Theorem erclwwlknsym 28125

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 28123	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
4	1, 2	erclwwlkneq 28122	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))))
5		simpl2 1194	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∈ 𝑊)
6		simpl1 1193	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑥 ∈ 𝑊)
7		eqid 2734	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
8	7	clwwlknbp 28090	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
9		eqcom 2741	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑥) = 𝑁 ↔ 𝑁 = (♯‘𝑥))
10	9	biimpi 219	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑥) = 𝑁 → 𝑁 = (♯‘𝑥))
11	8, 10	simpl2im 507	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 = (♯‘𝑥))
12	11, 1	eleq2s 2852	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑊 → 𝑁 = (♯‘𝑥))
13	12	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → 𝑁 = (♯‘𝑥))
14	13	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑁 = (♯‘𝑥))
15	7	clwwlknwrd 28089	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → 𝑦 ∈ Word (Vtx‘𝐺))
16	15, 1	eleq2s 2852	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ 𝑊 → 𝑦 ∈ Word (Vtx‘𝐺))
17	16	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → 𝑦 ∈ Word (Vtx‘𝐺))
18	17	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∈ Word (Vtx‘𝐺))
19	18	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → 𝑦 ∈ Word (Vtx‘𝐺))
20		simprr 773	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (♯‘𝑥) = (♯‘𝑦))
21	19, 20	cshwcshid 14375	. . . . . . . . . . . . . . . 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 7210	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (♯‘𝑥) → (0...𝑁) = (0...(♯‘𝑥)))
23		oveq2 7210	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑥) = (♯‘𝑦) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
24	23	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
25	22, 24	sylan9eq 2794	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (0...𝑁) = (0...(♯‘𝑦)))
26	25	eleq2d 2819	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(♯‘𝑦))))
27	26	anbi1d 633	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (0...𝑁) = (0...(♯‘𝑥)))
29	28	rexeqdv 3319	. . . . . . . . . . . . . . . 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 688	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
32	31	expd 419	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → (𝑛 ∈ (0...𝑁) → (𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
33	32	rexlimdv 3195	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
34	33	ex 416	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → ((♯‘𝑥) = (♯‘𝑦) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
35	34	com23 86	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((♯‘𝑥) = (♯‘𝑦) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
36	35	3impia 1119	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((♯‘𝑥) = (♯‘𝑦) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
37	36	imp 410	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))
38		oveq2 7210	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
39	38	eqeq2d 2745	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
40	39	cbvrexvw 3352	. . . . . . . 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 1130	. . . . . 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 28122	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
44	43	ancoms 462	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
45	42, 44	syl5ibr 249	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∼ 𝑥))
46	45	expd 419	. . . 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 43	. 2 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥))
49	48	el2v 3409	1 ⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∃wrex 3055  Vcvv 3401   class class class wbr 5043  {copab 5105  ‘cfv 6369  (class class class)co 7202  0cc0 10712  ...cfz 13078  ♯chash 13879  Word cword 14052   cyclShift ccsh 14336  Vtxcvtx 27059   ClWWalksN cclwwlkn 28079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789  ax-pre-sup 10790

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-1st 7750  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-er 8380  df-map 8499  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-sup 9047  df-inf 9048  df-card 9538  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-div 11473  df-nn 11814  df-2 11876  df-n0 12074  df-z 12160  df-uz 12422  df-rp 12570  df-fz 13079  df-fzo 13222  df-fl 13350  df-mod 13426  df-hash 13880  df-word 14053  df-concat 14109  df-substr 14189  df-pfx 14219  df-csh 14337  df-clwwlk 28037  df-clwwlkn 28080

This theorem is referenced by:  erclwwlkn  28127  eclclwwlkn1  28130