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Theorem erclwwlknsym 26354
 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 Alexander van der Vekens, 14-Jun-2018.)
Hypotheses
Ref Expression
erclwwlkn.w 𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)
erclwwlkn.r = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlknsym (𝑥 𝑦𝑦 𝑥)
Distinct variable groups:   𝑡,𝐸,𝑢   𝑡,𝑁,𝑢   𝑛,𝑉,𝑡,𝑢   𝑡,𝑊,𝑢   𝑥,𝑛,𝑡,𝑢   𝑛,𝑁   𝑦,𝑛,𝑡,𝑢,𝑥   𝑛,𝑊
Allowed substitution hints:   (𝑥,𝑦,𝑢,𝑡,𝑛)   𝐸(𝑥,𝑦,𝑛)   𝑁(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem erclwwlknsym
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 vex 3176 . 2 𝑥 ∈ V
2 vex 3176 . 2 𝑦 ∈ V
3 erclwwlkn.w . . . 4 𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)
4 erclwwlkn.r . . . 4 = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
53, 4erclwwlkneqlen 26352 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → (#‘𝑥) = (#‘𝑦)))
63, 4erclwwlkneq 26351 . . . 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 clwwlknprop 26300 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)))
10 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝑥) = 𝑁𝑁 = (#‘𝑥))
1110biimpi 205 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝑥) = 𝑁𝑁 = (#‘𝑥))
1211adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁) → 𝑁 = (#‘𝑥))
13123ad2ant3 1077 . . . . . . . . . . . . . . . . . . 19 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑥 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑥) = 𝑁)) → 𝑁 = (#‘𝑥))
149, 13syl 17 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → 𝑁 = (#‘𝑥))
1514, 3eleq2s 2706 . . . . . . . . . . . . . . . . 17 (𝑥𝑊𝑁 = (#‘𝑥))
1615adantr 480 . . . . . . . . . . . . . . . 16 ((𝑥𝑊𝑦𝑊) → 𝑁 = (#‘𝑥))
1716adantr 480 . . . . . . . . . . . . . . 15 (((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑁 = (#‘𝑥))
18 clwwlknprop 26300 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑦 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑦) = 𝑁)))
1918simp2d 1067 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → 𝑦 ∈ Word 𝑉)
2019, 3eleq2s 2706 . . . . . . . . . . . . . . . . . . . 20 (𝑦𝑊𝑦 ∈ Word 𝑉)
2120adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝑊𝑦𝑊) → 𝑦 ∈ Word 𝑉)
2221adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑦 ∈ Word 𝑉)
2322adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → 𝑦 ∈ Word 𝑉)
24 simprr 792 . . . . . . . . . . . . . . . . 17 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → (#‘𝑥) = (#‘𝑦))
2523, 24cshwcshid 13424 . . . . . . . . . . . . . . . 16 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → ((𝑛 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
26 oveq2 6557 . . . . . . . . . . . . . . . . . . 19 (𝑁 = (#‘𝑥) → (0...𝑁) = (0...(#‘𝑥)))
27 oveq2 6557 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑥) = (#‘𝑦) → (0...(#‘𝑥)) = (0...(#‘𝑦)))
2827adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → (0...(#‘𝑥)) = (0...(#‘𝑦)))
2926, 28sylan9eq 2664 . . . . . . . . . . . . . . . . . 18 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → (0...𝑁) = (0...(#‘𝑦)))
3029eleq2d 2673 . . . . . . . . . . . . . . . . 17 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(#‘𝑦))))
3130anbi1d 737 . . . . . . . . . . . . . . . 16 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) ↔ (𝑛 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛))))
3226adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → (0...𝑁) = (0...(#‘𝑥)))
3332rexeqdv 3122 . . . . . . . . . . . . . . . 16 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → (∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚) ↔ ∃𝑚 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
3425, 31, 333imtr4d 282 . . . . . . . . . . . . . . 15 ((𝑁 = (#‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦))) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
3517, 34mpancom 700 . . . . . . . . . . . . . 14 (((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
3635expd 451 . . . . . . . . . . . . 13 (((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → (𝑛 ∈ (0...𝑁) → (𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
3736rexlimdv 3012 . . . . . . . . . . . 12 (((𝑥𝑊𝑦𝑊) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
3837ex 449 . . . . . . . . . . 11 ((𝑥𝑊𝑦𝑊) → ((#‘𝑥) = (#‘𝑦) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
3938com23 84 . . . . . . . . . 10 ((𝑥𝑊𝑦𝑊) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((#‘𝑥) = (#‘𝑦) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
40393impia 1253 . . . . . . . . 9 ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((#‘𝑥) = (#‘𝑦) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
4140imp 444 . . . . . . . 8 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))
42 oveq2 6557 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
4342eqeq2d 2620 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
4443cbvrexv 3148 . . . . . . . 8 (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))
4541, 44sylibr 223 . . . . . . 7 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))
467, 8, 453jca 1235 . . . . . 6 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → (𝑦𝑊𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)))
473, 4erclwwlkneq 26351 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 𝑥 ↔ (𝑦𝑊𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
4847ancoms 468 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑦 𝑥 ↔ (𝑦𝑊𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
4946, 48syl5ibr 235 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (#‘𝑥) = (#‘𝑦)) → 𝑦 𝑥))
5049expd 451 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((#‘𝑥) = (#‘𝑦) → 𝑦 𝑥)))
516, 50sylbid 229 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → ((#‘𝑥) = (#‘𝑦) → 𝑦 𝑥)))
525, 51mpdd 42 . 2 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦𝑦 𝑥))
531, 2, 52mp2an 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  ℕ0cn0 11169  ...cfz 12197  #chash 12979  Word cword 13146   cyclShift ccsh 13385   ClWWalksN cclwwlkn 26277 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-clwwlk 26279  df-clwwlkn 26280 This theorem is referenced by:  erclwwlkn  26356  eclclwwlkn1  26359
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