Theorem erclwwlkref 30115

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 569	. . . 4 ⊢ ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺)) ↔ 𝑥 ∈ (ClWWalks‘𝐺))
2	1	anbi1i 630	. . 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 1094	. . 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 2740	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	clwwlkbp 30080	. . . . 5 ⊢ (𝑥 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅))
6		cshw0 14754	. . . . . . 7 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → (𝑥 cyclShift 0) = 𝑥)
7		0nn0 12450	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
8	7	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → 0 ∈ ℕ0)
9		lencl 14493	. . . . . . . . 9 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → (♯‘𝑥) ∈ ℕ0)
10		hashge0 14347	. . . . . . . . 9 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → 0 ≤ (♯‘𝑥))
11		elfz2nn0 13570	. . . . . . . . 9 ⊢ (0 ∈ (0...(♯‘𝑥)) ↔ (0 ∈ ℕ0 ∧ (♯‘𝑥) ∈ ℕ0 ∧ 0 ≤ (♯‘𝑥)))
12	8, 9, 10, 11	syl3anbrc 1350	. . . . . . . 8 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → 0 ∈ (0...(♯‘𝑥)))
13		eqcom 2747	. . . . . . . . 9 ⊢ ((𝑥 cyclShift 0) = 𝑥 ↔ 𝑥 = (𝑥 cyclShift 0))
14	13	biimpi 217	. . . . . . . 8 ⊢ ((𝑥 cyclShift 0) = 𝑥 → 𝑥 = (𝑥 cyclShift 0))
15		oveq2 7371	. . . . . . . . 9 ⊢ (𝑛 = 0 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 0))
16	15	rspceeqv 3590	. . . . . . . 8 ⊢ ((0 ∈ (0...(♯‘𝑥)) ∧ 𝑥 = (𝑥 cyclShift 0)) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
17	12, 14, 16	syl2an 602	. . . . . . 7 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (𝑥 cyclShift 0) = 𝑥) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
18	6, 17	mpdan 693	. . . . . 6 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
19	18	3ad2ant2 1140	. . . . 5 ⊢ ((𝐺 ∈ V ∧ 𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑥 ≠ ∅) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
20	5, 19	syl 17	. . . 4 ⊢ (𝑥 ∈ (ClWWalks‘𝐺) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))
21	20	pm4.71i 564	. . 3 ⊢ (𝑥 ∈ (ClWWalks‘𝐺) ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
22	2, 3, 21	3bitr4ri 305	. 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 30113	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛))))
25	24	el2v 3439	. 2 ⊢ (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑥 = (𝑥 cyclShift 𝑛)))
26	22, 25	bitr4i 279	1 ⊢ (𝑥 ∈ (ClWWalks‘𝐺) ↔ 𝑥 ∼ 𝑥)

Syntax hints:   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∃wrex 3064  Vcvv 3432  ∅c0 4268   class class class wbr 5079  {copab 5141  ‘cfv 6492  (class class class)co 7363  0cc0 11036   ≤ cle 11178  ℕ₀cn0 12435  ...cfz 13459  ♯chash 14290  Word cword 14473   cyclShift ccsh 14748  Vtxcvtx 29090  ClWWalkscclwwlk 30076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-pre-sup 11114

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-oadd 8406  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9352  df-inf 9353  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-div 11806  df-nn 12173  df-n0 12436  df-xnn0 12509  df-z 12523  df-uz 12787  df-rp 12941  df-fz 13460  df-fzo 13607  df-fl 13749  df-mod 13827  df-hash 14291  df-word 14474  df-concat 14531  df-substr 14602  df-pfx 14632  df-csh 14749  df-clwwlk 30077

This theorem is referenced by:  erclwwlk  30118