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Theorem erclwwlknsym 29063
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 29061 . . 3 ((𝑥 ∈ V ∧ 𝑊 ∈ V) → (𝑥 ∌ 𝑊 → (♯‘𝑥) = (♯‘𝑊)))
41, 2erclwwlkneq 29060 . . . 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 29028 . . . . . . . . . . . . . . . . . . 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 29027 . . . . . . . . . . . . . . . . . . . . 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 14725 . . . . . . . . . . . . . . . 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 7369 . . . . . . . . . . . . . . . . . . 19 (𝑁 = (♯‘𝑥) → (0...𝑁) = (0...(♯‘𝑥)))
23 oveq2 7369 . . . . . . . . . . . . . . . . . . . 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 3313 . . . . . . . . . . . . . . . 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 3147 . . . . . . . . . . . 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 7369 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
3938eqeq2d 2744 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑊 = (𝑥 cyclShift 𝑛) ↔ 𝑊 = (𝑥 cyclShift 𝑚)))
4039cbvrexvw 3225 . . . . . . . 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 29060 . . . . . . 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, 44syl5ibr 246 . . . . 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 3455 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 3070  Vcvv 3447   class class class wbr 5109  {copab 5171  â€˜cfv 6500  (class class class)co 7361  0cc0 11059  ...cfz 13433  â™¯chash 14239  Word cword 14411   cyclShift ccsh 14685  Vtxcvtx 27996   ClWWalksN cclwwlkn 29017
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-sup 9386  df-inf 9387  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-n0 12422  df-z 12508  df-uz 12772  df-rp 12924  df-fz 13434  df-fzo 13577  df-fl 13706  df-mod 13784  df-hash 14240  df-word 14412  df-concat 14468  df-substr 14538  df-pfx 14568  df-csh 14686  df-clwwlk 28975  df-clwwlkn 29018
This theorem is referenced by:  erclwwlkn  29065  eclclwwlkn1  29068
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