Theorem erclwwlkref 30644

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 AV, 29-Apr-2021.)

Hypothesis

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

Assertion

Ref	Expression
erclwwlkref	⊢ (𝑥 ∈ (ClWWalks‘𝐺) ↔ 𝑥 ∼ 𝑥)

Distinct variable groups:   𝑛,𝐺,𝑢,𝑤   𝑥,𝑛,𝑢,𝑤

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

Proof of Theorem erclwwlkref

Step	Hyp	Ref	Expression
1		anidm 575	. . . 4 ⊢ ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺)) ↔ 𝑥 ∈ (ClWWalks‘𝐺))
2	1	anbi1i 636	. . 3 ⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺)) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)) ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
3		df-3an 1105	. . 3 ⊢ ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)) ↔ ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺)) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
4		eqid 2763	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	clwwlkbp 30609	. . . . 5 ⊢ (𝑥 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅))
6		cshw0 14976	. . . . . . 7 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → (𝑥 cyclShift 0) = 𝑥)
7		0nn0 12651	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → 0 ∈ ℕ0)
9		lencl 14708	. . . . . . . . 9 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → (♯‘𝑥) ∈ ℕ0)
10		hashge0 14561	. . . . . . . . 9 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → 0 ≤ (♯‘𝑥))
11		elfz2nn0 13782	. . . . . . . . 9 ⊢ (0 ∈ (0...(♯‘𝑥)) ↔ (0 ∈ ℕ0 ∧ (♯‘𝑥) ∈ ℕ0 ∧ 0 ≤ (♯‘𝑥)))
12	8, 9, 10, 11	syl3anbrc 1362	. . . . . . . 8 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → 0 ∈ (0...(♯‘𝑥)))
13		eqcom 2770	. . . . . . . . 9 ⊢ ((𝑥 cyclShift 0) = 𝑥 ↔ 𝑥 = (𝑥 cyclShift 0))
14	13	biimpi 219	. . . . . . . 8 ⊢ ((𝑥 cyclShift 0) = 𝑥 → 𝑥 = (𝑥 cyclShift 0))
15		oveq2 7454	. . . . . . . . 9 ⊢ (𝑛 = 0 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 0))
16	15	rspceeqv 3602	. . . . . . . 8 ⊢ ((0 ∈ (0...(♯‘𝑥)) ∧ 𝑥 = (𝑥 cyclShift 0)) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
17	12, 14, 16	syl2an 608	. . . . . . 7 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (𝑥 cyclShift 0) = 𝑥) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
18	6, 17	mpdan 700	. . . . . 6 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
19	18	3ad2ant2 1152	. . . . 5 ⊢ ((𝐺 ∈ V ∧ 𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
20	5, 19	syl 18	. . . 4 ⊢ (𝑥 ∈ (ClWWalks‘𝐺) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
21	20	pm4.71i 569	. . 3 ⊢ (𝑥 ∈ (ClWWalks‘𝐺) ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
22	2, 3, 21	3bitr4ri 307	. 2 ⊢ (𝑥 ∈ (ClWWalks‘𝐺) ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
23		erclwwlk.r	. . . 4 ⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
24	23	erclwwlkeq 30642	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))))
25	24	el2v 3461	. 2 ⊢ (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
26	22, 25	bitr4i 281	1 ⊢ (𝑥 ∈ (ClWWalks‘𝐺) ↔ 𝑥 ∼ 𝑥)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090  Vcvv 3454  ∅c0 4281   class class class wbr 5106  {copab 5171  ‘cfv 6561  (class class class)co 7446  0cc0 11230   ≤ cle 11374  ℕ₀cn0 12636  ...cfz 13669  ♯chash 14504  Word cword 14688   cyclShift ccsh 14970  Vtxcvtx 29607  ClWWalkscclwwlk 30605

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 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307  ax-pre-sup 11308

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 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-oadd 8511  df-er 8748  df-map 8880  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-sup 9464  df-inf 9465  df-card 10050  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-div 12004  df-nn 12366  df-n0 12637  df-xnn0 12710  df-z 12724  df-uz 12996  df-rp 13151  df-fz 13670  df-fzo 13819  df-fl 13962  df-mod 14040  df-hash 14505  df-word 14689  df-concat 14746  df-substr 14820  df-pfx 14852  df-csh 14971  df-clwwlk 30606

This theorem is used by:  erclwwlk  30647