Theorem clwwlknscsh 29006

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 2740	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑊 cyclShift 𝑛) ↔ 𝑥 = (𝑊 cyclShift 𝑛)))
2	1	rexbidv 3175	. . 3 ⊢ (𝑦 = 𝑥 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)))
3	2	cbvrabv 3417	. 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 2736	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	clwwlknwrd 28978	. . . . . . 7 ⊢ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → 𝑤 ∈ Word (Vtx‘𝐺))
6	5	ad2antrl 726	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → 𝑤 ∈ Word (Vtx‘𝐺))
7		simprr 771	. . . . . 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 512	. . . . 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 771	. . . . . . . . . . . . 13 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺))) → 𝑊 ∈ (𝑁 ClWWalksN 𝐺))
10		simpllr 774	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → 𝑛 ∈ (0...𝑁))
11		clwwnisshclwwsn 29003	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑛 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺))
12	9, 10, 11	syl2an2r 683	. . . . . . . . . . . 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 2825	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑊 cyclShift 𝑛) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 cyclShift 𝑛) ∈ (𝑁 ClWWalksN 𝐺)))
14	13	adantl 482	. . . . . . . . . . . 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 256	. . . . . . . . . . 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 420	. . . . . . . . . 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 86	. . . . . . . . 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 3152	. . . . . . . 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 407	. . . . . . 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 408	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → 𝑤 ∈ (𝑁 ClWWalksN 𝐺))
21		simprr 771	. . . . . 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 512	. . . . 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 799	. . . 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 2740	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥 = (𝑊 cyclShift 𝑛) ↔ 𝑤 = (𝑊 cyclShift 𝑛)))
25	24	rexbidv 3175	. . . . 5 ⊢ (𝑥 = 𝑤 → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
26	25	elrab 3645	. . . 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 2740	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 = (𝑊 cyclShift 𝑛) ↔ 𝑤 = (𝑊 cyclShift 𝑛)))
28	27	rexbidv 3175	. . . . 5 ⊢ (𝑦 = 𝑤 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
29	28	elrab 3645	. . . 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 313	. . 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 2734	. 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 2788	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 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3073  {crab 3407  ‘cfv 6496  (class class class)co 7357  0cc0 11051  ℕ₀cn0 12413  ...cfz 13424  Word cword 14402   cyclShift ccsh 14676  Vtxcvtx 27947   ClWWalksN cclwwlkn 28968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-ico 13270  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-substr 14529  df-pfx 14559  df-csh 14677  df-clwwlk 28926  df-clwwlkn 28969

This theorem is referenced by:  hashecclwwlkn1  29021  umgrhashecclwwlk  29022