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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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