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Theorem erclwwlknsym 27222
 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
erclwwlkn.w 𝑊 = (𝑁 ClWWalksN 𝐺)
erclwwlkn.r = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlknsym (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 erclwwlknsym
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 vex 3343 . 2 𝑥 ∈ V
2 vex 3343 . 2 𝑦 ∈ V
3 erclwwlkn.w . . . 4 𝑊 = (𝑁 ClWWalksN 𝐺)
4 erclwwlkn.r . . . 4 = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
53, 4erclwwlkneqlen 27220 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
63, 4erclwwlkneq 27219 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 ↔ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))))
7 simpl2 1230 . . . . . . 7 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦𝑊)
8 simpl1 1228 . . . . . . 7 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑥𝑊)
9 eqid 2760 . . . . . . . . . . . . . . . . . . . 20 (Vtx‘𝐺) = (Vtx‘𝐺)
109clwwlknbp 27184 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
11 eqcom 2767 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑥) = 𝑁𝑁 = (♯‘𝑥))
1211biimpi 206 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑥) = 𝑁𝑁 = (♯‘𝑥))
1310, 12simpl2im 659 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 = (♯‘𝑥))
1413, 3eleq2s 2857 . . . . . . . . . . . . . . . . 17 (𝑥𝑊𝑁 = (♯‘𝑥))
1514adantr 472 . . . . . . . . . . . . . . . 16 ((𝑥𝑊𝑦𝑊) → 𝑁 = (♯‘𝑥))
1615adantr 472 . . . . . . . . . . . . . . 15 (((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑁 = (♯‘𝑥))
179clwwlknwrd 27183 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → 𝑦 ∈ Word (Vtx‘𝐺))
1817, 3eleq2s 2857 . . . . . . . . . . . . . . . . . . . 20 (𝑦𝑊𝑦 ∈ Word (Vtx‘𝐺))
1918adantl 473 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝑊𝑦𝑊) → 𝑦 ∈ Word (Vtx‘𝐺))
2019adantr 472 . . . . . . . . . . . . . . . . . 18 (((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∈ Word (Vtx‘𝐺))
2120adantl 473 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → 𝑦 ∈ Word (Vtx‘𝐺))
22 simprr 813 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (♯‘𝑥) = (♯‘𝑦))
2321, 22cshwcshid 13793 . . . . . . . . . . . . . . . 16 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → ((𝑛 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
24 oveq2 6822 . . . . . . . . . . . . . . . . . . 19 (𝑁 = (♯‘𝑥) → (0...𝑁) = (0...(♯‘𝑥)))
25 oveq2 6822 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑥) = (♯‘𝑦) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
2625adantl 473 . . . . . . . . . . . . . . . . . . 19 (((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
2724, 26sylan9eq 2814 . . . . . . . . . . . . . . . . . 18 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (0...𝑁) = (0...(♯‘𝑦)))
2827eleq2d 2825 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(♯‘𝑦))))
2928anbi1d 743 . . . . . . . . . . . . . . . 16 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) ↔ (𝑛 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛))))
3024adantr 472 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (0...𝑁) = (0...(♯‘𝑥)))
3130rexeqdv 3284 . . . . . . . . . . . . . . . 16 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚) ↔ ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
3223, 29, 313imtr4d 283 . . . . . . . . . . . . . . 15 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
3316, 32mpancom 706 . . . . . . . . . . . . . 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 3168 . . . . . . . . . . . 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 86 . . . . . . . . . 10 ((𝑥𝑊𝑦𝑊) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((♯‘𝑥) = (♯‘𝑦) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
38373impia 1110 . . . . . . . . 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 6822 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
4140eqeq2d 2770 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
4241cbvrexv 3311 . . . . . . . 8 (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))
4339, 42sylibr 224 . . . . . . 7 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))
447, 8, 433jca 1123 . . . . . 6 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → (𝑦𝑊𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)))
453, 4erclwwlkneq 27219 . . . . . . 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 236 . . . . 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 230 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → ((♯‘𝑥) = (♯‘𝑦) → 𝑦 𝑥)))
505, 49mpdd 43 . 2 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦𝑦 𝑥))
511, 2, 50mp2an 710 1 (𝑥 𝑦𝑦 𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∃wrex 3051  Vcvv 3340   class class class wbr 4804  {copab 4864  ‘cfv 6049  (class class class)co 6814  0cc0 10148  ...cfz 12539  ♯chash 13331  Word cword 13497   cyclShift ccsh 13754  Vtxcvtx 26094   ClWWalksN cclwwlkn 27168 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-pm 8028  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-inf 8516  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-n0 11505  df-z 11590  df-uz 11900  df-rp 12046  df-fz 12540  df-fzo 12680  df-fl 12807  df-mod 12883  df-hash 13332  df-word 13505  df-concat 13507  df-substr 13509  df-csh 13755  df-clwwlk 27126  df-clwwlkn 27170 This theorem is referenced by:  erclwwlkn  27224  eclclwwlkn1  27227
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