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Theorem erclwwlksym 29805
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
erclwwlk.r ∼ = {βŸ¨π‘’, π‘€βŸ© ∣ (𝑒 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑀 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘€))𝑒 = (𝑀 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlksym (π‘₯ ∼ 𝑦 β†’ 𝑦 ∼ π‘₯)
Distinct variable groups:   𝑛,𝐺,𝑒,𝑀   π‘₯,𝑛,𝑒,𝑀,𝑦
Allowed substitution hints:   ∼ (π‘₯,𝑦,𝑀,𝑒,𝑛)   𝐺(π‘₯,𝑦)

Proof of Theorem erclwwlksym
Dummy variable π‘š is distinct from all other variables.
StepHypRef Expression
1 erclwwlk.r . . . 4 ∼ = {βŸ¨π‘’, π‘€βŸ© ∣ (𝑒 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑀 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘€))𝑒 = (𝑀 cyclShift 𝑛))}
21erclwwlkeqlen 29803 . . 3 ((π‘₯ ∈ V ∧ 𝑦 ∈ V) β†’ (π‘₯ ∼ 𝑦 β†’ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)))
31erclwwlkeq 29802 . . . 4 ((π‘₯ ∈ V ∧ 𝑦 ∈ V) β†’ (π‘₯ ∼ 𝑦 ↔ (π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛))))
4 simpl2 1190 . . . . . . 7 (((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛)) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ 𝑦 ∈ (ClWWalksβ€˜πΊ))
5 simpl1 1189 . . . . . . 7 (((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛)) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ π‘₯ ∈ (ClWWalksβ€˜πΊ))
6 eqid 2727 . . . . . . . . . . . . . . . . . 18 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
76clwwlkbp 29769 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ClWWalksβ€˜πΊ) β†’ (𝐺 ∈ V ∧ 𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑦 β‰  βˆ…))
87simp2d 1141 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (ClWWalksβ€˜πΊ) β†’ 𝑦 ∈ Word (Vtxβ€˜πΊ))
98ad2antlr 726 . . . . . . . . . . . . . . 15 (((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ 𝑦 ∈ Word (Vtxβ€˜πΊ))
10 simpr 484 . . . . . . . . . . . . . . 15 (((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ (β™―β€˜π‘₯) = (β™―β€˜π‘¦))
119, 10cshwcshid 14796 . . . . . . . . . . . . . 14 (((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ ((𝑛 ∈ (0...(β™―β€˜π‘¦)) ∧ π‘₯ = (𝑦 cyclShift 𝑛)) β†’ βˆƒπ‘š ∈ (0...(β™―β€˜π‘₯))𝑦 = (π‘₯ cyclShift π‘š)))
1211expd 415 . . . . . . . . . . . . 13 (((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ (𝑛 ∈ (0...(β™―β€˜π‘¦)) β†’ (π‘₯ = (𝑦 cyclShift 𝑛) β†’ βˆƒπ‘š ∈ (0...(β™―β€˜π‘₯))𝑦 = (π‘₯ cyclShift π‘š))))
1312rexlimdv 3148 . . . . . . . . . . . 12 (((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ (βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛) β†’ βˆƒπ‘š ∈ (0...(β™―β€˜π‘₯))𝑦 = (π‘₯ cyclShift π‘š)))
1413ex 412 . . . . . . . . . . 11 ((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) β†’ ((β™―β€˜π‘₯) = (β™―β€˜π‘¦) β†’ (βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛) β†’ βˆƒπ‘š ∈ (0...(β™―β€˜π‘₯))𝑦 = (π‘₯ cyclShift π‘š))))
1514com23 86 . . . . . . . . . 10 ((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) β†’ (βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛) β†’ ((β™―β€˜π‘₯) = (β™―β€˜π‘¦) β†’ βˆƒπ‘š ∈ (0...(β™―β€˜π‘₯))𝑦 = (π‘₯ cyclShift π‘š))))
16153impia 1115 . . . . . . . . 9 ((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛)) β†’ ((β™―β€˜π‘₯) = (β™―β€˜π‘¦) β†’ βˆƒπ‘š ∈ (0...(β™―β€˜π‘₯))𝑦 = (π‘₯ cyclShift π‘š)))
1716imp 406 . . . . . . . 8 (((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛)) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ βˆƒπ‘š ∈ (0...(β™―β€˜π‘₯))𝑦 = (π‘₯ cyclShift π‘š))
18 oveq2 7422 . . . . . . . . . 10 (𝑛 = π‘š β†’ (π‘₯ cyclShift 𝑛) = (π‘₯ cyclShift π‘š))
1918eqeq2d 2738 . . . . . . . . 9 (𝑛 = π‘š β†’ (𝑦 = (π‘₯ cyclShift 𝑛) ↔ 𝑦 = (π‘₯ cyclShift π‘š)))
2019cbvrexvw 3230 . . . . . . . 8 (βˆƒπ‘› ∈ (0...(β™―β€˜π‘₯))𝑦 = (π‘₯ cyclShift 𝑛) ↔ βˆƒπ‘š ∈ (0...(β™―β€˜π‘₯))𝑦 = (π‘₯ cyclShift π‘š))
2117, 20sylibr 233 . . . . . . 7 (((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛)) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘₯))𝑦 = (π‘₯ cyclShift 𝑛))
224, 5, 213jca 1126 . . . . . 6 (((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛)) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ (𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘₯))𝑦 = (π‘₯ cyclShift 𝑛)))
231erclwwlkeq 29802 . . . . . . 7 ((𝑦 ∈ V ∧ π‘₯ ∈ V) β†’ (𝑦 ∼ π‘₯ ↔ (𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘₯))𝑦 = (π‘₯ cyclShift 𝑛))))
2423ancoms 458 . . . . . 6 ((π‘₯ ∈ V ∧ 𝑦 ∈ V) β†’ (𝑦 ∼ π‘₯ ↔ (𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘₯))𝑦 = (π‘₯ cyclShift 𝑛))))
2522, 24imbitrrid 245 . . . . 5 ((π‘₯ ∈ V ∧ 𝑦 ∈ V) β†’ (((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛)) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ 𝑦 ∼ π‘₯))
2625expd 415 . . . 4 ((π‘₯ ∈ V ∧ 𝑦 ∈ V) β†’ ((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛)) β†’ ((β™―β€˜π‘₯) = (β™―β€˜π‘¦) β†’ 𝑦 ∼ π‘₯)))
273, 26sylbid 239 . . 3 ((π‘₯ ∈ V ∧ 𝑦 ∈ V) β†’ (π‘₯ ∼ 𝑦 β†’ ((β™―β€˜π‘₯) = (β™―β€˜π‘¦) β†’ 𝑦 ∼ π‘₯)))
282, 27mpdd 43 . 2 ((π‘₯ ∈ V ∧ 𝑦 ∈ V) β†’ (π‘₯ ∼ 𝑦 β†’ 𝑦 ∼ π‘₯))
2928el2v 3477 1 (π‘₯ ∼ 𝑦 β†’ 𝑦 ∼ π‘₯)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  βˆƒwrex 3065  Vcvv 3469  βˆ…c0 4318   class class class wbr 5142  {copab 5204  β€˜cfv 6542  (class class class)co 7414  0cc0 11124  ...cfz 13502  β™―chash 14307  Word cword 14482   cyclShift ccsh 14756  Vtxcvtx 28783  ClWWalkscclwwlk 29765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-map 8836  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-sup 9451  df-inf 9452  df-card 9948  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-div 11888  df-nn 12229  df-2 12291  df-n0 12489  df-z 12575  df-uz 12839  df-rp 12993  df-fz 13503  df-fzo 13646  df-fl 13775  df-mod 13853  df-hash 14308  df-word 14483  df-concat 14539  df-substr 14609  df-pfx 14639  df-csh 14757  df-clwwlk 29766
This theorem is referenced by:  erclwwlk  29807
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