Theorem erclwwlkref 26341

Description: ∼ is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by Alexander van der Vekens, 11-Jun-2018.)

Hypothesis

Ref	Expression
erclwwlk.r	⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}

Assertion

Ref	Expression
erclwwlkref	⊢ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ↔ 𝑥 ∼ 𝑥)

Distinct variable groups:   𝑛,𝐸,𝑢,𝑤   𝑛,𝑉,𝑢,𝑤   𝑥,𝑛,𝑢,𝑤

Allowed substitution hints:   ∼ (𝑥,𝑤,𝑢,𝑛)   𝐸(𝑥)   𝑉(𝑥)

Proof of Theorem erclwwlkref

Step	Hyp	Ref	Expression
1		anidm 674	. . . 4 ⊢ ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑥 ∈ (𝑉 ClWWalks 𝐸)) ↔ 𝑥 ∈ (𝑉 ClWWalks 𝐸))
2	1	anbi1i 727	. . 3 ⊢ (((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑥 ∈ (𝑉 ClWWalks 𝐸)) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)) ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
3		df-3an 1033	. . 3 ⊢ ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)) ↔ ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑥 ∈ (𝑉 ClWWalks 𝐸)) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
4		clwwlkprop 26298	. . . . 5 ⊢ (𝑥 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑥 ∈ Word 𝑉))
5		cshw0 13391	. . . . . . 7 ⊢ (𝑥 ∈ Word 𝑉 → (𝑥 cyclShift 0) = 𝑥)
6		0nn0 11184	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
7	6	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ Word 𝑉 → 0 ∈ ℕ0)
8		lencl 13179	. . . . . . . . 9 ⊢ (𝑥 ∈ Word 𝑉 → (#‘𝑥) ∈ ℕ0)
9		hashge0 13037	. . . . . . . . 9 ⊢ (𝑥 ∈ Word 𝑉 → 0 ≤ (#‘𝑥))
10		elfz2nn0 12300	. . . . . . . . 9 ⊢ (0 ∈ (0...(#‘𝑥)) ↔ (0 ∈ ℕ0 ∧ (#‘𝑥) ∈ ℕ0 ∧ 0 ≤ (#‘𝑥)))
11	7, 8, 9, 10	syl3anbrc 1239	. . . . . . . 8 ⊢ (𝑥 ∈ Word 𝑉 → 0 ∈ (0...(#‘𝑥)))
12		eqcom 2617	. . . . . . . . 9 ⊢ ((𝑥 cyclShift 0) = 𝑥 ↔ 𝑥 = (𝑥 cyclShift 0))
13	12	biimpi 205	. . . . . . . 8 ⊢ ((𝑥 cyclShift 0) = 𝑥 → 𝑥 = (𝑥 cyclShift 0))
14		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑛 = 0 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 0))
15	14	eqeq2d 2620	. . . . . . . . 9 ⊢ (𝑛 = 0 → (𝑥 = (𝑥 cyclShift 𝑛) ↔ 𝑥 = (𝑥 cyclShift 0)))
16	15	rspcev 3282	. . . . . . . 8 ⊢ ((0 ∈ (0...(#‘𝑥)) ∧ 𝑥 = (𝑥 cyclShift 0)) → ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
17	11, 13, 16	syl2an 493	. . . . . . 7 ⊢ ((𝑥 ∈ Word 𝑉 ∧ (𝑥 cyclShift 0) = 𝑥) → ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
18	5, 17	mpdan 699	. . . . . 6 ⊢ (𝑥 ∈ Word 𝑉 → ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
19	18	3ad2ant3 1077	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑥 ∈ Word 𝑉) → ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
20	4, 19	syl 17	. . . 4 ⊢ (𝑥 ∈ (𝑉 ClWWalks 𝐸) → ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
21	20	pm4.71i 662	. . 3 ⊢ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
22	2, 3, 21	3bitr4ri 292	. 2 ⊢ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
23		vex 3176	. . 3 ⊢ 𝑥 ∈ V
24		erclwwlk.r	. . . 4 ⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
25	24	erclwwlkeq 26339	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))))
26	23, 23, 25	mp2an 704	. 2 ⊢ (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
27	22, 26	bitr4i 266	1 ⊢ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ↔ 𝑥 ∼ 𝑥)

Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   class class class wbr 4583  {copab 4642  ‘cfv 5804  (class class class)co 6549  0cc0 9815   ≤ cle 9954  ℕ₀cn0 11169  ...cfz 12197  #chash 12979  Word cword 13146   cyclShift ccsh 13385   ClWWalks cclwwlk 26276

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-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-csh 13386  df-clwwlk 26279

This theorem is referenced by:  erclwwlk  26344