MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwwlknscsh Structured version   Visualization version   GIF version

Theorem clwwlknscsh 29055
Description: The set of cyclical shifts of a word representing a closed walk is the set of closed walks represented by cyclical shifts of a word. (Contributed by Alexander van der Vekens, 15-Jun-2018.) (Revised by AV, 30-Apr-2021.)
Assertion
Ref Expression
clwwlknscsh ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → {𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑊 cyclShift 𝑛)} = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑊 cyclShift 𝑛)})
Distinct variable groups:   𝑛,𝐺,𝑊   𝑛,𝑁,𝑊   𝑛,𝑊,𝑊

Proof of Theorem clwwlknscsh
Dummy variables 𝑀 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2737 . . . 4 (𝑊 = 𝑥 → (𝑊 = (𝑊 cyclShift 𝑛) ↔ 𝑥 = (𝑊 cyclShift 𝑛)))
21rexbidv 3172 . . 3 (𝑊 = 𝑥 → (∃𝑛 ∈ (0...𝑁)𝑊 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)))
32cbvrabv 3416 . 2 {𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑊 cyclShift 𝑛)} = {𝑥 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)}
4 eqid 2733 . . . . . . . 8 (Vtx‘𝐺) = (Vtx‘𝐺)
54clwwlknwrd 29027 . . . . . . 7 (𝑀 ∈ (𝑁 ClWWalksN 𝐺) → 𝑀 ∈ Word (Vtx‘𝐺))
65ad2antrl 727 . . . . . 6 (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛))) → 𝑀 ∈ Word (Vtx‘𝐺))
7 simprr 772 . . . . . 6 (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛))
86, 7jca 513 . . . . 5 (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛))) → (𝑀 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛)))
9 simprr 772 . . . . . . . . . . . . 13 (((𝑀 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺))) → 𝑊 ∈ (𝑁 ClWWalksN 𝐺))
10 simpllr 775 . . . . . . . . . . . . 13 ((((𝑀 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺))) ∧ 𝑀 = (𝑊 cyclShift 𝑛)) → 𝑛 ∈ (0...𝑁))
11 clwwnisshclwwsn 29052 . . . . . . . . . . . . 13 ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑛 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺))
129, 10, 11syl2an2r 684 . . . . . . . . . . . 12 ((((𝑀 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺))) ∧ 𝑀 = (𝑊 cyclShift 𝑛)) → (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺))
13 eleq1 2822 . . . . . . . . . . . . 13 (𝑀 = (𝑊 cyclShift 𝑛) → (𝑀 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺)))
1413adantl 483 . . . . . . . . . . . 12 ((((𝑀 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺))) ∧ 𝑀 = (𝑊 cyclShift 𝑛)) → (𝑀 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺)))
1512, 14mpbird 257 . . . . . . . . . . 11 ((((𝑀 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺))) ∧ 𝑀 = (𝑊 cyclShift 𝑛)) → 𝑀 ∈ (𝑁 ClWWalksN 𝐺))
1615exp31 421 . . . . . . . . . 10 ((𝑀 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑀 = (𝑊 cyclShift 𝑛) → 𝑀 ∈ (𝑁 ClWWalksN 𝐺))))
1716com23 86 . . . . . . . . 9 ((𝑀 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) → (𝑀 = (𝑊 cyclShift 𝑛) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → 𝑀 ∈ (𝑁 ClWWalksN 𝐺))))
1817rexlimdva 3149 . . . . . . . 8 (𝑀 ∈ Word (Vtx‘𝐺) → (∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → 𝑀 ∈ (𝑁 ClWWalksN 𝐺))))
1918imp 408 . . . . . . 7 ((𝑀 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛)) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → 𝑀 ∈ (𝑁 ClWWalksN 𝐺)))
2019impcom 409 . . . . . 6 (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑀 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛))) → 𝑀 ∈ (𝑁 ClWWalksN 𝐺))
21 simprr 772 . . . . . 6 (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑀 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛))
2220, 21jca 513 . . . . 5 (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑀 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛))) → (𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛)))
238, 22impbida 800 . . . 4 ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → ((𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛)) ↔ (𝑀 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛))))
24 eqeq1 2737 . . . . . 6 (𝑥 = 𝑀 → (𝑥 = (𝑊 cyclShift 𝑛) ↔ 𝑀 = (𝑊 cyclShift 𝑛)))
2524rexbidv 3172 . . . . 5 (𝑥 = 𝑀 → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛)))
2625elrab 3649 . . . 4 (𝑀 ∈ {𝑥 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)} ↔ (𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛)))
27 eqeq1 2737 . . . . . 6 (𝑊 = 𝑀 → (𝑊 = (𝑊 cyclShift 𝑛) ↔ 𝑀 = (𝑊 cyclShift 𝑛)))
2827rexbidv 3172 . . . . 5 (𝑊 = 𝑀 → (∃𝑛 ∈ (0...𝑁)𝑊 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛)))
2928elrab 3649 . . . 4 (𝑀 ∈ {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑊 cyclShift 𝑛)} ↔ (𝑀 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑀 = (𝑊 cyclShift 𝑛)))
3023, 26, 293bitr4g 314 . . 3 ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑀 ∈ {𝑥 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)} ↔ 𝑀 ∈ {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑊 cyclShift 𝑛)}))
3130eqrdv 2731 . 2 ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → {𝑥 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)} = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑊 cyclShift 𝑛)})
323, 31eqtrid 2785 1 ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → {𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑊 cyclShift 𝑛)} = {𝑊 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑊 cyclShift 𝑛)})
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  âˆƒwrex 3070  {crab 3406  â€˜cfv 6500  (class class class)co 7361  0cc0 11059  â„•0cn0 12421  ...cfz 13433  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-ico 13279  df-fz 13434  df-fzo 13577  df-fl 13706  df-mod 13784  df-hash 14240  df-word 14412  df-lsw 14460  df-concat 14468  df-substr 14538  df-pfx 14568  df-csh 14686  df-clwwlk 28975  df-clwwlkn 29018
This theorem is referenced by:  hashecclwwlkn1  29070  umgrhashecclwwlk  29071
  Copyright terms: Public domain W3C validator