Theorem erclwwlkref 30349

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 574	. . . 4 ⊢ ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺)) ↔ 𝑥 ∈ (ClWWalks‘𝐺))
2	1	anbi1i 635	. . 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 30314	. . . . 5 ⊢ (𝑥 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅))
6		cshw0 14833	. . . . . . 7 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → (𝑥 cyclShift 0) = 𝑥)
7		0nn0 12520	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → 0 ∈ ℕ0)
9		lencl 14572	. . . . . . . . 9 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → (♯‘𝑥) ∈ ℕ0)
10		hashge0 14425	. . . . . . . . 9 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → 0 ≤ (♯‘𝑥))
11		elfz2nn0 13648	. . . . . . . . 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 7420	. . . . . . . . 9 ⊢ (𝑛 = 0 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 0))
16	15	rspceeqv 3605	. . . . . . . 8 ⊢ ((0 ∈ (0...(♯‘𝑥)) ∧ 𝑥 = (𝑥 cyclShift 0)) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
17	12, 14, 16	syl2an 607	. . . . . . 7 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (𝑥 cyclShift 0) = 𝑥) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
18	6, 17	mpdan 699	. . . . . 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 568	. . 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 30347	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))))
25	24	el2v 3462	. 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)

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089  Vcvv 3455  ∅c0 4287   class class class wbr 5110  {copab 5174  ‘cfv 6538  (class class class)co 7412  0cc0 11101   ≤ cle 11245  ℕ₀cn0 12505  ...cfz 13536  ♯chash 14368  Word cword 14552   cyclShift ccsh 14827  Vtxcvtx 29324  ClWWalkscclwwlk 30310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178  ax-pre-sup 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-oadd 8458  df-er 8695  df-map 8827  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-sup 9403  df-inf 9404  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-div 11873  df-nn 12235  df-n0 12506  df-xnn0 12579  df-z 12593  df-uz 12864  df-rp 13018  df-fz 13537  df-fzo 13685  df-fl 13827  df-mod 13905  df-hash 14369  df-word 14553  df-concat 14610  df-substr 14681  df-pfx 14711  df-csh 14828  df-clwwlk 30311

This theorem is referenced by:  erclwwlk  30352