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

Theorem erclwwlksnsym 41254
 Description: ∼ is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.)
Hypotheses
Ref Expression
erclwwlksn.w 𝑊 = (𝑁 ClWWalkSN 𝐺)
erclwwlksn.r = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlksnsym (𝑥 𝑦𝑦 𝑥)
Distinct variable groups:   𝑡,𝑊,𝑢   𝑛,𝑁,𝑢,𝑡,𝑥   𝑦,𝑛,𝑡,𝑢,𝑥   𝑛,𝑊
Allowed substitution hints:   (𝑥,𝑦,𝑢,𝑡,𝑛)   𝐺(𝑥,𝑦,𝑢,𝑡,𝑛)   𝑁(𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem erclwwlksnsym
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 vex 3176 . 2 𝑥 ∈ V
2 vex 3176 . 2 𝑦 ∈ V
3 erclwwlksn.w . . . 4 𝑊 = (𝑁 ClWWalkSN 𝐺)
4 erclwwlksn.r . . . 4 = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
53, 4erclwwlksneqlen 41252 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → (#‘𝑥) = (#‘𝑦)))
63, 4erclwwlksneq 41251 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 ↔ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))))
7 simpl2 1058 . . . . . . 7 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑦𝑊)
8 simpl1 1057 . . . . . . 7 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑥𝑊)
9 eqid 2610 . . . . . . . . . . . . . . . . . . . 20 (Vtx‘𝐺) = (Vtx‘𝐺)
109clwwlknbp 41193 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝑁 ClWWalkSN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁))
11 eqcom 2617 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑥) = 𝑁𝑁 = (#‘𝑥))
1211biimpi 205 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑥) = 𝑁𝑁 = (#‘𝑥))
1310, 12simpl2im 656 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑁 ClWWalkSN 𝐺) → 𝑁 = (#‘𝑥))
1413, 3eleq2s 2706 . . . . . . . . . . . . . . . . 17 (𝑥𝑊𝑁 = (#‘𝑥))
1514adantr 480 . . . . . . . . . . . . . . . 16 ((𝑥𝑊𝑦𝑊) → 𝑁 = (#‘𝑥))
1615adantr 480 . . . . . . . . . . . . . . 15 (((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑁 = (#‘𝑥))
179clwwlksnwrd 41194 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝑁 ClWWalkSN 𝐺) → 𝑦 ∈ Word (Vtx‘𝐺))
1817, 3eleq2s 2706 . . . . . . . . . . . . . . . . . . . 20 (𝑦𝑊𝑦 ∈ Word (Vtx‘𝐺))
1918adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝑊𝑦𝑊) → 𝑦 ∈ Word (Vtx‘𝐺))
2019adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑦 ∈ Word (Vtx‘𝐺))
2120adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → 𝑦 ∈ Word (Vtx‘𝐺))
22 simprr 792 . . . . . . . . . . . . . . . . 17 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → (#‘𝑥) = (#‘𝑦))
2321, 22cshwcshid 13424 . . . . . . . . . . . . . . . 16 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → ((𝑛 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
24 oveq2 6557 . . . . . . . . . . . . . . . . . . 19 (𝑁 = (#‘𝑥) → (0...𝑁) = (0...(#‘𝑥)))
25 oveq2 6557 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑥) = (#‘𝑦) → (0...(#‘𝑥)) = (0...(#‘𝑦)))
2625adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → (0...(#‘𝑥)) = (0...(#‘𝑦)))
2724, 26sylan9eq 2664 . . . . . . . . . . . . . . . . . 18 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → (0...𝑁) = (0...(#‘𝑦)))
2827eleq2d 2673 . . . . . . . . . . . . . . . . 17 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(#‘𝑦))))
2928anbi1d 737 . . . . . . . . . . . . . . . 16 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) ↔ (𝑛 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛))))
3024adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → (0...𝑁) = (0...(#‘𝑥)))
3130rexeqdv 3122 . . . . . . . . . . . . . . . 16 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → (∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚) ↔ ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
3223, 29, 313imtr4d 282 . . . . . . . . . . . . . . 15 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
3316, 32mpancom 700 . . . . . . . . . . . . . 14 (((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
3433expd 451 . . . . . . . . . . . . 13 (((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → (𝑛 ∈ (0...𝑁) → (𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
3534rexlimdv 3012 . . . . . . . . . . . 12 (((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
3635ex 449 . . . . . . . . . . 11 ((𝑥𝑊𝑦𝑊) → ((#‘𝑥) = (#‘𝑦) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
3736com23 84 . . . . . . . . . 10 ((𝑥𝑊𝑦𝑊) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((#‘𝑥) = (#‘𝑦) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
38373impia 1253 . . . . . . . . 9 ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((#‘𝑥) = (#‘𝑦) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
3938imp 444 . . . . . . . 8 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))
40 oveq2 6557 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
4140eqeq2d 2620 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
4241cbvrexv 3148 . . . . . . . 8 (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))
4339, 42sylibr 223 . . . . . . 7 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))
447, 8, 433jca 1235 . . . . . 6 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → (𝑦𝑊𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)))
453, 4erclwwlksneq 41251 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 𝑥 ↔ (𝑦𝑊𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
4645ancoms 468 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑦 𝑥 ↔ (𝑦𝑊𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
4744, 46syl5ibr 235 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑦 𝑥))
4847expd 451 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((#‘𝑥) = (#‘𝑦) → 𝑦 𝑥)))
496, 48sylbid 229 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → ((#‘𝑥) = (#‘𝑦) → 𝑦 𝑥)))
505, 49mpdd 42 . 2 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦𝑦 𝑥))
511, 2, 50mp2an 704 1 (𝑥 𝑦𝑦 𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   class class class wbr 4583  {copab 4642  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ...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:  erclwwlksn  41256  eclclwwlksn1  41259
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