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Theorem erclwwlknsym 29323
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 29321 . . 3 ((𝑥 ∈ V ∧ 𝑊 ∈ V) → (𝑥 ∌ 𝑊 → (♯‘𝑥) = (♯‘𝑊)))
41, 2erclwwlkneq 29320 . . . 4 ((𝑥 ∈ V ∧ 𝑊 ∈ V) → (𝑥 ∌ 𝑊 ↔ (𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛))))
5 simpl2 1193 . . . . . . 7 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑊)) → 𝑊 ∈ 𝑊)
6 simpl1 1192 . . . . . . 7 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑊)) → 𝑥 ∈ 𝑊)
7 eqid 2733 . . . . . . . . . . . . . . . . . . . 20 (Vtx‘𝐺) = (Vtx‘𝐺)
87clwwlknbp 29288 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
9 eqcom 2740 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑥) = 𝑁 ↔ 𝑁 = (♯‘𝑥))
109biimpi 215 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑥) = 𝑁 → 𝑁 = (♯‘𝑥))
118, 10simpl2im 505 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 = (♯‘𝑥))
1211, 1eleq2s 2852 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ 𝑊 → 𝑁 = (♯‘𝑥))
1312adantr 482 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) → 𝑁 = (♯‘𝑥))
1413adantr 482 . . . . . . . . . . . . . . 15 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊)) → 𝑁 = (♯‘𝑥))
157clwwlknwrd 29287 . . . . . . . . . . . . . . . . . . . . 21 (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ Word (Vtx‘𝐺))
1615, 1eleq2s 2852 . . . . . . . . . . . . . . . . . . . 20 (𝑊 ∈ 𝑊 → 𝑊 ∈ Word (Vtx‘𝐺))
1716adantl 483 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) → 𝑊 ∈ Word (Vtx‘𝐺))
1817adantr 482 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊)) → 𝑊 ∈ Word (Vtx‘𝐺))
1918adantl 483 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊))) → 𝑊 ∈ Word (Vtx‘𝐺))
20 simprr 772 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊))) → (♯‘𝑥) = (♯‘𝑊))
2119, 20cshwcshid 14778 . . . . . . . . . . . . . . . 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 7417 . . . . . . . . . . . . . . . . . . 19 (𝑁 = (♯‘𝑥) → (0...𝑁) = (0...(♯‘𝑥)))
23 oveq2 7417 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑥) = (♯‘𝑊) → (0...(♯‘𝑥)) = (0...(♯‘𝑊)))
2423adantl 483 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊)) → (0...(♯‘𝑥)) = (0...(♯‘𝑊)))
2522, 24sylan9eq 2793 . . . . . . . . . . . . . . . . . 18 ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊))) → (0...𝑁) = (0...(♯‘𝑊)))
2625eleq2d 2820 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊))) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(♯‘𝑊))))
2726anbi1d 631 . . . . . . . . . . . . . . . 16 ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊))) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑊 cyclShift 𝑛)) ↔ (𝑛 ∈ (0...(♯‘𝑊)) ∧ 𝑥 = (𝑊 cyclShift 𝑛))))
2822adantr 482 . . . . . . . . . . . . . . . . 17 ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊))) → (0...𝑁) = (0...(♯‘𝑥)))
2928rexeqdv 3327 . . . . . . . . . . . . . . . 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 294 . . . . . . . . . . . . . . 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 687 . . . . . . . . . . . . . 14 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊)) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑊 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑚)))
3231expd 417 . . . . . . . . . . . . 13 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊)) → (𝑛 ∈ (0...𝑁) → (𝑥 = (𝑊 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑚))))
3332rexlimdv 3154 . . . . . . . . . . . 12 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑊)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑚)))
3433ex 414 . . . . . . . . . . 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 1118 . . . . . . . . 9 ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) → ((♯‘𝑥) = (♯‘𝑊) → ∃𝑚 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑚)))
3736imp 408 . . . . . . . 8 (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑊)) → ∃𝑚 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑚))
38 oveq2 7417 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
3938eqeq2d 2744 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑊 = (𝑥 cyclShift 𝑛) ↔ 𝑊 = (𝑥 cyclShift 𝑚)))
4039cbvrexvw 3236 . . . . . . . 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 1129 . . . . . 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 29320 . . . . . . 7 ((𝑊 ∈ V ∧ 𝑥 ∈ V) → (𝑊 ∌ 𝑥 ↔ (𝑊 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑥 cyclShift 𝑛))))
4443ancoms 460 . . . . . 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 417 . . . 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 3483 1 (𝑥 ∌ 𝑊 → 𝑊 ∌ 𝑥)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  âˆƒwrex 3071  Vcvv 3475   class class class wbr 5149  {copab 5211  â€˜cfv 6544  (class class class)co 7409  0cc0 11110  ...cfz 13484  â™¯chash 14290  Word cword 14464   cyclShift ccsh 14738  Vtxcvtx 28256   ClWWalksN cclwwlkn 29277
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  df-clwwlkn 29278
This theorem is referenced by:  erclwwlkn  29325  eclclwwlkn1  29328
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