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Theorem erclwwlknsym 29320
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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlknsym (𝑥 ∌ 𝑊 → 𝑊 ∌ 𝑥)
Distinct variable groups:   𝑡,𝑊,𝑢   𝑛,𝑁,𝑢,𝑡,𝑥   𝑊,𝑛,𝑡,𝑢,𝑥   𝑛,𝑊
Allowed substitution hints:   ∌ (𝑥,𝑊,𝑢,𝑡,𝑛)   𝐺(𝑥,𝑊,𝑢,𝑡,𝑛)   𝑁(𝑊)   𝑊(𝑥,𝑊)

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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
31, 2erclwwlkneqlen 29318 . . 3 ((𝑥 ∈ V ∧ 𝑊 ∈ V) → (𝑥 ∌ 𝑊 → (♯‘𝑥) = (♯‘𝑊)))
41, 2erclwwlkneq 29317 . . . 4 ((𝑥 ∈ V ∧ 𝑊 ∈ V) → (𝑥 ∌ 𝑊 ↔ (𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛))))
5 simpl2 1192 . . . . . . 7 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑊)) → 𝑊 ∈ 𝑊)
6 simpl1 1191 . . . . . . 7 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑊)) → 𝑥 ∈ 𝑊)
7 eqid 2732 . . . . . . . . . . . . . . . . . . . 20 (Vtx‘𝐺) = (Vtx‘𝐺)
87clwwlknbp 29285 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
9 eqcom 2739 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑥) = 𝑁 ↔ 𝑁 = (♯‘𝑥))
109biimpi 215 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑥) = 𝑁 → 𝑁 = (♯‘𝑥))
118, 10simpl2im 504 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 = (♯‘𝑥))
1211, 1eleq2s 2851 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ 𝑊 → 𝑁 = (♯‘𝑥))
1312adantr 481 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) → 𝑁 = (♯‘𝑥))
1413adantr 481 . . . . . . . . . . . . . . 15 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊)) → 𝑁 = (♯‘𝑥))
157clwwlknwrd 29284 . . . . . . . . . . . . . . . . . . . . 21 (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ Word (Vtx‘𝐺))
1615, 1eleq2s 2851 . . . . . . . . . . . . . . . . . . . 20 (𝑊 ∈ 𝑊 → 𝑊 ∈ Word (Vtx‘𝐺))
1716adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) → 𝑊 ∈ Word (Vtx‘𝐺))
1817adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊)) → 𝑊 ∈ Word (Vtx‘𝐺))
1918adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊))) → 𝑊 ∈ Word (Vtx‘𝐺))
20 simprr 771 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊))) → (♯‘𝑥) = (♯‘𝑊))
2119, 20cshwcshid 14777 . . . . . . . . . . . . . . . 16 ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊))) → ((𝑛 ∈ (0...(♯‘𝑊)) ∧ 𝑥 = (𝑊 cyclShift 𝑛)) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑚)))
22 oveq2 7416 . . . . . . . . . . . . . . . . . . 19 (𝑁 = (♯‘𝑥) → (0...𝑁) = (0...(♯‘𝑥)))
23 oveq2 7416 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑥) = (♯‘𝑊) → (0...(♯‘𝑥)) = (0...(♯‘𝑊)))
2423adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊)) → (0...(♯‘𝑥)) = (0...(♯‘𝑊)))
2522, 24sylan9eq 2792 . . . . . . . . . . . . . . . . . 18 ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊))) → (0...𝑁) = (0...(♯‘𝑊)))
2625eleq2d 2819 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊))) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(♯‘𝑊))))
2726anbi1d 630 . . . . . . . . . . . . . . . 16 ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊))) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑊 cyclShift 𝑛)) ↔ (𝑛 ∈ (0...(♯‘𝑊)) ∧ 𝑥 = (𝑊 cyclShift 𝑛))))
2822adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊))) → (0...𝑁) = (0...(♯‘𝑥)))
2928rexeqdv 3326 . . . . . . . . . . . . . . . 16 ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊))) → (∃𝑚 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑚) ↔ ∃𝑚 ∈ (0...(♯‘𝑥))𝑊 = (𝑥 cyclShift 𝑚)))
3021, 27, 293imtr4d 293 . . . . . . . . . . . . . . 15 ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊))) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑊 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑚)))
3114, 30mpancom 686 . . . . . . . . . . . . . 14 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊)) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑊 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑚)))
3231expd 416 . . . . . . . . . . . . 13 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊)) → (𝑛 ∈ (0...𝑁) → (𝑥 = (𝑊 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑚))))
3332rexlimdv 3153 . . . . . . . . . . . 12 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑚)))
3433ex 413 . . . . . . . . . . 11 ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) → ((♯‘𝑥) = (♯‘𝑊) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑚))))
3534com23 86 . . . . . . . . . 10 ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) → ((♯‘𝑥) = (♯‘𝑊) → ∃𝑚 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑚))))
36353impia 1117 . . . . . . . . 9 ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) → ((♯‘𝑥) = (♯‘𝑊) → ∃𝑚 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑚)))
3736imp 407 . . . . . . . 8 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑊)) → ∃𝑚 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑚))
38 oveq2 7416 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
3938eqeq2d 2743 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑊 = (𝑥 cyclShift 𝑛) ↔ 𝑊 = (𝑥 cyclShift 𝑚)))
4039cbvrexvw 3235 . . . . . . . 8 (∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑚))
4137, 40sylibr 233 . . . . . . 7 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑊)) → ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛))
425, 6, 413jca 1128 . . . . . 6 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑊)) → (𝑊 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛)))
431, 2erclwwlkneq 29317 . . . . . . 7 ((𝑊 ∈ V ∧ 𝑥 ∈ V) → (𝑊 ∌ 𝑥 ↔ (𝑊 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛))))
4443ancoms 459 . . . . . 6 ((𝑥 ∈ V ∧ 𝑊 ∈ V) → (𝑊 ∌ 𝑥 ↔ (𝑊 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛))))
4542, 44imbitrrid 245 . . . . 5 ((𝑥 ∈ V ∧ 𝑊 ∈ V) → (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑊)) → 𝑊 ∌ 𝑥))
4645expd 416 . . . 4 ((𝑥 ∈ V ∧ 𝑊 ∈ V) → ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) → ((♯‘𝑥) = (♯‘𝑊) → 𝑊 ∌ 𝑥)))
474, 46sylbid 239 . . 3 ((𝑥 ∈ V ∧ 𝑊 ∈ V) → (𝑥 ∌ 𝑊 → ((♯‘𝑥) = (♯‘𝑊) → 𝑊 ∌ 𝑥)))
483, 47mpdd 43 . 2 ((𝑥 ∈ V ∧ 𝑊 ∈ V) → (𝑥 ∌ 𝑊 → 𝑊 ∌ 𝑥))
4948el2v 3482 1 (𝑥 ∌ 𝑊 → 𝑊 ∌ 𝑥)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  âˆƒwrex 3070  Vcvv 3474   class class class wbr 5148  {copab 5210  â€˜cfv 6543  (class class class)co 7408  0cc0 11109  ...cfz 13483  â™¯chash 14289  Word cword 14463   cyclShift ccsh 14737  Vtxcvtx 28253   ClWWalksN cclwwlkn 29274
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-sup 9436  df-inf 9437  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-n0 12472  df-z 12558  df-uz 12822  df-rp 12974  df-fz 13484  df-fzo 13627  df-fl 13756  df-mod 13834  df-hash 14290  df-word 14464  df-concat 14520  df-substr 14590  df-pfx 14620  df-csh 14738  df-clwwlk 29232  df-clwwlkn 29275
This theorem is referenced by:  erclwwlkn  29322  eclclwwlkn1  29325
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