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Theorem erclwwlkssym 41242
 Description: ∼ is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by AV, 29-Apr-2021.)
Hypothesis
Ref Expression
erclwwlks.r = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalkS‘𝐺) ∧ 𝑤 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlkssym (𝑥 𝑦𝑦 𝑥)
Distinct variable groups:   𝑛,𝐺,𝑢,𝑤   𝑥,𝑛,𝑢,𝑤,𝑦
Allowed substitution hints:   (𝑥,𝑦,𝑤,𝑢,𝑛)   𝐺(𝑥,𝑦)

Proof of Theorem erclwwlkssym
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 vex 3176 . 2 𝑥 ∈ V
2 vex 3176 . 2 𝑦 ∈ V
3 erclwwlks.r . . . 4 = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalkS‘𝐺) ∧ 𝑤 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
43erclwwlkseqlen 41240 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → (#‘𝑥) = (#‘𝑦)))
53erclwwlkseq 41239 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 ↔ (𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))))
6 simpl2 1058 . . . . . . 7 (((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑦 ∈ (ClWWalkS‘𝐺))
7 simpl1 1057 . . . . . . 7 (((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑥 ∈ (ClWWalkS‘𝐺))
8 eqid 2610 . . . . . . . . . . . . . . . . . 18 (Vtx‘𝐺) = (Vtx‘𝐺)
98clwwlkbp 41191 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ClWWalkS‘𝐺) → (𝐺 ∈ V ∧ 𝑦 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ≠ ∅))
109simp2d 1067 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (ClWWalkS‘𝐺) → 𝑦 ∈ Word (Vtx‘𝐺))
1110ad2antlr 759 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺)) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑦 ∈ Word (Vtx‘𝐺))
12 simpr 476 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺)) ∧ (#‘𝑥) = (#‘𝑦)) → (#‘𝑥) = (#‘𝑦))
1311, 12cshwcshid 13424 . . . . . . . . . . . . . 14 (((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺)) ∧ (#‘𝑥) = (#‘𝑦)) → ((𝑛 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
1413expd 451 . . . . . . . . . . . . 13 (((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺)) ∧ (#‘𝑥) = (#‘𝑦)) → (𝑛 ∈ (0...(#‘𝑦)) → (𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))))
1514rexlimdv 3012 . . . . . . . . . . . 12 (((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺)) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
1615ex 449 . . . . . . . . . . 11 ((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺)) → ((#‘𝑥) = (#‘𝑦) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))))
1716com23 84 . . . . . . . . . 10 ((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((#‘𝑥) = (#‘𝑦) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))))
18173impia 1253 . . . . . . . . 9 ((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((#‘𝑥) = (#‘𝑦) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
1918imp 444 . . . . . . . 8 (((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))
20 oveq2 6557 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
2120eqeq2d 2620 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
2221cbvrexv 3148 . . . . . . . 8 (∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))
2319, 22sylibr 223 . . . . . . 7 (((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
246, 7, 233jca 1235 . . . . . 6 (((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → (𝑦 ∈ (ClWWalkS‘𝐺) ∧ 𝑥 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))
253erclwwlkseq 41239 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 𝑥 ↔ (𝑦 ∈ (ClWWalkS‘𝐺) ∧ 𝑥 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))))
2625ancoms 468 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑦 𝑥 ↔ (𝑦 ∈ (ClWWalkS‘𝐺) ∧ 𝑥 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))))
2724, 26syl5ibr 235 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑦 𝑥))
2827expd 451 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ((𝑥 ∈ (ClWWalkS‘𝐺) ∧ 𝑦 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((#‘𝑥) = (#‘𝑦) → 𝑦 𝑥)))
295, 28sylbid 229 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → ((#‘𝑥) = (#‘𝑦) → 𝑦 𝑥)))
304, 29mpdd 42 . 2 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦𝑦 𝑥))
311, 2, 30mp2an 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   ≠ wne 2780  ∃wrex 2897  Vcvv 3173  ∅c0 3874   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  ClWWalkScclwwlks 41183 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 This theorem is referenced by:  erclwwlks  41244
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