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Theorem erclwwlknsym 27776
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 erclwwlkn.w . . . 4 𝑊 = (𝑁 ClWWalksN 𝐺)
2 erclwwlkn.r . . . 4 = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
31, 2erclwwlkneqlen 27774 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
41, 2erclwwlkneq 27773 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 ↔ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))))
5 simpl2 1184 . . . . . . 7 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦𝑊)
6 simpl1 1183 . . . . . . 7 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑥𝑊)
7 eqid 2818 . . . . . . . . . . . . . . . . . . . 20 (Vtx‘𝐺) = (Vtx‘𝐺)
87clwwlknbp 27740 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
9 eqcom 2825 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑥) = 𝑁𝑁 = (♯‘𝑥))
109biimpi 217 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑥) = 𝑁𝑁 = (♯‘𝑥))
118, 10simpl2im 504 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 = (♯‘𝑥))
1211, 1eleq2s 2928 . . . . . . . . . . . . . . . . 17 (𝑥𝑊𝑁 = (♯‘𝑥))
1312adantr 481 . . . . . . . . . . . . . . . 16 ((𝑥𝑊𝑦𝑊) → 𝑁 = (♯‘𝑥))
1413adantr 481 . . . . . . . . . . . . . . 15 (((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑁 = (♯‘𝑥))
157clwwlknwrd 27739 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → 𝑦 ∈ Word (Vtx‘𝐺))
1615, 1eleq2s 2928 . . . . . . . . . . . . . . . . . . . 20 (𝑦𝑊𝑦 ∈ Word (Vtx‘𝐺))
1716adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝑊𝑦𝑊) → 𝑦 ∈ Word (Vtx‘𝐺))
1817adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∈ Word (Vtx‘𝐺))
1918adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → 𝑦 ∈ Word (Vtx‘𝐺))
20 simprr 769 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (♯‘𝑥) = (♯‘𝑦))
2119, 20cshwcshid 14177 . . . . . . . . . . . . . . . 16 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → ((𝑛 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
22 oveq2 7153 . . . . . . . . . . . . . . . . . . 19 (𝑁 = (♯‘𝑥) → (0...𝑁) = (0...(♯‘𝑥)))
23 oveq2 7153 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑥) = (♯‘𝑦) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
2423adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
2522, 24sylan9eq 2873 . . . . . . . . . . . . . . . . . 18 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (0...𝑁) = (0...(♯‘𝑦)))
2625eleq2d 2895 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(♯‘𝑦))))
2726anbi1d 629 . . . . . . . . . . . . . . . 16 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) ↔ (𝑛 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛))))
2822adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (0...𝑁) = (0...(♯‘𝑥)))
2928rexeqdv 3414 . . . . . . . . . . . . . . . 16 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚) ↔ ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
3021, 27, 293imtr4d 295 . . . . . . . . . . . . . . 15 ((𝑁 = (♯‘𝑥) ∧ ((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
3114, 30mpancom 684 . . . . . . . . . . . . . 14 (((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
3231expd 416 . . . . . . . . . . . . 13 (((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → (𝑛 ∈ (0...𝑁) → (𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
3332rexlimdv 3280 . . . . . . . . . . . 12 (((𝑥𝑊𝑦𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
3433ex 413 . . . . . . . . . . 11 ((𝑥𝑊𝑦𝑊) → ((♯‘𝑥) = (♯‘𝑦) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
3534com23 86 . . . . . . . . . 10 ((𝑥𝑊𝑦𝑊) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((♯‘𝑥) = (♯‘𝑦) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
36353impia 1109 . . . . . . . . 9 ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((♯‘𝑥) = (♯‘𝑦) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
3736imp 407 . . . . . . . 8 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))
38 oveq2 7153 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
3938eqeq2d 2829 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
4039cbvrexvw 3448 . . . . . . . 8 (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))
4137, 40sylibr 235 . . . . . . 7 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))
425, 6, 413jca 1120 . . . . . 6 (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → (𝑦𝑊𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)))
431, 2erclwwlkneq 27773 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 𝑥 ↔ (𝑦𝑊𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
4443ancoms 459 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑦 𝑥 ↔ (𝑦𝑊𝑥𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
4542, 44syl5ibr 247 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 𝑥))
4645expd 416 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((♯‘𝑥) = (♯‘𝑦) → 𝑦 𝑥)))
474, 46sylbid 241 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → ((♯‘𝑥) = (♯‘𝑦) → 𝑦 𝑥)))
483, 47mpdd 43 . 2 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦𝑦 𝑥))
4948el2v 3499 1 (𝑥 𝑦𝑦 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3136  Vcvv 3492   class class class wbr 5057  {copab 5119  cfv 6348  (class class class)co 7145  0cc0 10525  ...cfz 12880  chash 13678  Word cword 13849   cyclShift ccsh 14138  Vtxcvtx 26708   ClWWalksN cclwwlkn 27729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-hash 13679  df-word 13850  df-concat 13911  df-substr 13991  df-pfx 14021  df-csh 14139  df-clwwlk 27687  df-clwwlkn 27730
This theorem is referenced by:  erclwwlkn  27778  eclclwwlkn1  27781
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