Theorem clwwlknscsh 30710

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 2767	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑊 cyclShift 𝑛) ↔ 𝑥 = (𝑊 cyclShift 𝑛)))
2	1	rexbidv 3190	. . 3 ⊢ (𝑦 = 𝑥 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)))
3	2	cbvrabv 3426	. 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 2763	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	clwwlknwrd 30682	. . . . . . 7 ⊢ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → 𝑤 ∈ Word (Vtx‘𝐺))
6	5	ad2antrl 741	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → 𝑤 ∈ Word (Vtx‘𝐺))
7		simprr 785	. . . . . 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 521	. . . . 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 785	. . . . . . . . . . . . 13 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺))) → 𝑊 ∈ (𝑁 ClWWalksN 𝐺))
10		simpllr 788	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → 𝑛 ∈ (0...𝑁))
11		clwwnisshclwwsn 30707	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑛 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺))
12	9, 10, 11	syl2an2r 698	. . . . . . . . . . . 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 2852	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑊 cyclShift 𝑛) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺)))
14	13	adantl 487	. . . . . . . . . . . 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 425	. . . . . . . . . 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 3167	. . . . . . . 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 412	. . . . . . 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 413	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → 𝑤 ∈ (𝑁 ClWWalksN 𝐺))
21		simprr 785	. . . . . 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 521	. . . . 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 813	. . . 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 2767	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥 = (𝑊 cyclShift 𝑛) ↔ 𝑤 = (𝑊 cyclShift 𝑛)))
25	24	rexbidv 3190	. . . . 5 ⊢ (𝑥 = 𝑤 → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
26	25	elrab 3648	. . . 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 2767	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 = (𝑊 cyclShift 𝑛) ↔ 𝑤 = (𝑊 cyclShift 𝑛)))
28	27	rexbidv 3190	. . . . 5 ⊢ (𝑦 = 𝑤 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
29	28	elrab 3648	. . . 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 2761	. 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 2810	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)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {crab 3416  ‘cfv 6566  (class class class)co 7451  0cc0 11239  ℕ₀cn0 12646  ...cfz 13679  Word cword 14699   cyclShift ccsh 14981  Vtxcvtx 29631   ClWWalksN cclwwlkn 30672

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-sup 9471  df-inf 9472  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-n0 12647  df-z 12734  df-uz 13006  df-rp 13161  df-ico 13522  df-fz 13680  df-fzo 13829  df-fl 13972  df-mod 14050  df-hash 14516  df-word 14700  df-lsw 14749  df-concat 14757  df-substr 14831  df-pfx 14863  df-csh 14982  df-clwwlk 30630  df-clwwlkn 30673

This theorem is used by:  hashecclwwlkn1  30725  umgrhashecclwwlk  30726