Theorem erclwwlknsym 30490

Description: ∼ is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.)

Hypotheses

Ref	Expression
erclwwlkn.w	⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)
erclwwlkn.r	⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}

Assertion

Ref	Expression
erclwwlknsym	⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥)

Distinct variable groups:   𝑡,𝑊,𝑢   𝑛,𝑁,𝑢,𝑡,𝑥   𝑦,𝑛,𝑡,𝑢,𝑥   𝑛,𝑊

Allowed substitution hints:   ∼ (𝑥, 𝑦, 𝑢, 𝑡, 𝑛)   𝐺(𝑥, 𝑦, 𝑢, 𝑡, 𝑛)   𝑁(𝑦)   𝑊(𝑥, 𝑦)

Proof of Theorem erclwwlknsym

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		erclwwlkn.w	. . . 4 ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)
2		erclwwlkn.r	. . . 4 ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
3	1, 2	erclwwlkneqlen 30488	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
4	1, 2	erclwwlkneq 30487	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))))
5		simpl2 1211	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∈ 𝑊)
6		simpl1 1210	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑥 ∈ 𝑊)
7		eqid 2765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
8	7	clwwlknbp 30455	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
9		eqcom 2772	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑥) = 𝑁 ↔ 𝑁 = (♯‘𝑥))
10	9	biimpi 219	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑥) = 𝑁 → 𝑁 = (♯‘𝑥))
11	8, 10	simpl2im 513	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 = (♯‘𝑥))
12	11, 1	eleq2s 2883	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑊 → 𝑁 = (♯‘𝑥))
13	12	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → 𝑁 = (♯‘𝑥))
14	13	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑁 = (♯‘𝑥))
15	7	clwwlknwrd 30454	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → 𝑦 ∈ Word (Vtx‘𝐺))
16	15, 1	eleq2s 2883	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ 𝑊 → 𝑦 ∈ Word (Vtx‘𝐺))
17	16	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → 𝑦 ∈ Word (Vtx‘𝐺))
18	17	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∈ Word (Vtx‘𝐺))
19	18	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → 𝑦 ∈ Word (Vtx‘𝐺))
20		simprr 785	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (♯‘𝑥) = (♯‘𝑦))
21	19, 20	cshwcshid 14890	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → ((𝑛 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
22		oveq2 7427	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (♯‘𝑥) → (0...𝑁) = (0...(♯‘𝑥)))
23		oveq2 7427	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑥) = (♯‘𝑦) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
24	23	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
25	22, 24	sylan9eq 2820	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (0...𝑁) = (0...(♯‘𝑦)))
26	25	eleq2d 2851	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(♯‘𝑦))))
27	26	anbi1d 643	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) ↔ (𝑛 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛))))
28	22	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (0...𝑁) = (0...(♯‘𝑥)))
29	28	rexeqdv 3326	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚) ↔ ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
30	21, 27, 29	3imtr4d 297	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
31	14, 30	mpancom 701	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
32	31	expd 421	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → (𝑛 ∈ (0...𝑁) → (𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
33	32	rexlimdv 3166	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
34	33	ex 418	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → ((♯‘𝑥) = (♯‘𝑦) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
35	34	com23 87	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((♯‘𝑥) = (♯‘𝑦) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
36	35	3impia 1135	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((♯‘𝑥) = (♯‘𝑦) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
37	36	imp 412	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))
38		oveq2 7427	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
39	38	eqeq2d 2776	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
40	39	cbvrexvw 3246	. . . . . . . 8 ⊢ (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))
41	37, 40	sylibr 237	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))
42	5, 6, 41	3jca 1146	. . . . . 6 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)))
43	1, 2	erclwwlkneq 30487	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
44	43	ancoms 464	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
45	42, 44	imbitrrid 249	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∼ 𝑥))
46	45	expd 421	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((♯‘𝑥) = (♯‘𝑦) → 𝑦 ∼ 𝑥)))
47	4, 46	sylbid 243	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → ((♯‘𝑥) = (♯‘𝑦) → 𝑦 ∼ 𝑥)))
48	3, 47	mpdd 44	. 2 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥))
49	48	el2v 3464	1 ⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃wrex 3091  Vcvv 3457   class class class wbr 5111  {copab 5175  ‘cfv 6540  (class class class)co 7419  0cc0 11117  ...cfz 13553  ♯chash 14386  Word cword 14570   cyclShift ccsh 14851  Vtxcvtx 29403   ClWWalksN cclwwlkn 30444

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194  ax-pre-sup 11195

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  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 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-sup 9409  df-inf 9410  df-card 9941  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-div 11889  df-nn 12251  df-2 12320  df-n0 12522  df-z 12609  df-uz 12881  df-rp 13035  df-fz 13554  df-fzo 13702  df-fl 13845  df-mod 13923  df-hash 14387  df-word 14571  df-concat 14628  df-substr 14701  df-pfx 14733  df-csh 14852  df-clwwlk 30402  df-clwwlkn 30445

This theorem is used by:  erclwwlkn  30492  eclclwwlkn1  30495