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Theorem erclwwlksym 29274
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 29272 . . 3 ((π‘₯ ∈ V ∧ 𝑦 ∈ V) β†’ (π‘₯ ∼ 𝑦 β†’ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)))
31erclwwlkeq 29271 . . . 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 1193 . . . . . . 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 1192 . . . . . . 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 2733 . . . . . . . . . . . . . . . . . 18 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
76clwwlkbp 29238 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ClWWalksβ€˜πΊ) β†’ (𝐺 ∈ V ∧ 𝑦 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑦 β‰  βˆ…))
87simp2d 1144 . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . 15 (((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ (β™―β€˜π‘₯) = (β™―β€˜π‘¦))
119, 10cshwcshid 14778 . . . . . . . . . . . . . 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 417 . . . . . . . . . . . . 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 3154 . . . . . . . . . . . 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 414 . . . . . . . . . . 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 1118 . . . . . . . . 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 408 . . . . . . . 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 7417 . . . . . . . . . 10 (𝑛 = π‘š β†’ (π‘₯ cyclShift 𝑛) = (π‘₯ cyclShift π‘š))
1918eqeq2d 2744 . . . . . . . . 9 (𝑛 = π‘š β†’ (𝑦 = (π‘₯ cyclShift 𝑛) ↔ 𝑦 = (π‘₯ cyclShift π‘š)))
2019cbvrexvw 3236 . . . . . . . 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 1129 . . . . . 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 29271 . . . . . . 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 460 . . . . . 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 417 . . . 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 3483 1 (π‘₯ ∼ 𝑦 β†’ 𝑦 ∼ π‘₯)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆƒwrex 3071  Vcvv 3475  βˆ…c0 4323   class class class wbr 5149  {copab 5211  β€˜cfv 6544  (class class class)co 7409  0cc0 11110  ...cfz 13484  β™―chash 14290  Word cword 14464   cyclShift ccsh 14738  Vtxcvtx 28256  ClWWalkscclwwlk 29234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-rp 12975  df-fz 13485  df-fzo 13628  df-fl 13757  df-mod 13835  df-hash 14291  df-word 14465  df-concat 14521  df-substr 14591  df-pfx 14621  df-csh 14739  df-clwwlk 29235
This theorem is referenced by:  erclwwlk  29276
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