𝑥 = (𝑋 cyclShift ((#‘𝑥) − 𝑘)))
5150anim1i 590 . . 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 41245 . . 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 63 . 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 873 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ‘cfv 5804  (class class class)co 6549  0cc0 9815   − cmin 10145  ℤcz 11254  ...cfz 12197  #chash 12979  Word cword 13146   cyclShift ccsh 13385  Vtxcvtx 25673   ClWWalkSN cclwwlksn 41184 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-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-csh 13386  df-clwwlks 41185  df-clwwlksn 41186 This theorem is referenced by:  eleclclwwlksn  41260
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