Theorem clwwlknscsh 30353

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)) → {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)})

Distinct variable groups:   𝑛,𝐺,𝑦   𝑛,𝑁,𝑦   𝑛,𝑊,𝑦

Proof of Theorem clwwlknscsh

Dummy variables 𝑤 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2773	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑊 cyclShift 𝑛) ↔ 𝑥 = (𝑊 cyclShift 𝑛)))
2	1	rexbidv 3195	. . 3 ⊢ (𝑦 = 𝑥 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)))
3	2	cbvrabv 3433	. 2 ⊢ {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑥 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)}
4		eqid 2769	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	clwwlknwrd 30325	. . . . . . 7 ⊢ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → 𝑤 ∈ Word (Vtx‘𝐺))
6	5	ad2antrl 740	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → 𝑤 ∈ Word (Vtx‘𝐺))
7		simprr 784	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))
8	6, 7	jca 520	. . . . 5 ⊢ (((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)))
9		simprr 784	. . . . . . . . . . . . 13 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺))) → 𝑊 ∈ (𝑁 ClWWalksN 𝐺))
10		simpllr 787	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → 𝑛 ∈ (0...𝑁))
11		clwwnisshclwwsn 30350	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑛 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺))
12	9, 10, 11	syl2an2r 697	. . . . . . . . . . . 12 ⊢ ((((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺))
13		eleq1 2857	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑊 cyclShift 𝑛) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺)))
14	13	adantl 486	. . . . . . . . . . . 12 ⊢ ((((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺)))
15	12, 14	mpbird 260	. . . . . . . . . . 11 ⊢ ((((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → 𝑤 ∈ (𝑁 ClWWalksN 𝐺))
16	15	exp31 424	. . . . . . . . . 10 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑤 = (𝑊 cyclShift 𝑛) → 𝑤 ∈ (𝑁 ClWWalksN 𝐺))))
17	16	com23 87	. . . . . . . . 9 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) → (𝑤 = (𝑊 cyclShift 𝑛) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → 𝑤 ∈ (𝑁 ClWWalksN 𝐺))))
18	17	rexlimdva 3172	. . . . . . . 8 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → 𝑤 ∈ (𝑁 ClWWalksN 𝐺))))
19	18	imp 411	. . . . . . 7 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → 𝑤 ∈ (𝑁 ClWWalksN 𝐺)))
20	19	impcom 412	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → 𝑤 ∈ (𝑁 ClWWalksN 𝐺))
21		simprr 784	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))
22	20, 21	jca 520	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
23	8, 22	impbida 812	. . . 4 ⊢ ((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))))
24		eqeq1 2773	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥 = (𝑊 cyclShift 𝑛) ↔ 𝑤 = (𝑊 cyclShift 𝑛)))
25	24	rexbidv 3195	. . . . 5 ⊢ (𝑥 = 𝑤 → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
26	25	elrab 3659	. . . 4 ⊢ (𝑤 ∈ {𝑥 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)} ↔ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
27		eqeq1 2773	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 = (𝑊 cyclShift 𝑛) ↔ 𝑤 = (𝑊 cyclShift 𝑛)))
28	27	rexbidv 3195	. . . . 5 ⊢ (𝑦 = 𝑤 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
29	28	elrab 3659	. . . 4 ⊢ (𝑤 ∈ {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
30	23, 26, 29	3bitr4g 317	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑤 ∈ {𝑥 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)} ↔ 𝑤 ∈ {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)}))
31	30	eqrdv 2767	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → {𝑥 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)})
32	3, 31	eqtrid 2816	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  {crab 3423  ‘cfv 6537  (class class class)co 7411  0cc0 11099  ℕ₀cn0 12503  ...cfz 13534  Word cword 14549   cyclShift ccsh 14824  Vtxcvtx 29286   ClWWalksN cclwwlkn 30315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9401  df-inf 9402  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-n0 12504  df-z 12591  df-uz 12862  df-rp 13016  df-ico 13377  df-fz 13535  df-fzo 13682  df-fl 13824  df-mod 13902  df-hash 14366  df-word 14550  df-lsw 14599  df-concat 14607  df-substr 14678  df-pfx 14708  df-csh 14825  df-clwwlk 30273  df-clwwlkn 30316

This theorem is referenced by:  hashecclwwlkn1  30368  umgrhashecclwwlk  30369