Theorem clwwlknscsh 26347

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.)

Assertion

Ref	Expression
clwwlknscsh	⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)})

Distinct variable groups:   𝑛,𝐸,𝑦   𝑛,𝑁,𝑦   𝑛,𝑉,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 2614	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑊 cyclShift 𝑛) ↔ 𝑥 = (𝑊 cyclShift 𝑛)))
2	1	rexbidv 3034	. . 3 ⊢ (𝑦 = 𝑥 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)))
3	2	cbvrabv 3172	. 2 ⊢ {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)}
4		clwwlknprop 26300	. . . . . . . 8 ⊢ (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)))
5	4	simp2d 1067	. . . . . . 7 ⊢ (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → 𝑤 ∈ Word 𝑉)
6	5	ad2antrl 760	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → 𝑤 ∈ Word 𝑉)
7		simprr 792	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))
8	6, 7	jca 553	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
9		simplr 788	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))
10		simpllr 795	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → 𝑛 ∈ (0...𝑁))
11		clwwnisshclwwn 26337	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑛 ∈ (0...𝑁) → (𝑊 cyclShift 𝑛) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))
12	9, 10, 11	sylc 63	. . . . . . . . . . . 12 ⊢ ((((𝑤 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → (𝑊 cyclShift 𝑛) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))
13		eleq1 2676	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑊 cyclShift 𝑛) → (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝑊 cyclShift 𝑛) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))
14	13	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝑤 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝑊 cyclShift 𝑛) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))
15	12, 14	mpbird 246	. . . . . . . . . . 11 ⊢ ((((𝑤 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...𝑁)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))) ∧ 𝑤 = (𝑊 cyclShift 𝑛)) → 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))
16	15	exp31 628	. . . . . . . . . 10 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...𝑁)) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑤 = (𝑊 cyclShift 𝑛) → 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))))
17	16	com23 84	. . . . . . . . 9 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...𝑁)) → (𝑤 = (𝑊 cyclShift 𝑛) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))))
18	17	rexlimdva 3013	. . . . . . . 8 ⊢ (𝑤 ∈ Word 𝑉 → (∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))))
19	18	imp 444	. . . . . . 7 ⊢ ((𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))
20	19	impcom 445	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))
21		simprr 792	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))
22	20, 21	jca 553	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))) → (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
23	8, 22	impbida 873	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → ((𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)) ↔ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛))))
24		eqeq1 2614	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥 = (𝑊 cyclShift 𝑛) ↔ 𝑤 = (𝑊 cyclShift 𝑛)))
25	24	rexbidv 3034	. . . . 5 ⊢ (𝑥 = 𝑤 → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
26	25	elrab 3331	. . . 4 ⊢ (𝑤 ∈ {𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)} ↔ (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
27		eqeq1 2614	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 = (𝑊 cyclShift 𝑛) ↔ 𝑤 = (𝑊 cyclShift 𝑛)))
28	27	rexbidv 3034	. . . . 5 ⊢ (𝑦 = 𝑤 → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
29	28	elrab 3331	. . . 4 ⊢ (𝑤 ∈ {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} ↔ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0...𝑁)𝑤 = (𝑊 cyclShift 𝑛)))
30	23, 26, 29	3bitr4g 302	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑤 ∈ {𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)} ↔ 𝑤 ∈ {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)}))
31	30	eqrdv 2608	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → {𝑥 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)})
32	3, 31	syl5eq 2656	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℕ₀cn0 11169  ...cfz 12197  #chash 12979  Word cword 13146   cyclShift ccsh 13385   ClWWalksN cclwwlkn 26277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-substr 13158  df-csh 13386  df-clwwlk 26279  df-clwwlkn 26280

This theorem is referenced by:  hashecclwwlkn1  26361  usghashecclwwlk  26362