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Theorem erclwwlksym 29263
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 = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlksym (𝑥 𝑦𝑦 𝑥)
Distinct variable groups:   𝑛,𝐺,𝑢,𝑤   𝑥,𝑛,𝑢,𝑤,𝑦
Allowed substitution hints:   (𝑥,𝑦,𝑤,𝑢,𝑛)   𝐺(𝑥,𝑦)

Proof of Theorem erclwwlksym
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 erclwwlk.r . . . 4 = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
21erclwwlkeqlen 29261 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
31erclwwlkeq 29260 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))))
4 simpl2 1192 . . . . . . 7 (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∈ (ClWWalks‘𝐺))
5 simpl1 1191 . . . . . . 7 (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑥 ∈ (ClWWalks‘𝐺))
6 eqid 2732 . . . . . . . . . . . . . . . . . 18 (Vtx‘𝐺) = (Vtx‘𝐺)
76clwwlkbp 29227 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑦 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ≠ ∅))
87simp2d 1143 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (ClWWalks‘𝐺) → 𝑦 ∈ Word (Vtx‘𝐺))
98ad2antlr 725 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∈ Word (Vtx‘𝐺))
10 simpr 485 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ (♯‘𝑥) = (♯‘𝑦)) → (♯‘𝑥) = (♯‘𝑦))
119, 10cshwcshid 14774 . . . . . . . . . . . . . 14 (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ (♯‘𝑥) = (♯‘𝑦)) → ((𝑛 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
1211expd 416 . . . . . . . . . . . . 13 (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ (♯‘𝑥) = (♯‘𝑦)) → (𝑛 ∈ (0...(♯‘𝑦)) → (𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))))
1312rexlimdv 3153 . . . . . . . . . . . 12 (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
1413ex 413 . . . . . . . . . . 11 ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → ((♯‘𝑥) = (♯‘𝑦) → (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))))
1514com23 86 . . . . . . . . . 10 ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((♯‘𝑥) = (♯‘𝑦) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))))
16153impia 1117 . . . . . . . . 9 ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((♯‘𝑥) = (♯‘𝑦) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
1716imp 407 . . . . . . . 8 (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))
18 oveq2 7413 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
1918eqeq2d 2743 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
2019cbvrexvw 3235 . . . . . . . 8 (∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))
2117, 20sylibr 233 . . . . . . 7 (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
224, 5, 213jca 1128 . . . . . 6 (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))
231erclwwlkeq 29260 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 𝑥 ↔ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))))
2423ancoms 459 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑦 𝑥 ↔ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))))
2522, 24imbitrrid 245 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 𝑥))
2625expd 416 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((♯‘𝑥) = (♯‘𝑦) → 𝑦 𝑥)))
273, 26sylbid 239 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → ((♯‘𝑥) = (♯‘𝑦) → 𝑦 𝑥)))
282, 27mpdd 43 . 2 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦𝑦 𝑥))
2928el2v 3482 1 (𝑥 𝑦𝑦 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wrex 3070  Vcvv 3474  c0 4321   class class class wbr 5147  {copab 5209  cfv 6540  (class class class)co 7405  0cc0 11106  ...cfz 13480  chash 14286  Word cword 14460   cyclShift ccsh 14734  Vtxcvtx 28245  ClWWalkscclwwlk 29223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-fz 13481  df-fzo 13624  df-fl 13753  df-mod 13831  df-hash 14287  df-word 14461  df-concat 14517  df-substr 14587  df-pfx 14617  df-csh 14735  df-clwwlk 29224
This theorem is referenced by:  erclwwlk  29265
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