Theorem erclwwlknsym 30158

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 30156	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
4	1, 2	erclwwlkneq 30155	. . . 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 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
8	7	clwwlknbp 30123	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
9		eqcom 2744	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑥) = 𝑁 ↔ 𝑁 = (♯‘𝑥))
10	9	biimpi 216	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑥) = 𝑁 → 𝑁 = (♯‘𝑥))
11	8, 10	simpl2im 503	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 = (♯‘𝑥))
12	11, 1	eleq2s 2855	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑊 → 𝑁 = (♯‘𝑥))
13	12	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → 𝑁 = (♯‘𝑥))
14	13	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑁 = (♯‘𝑥))
15	7	clwwlknwrd 30122	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → 𝑦 ∈ Word (Vtx‘𝐺))
16	15, 1	eleq2s 2855	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ 𝑊 → 𝑦 ∈ Word (Vtx‘𝐺))
17	16	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → 𝑦 ∈ Word (Vtx‘𝐺))
18	17	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∈ Word (Vtx‘𝐺))
19	18	adantl 481	. . . . . . . . . . . . . . . . 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 14783	. . . . . . . . . . . . . . . 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 7369	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (♯‘𝑥) → (0...𝑁) = (0...(♯‘𝑥)))
23		oveq2 7369	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑥) = (♯‘𝑦) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
24	23	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
25	22, 24	sylan9eq 2792	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (0...𝑁) = (0...(♯‘𝑦)))
26	25	eleq2d 2823	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(♯‘𝑦))))
27	26	anbi1d 632	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (0...𝑁) = (0...(♯‘𝑥)))
29	28	rexeqdv 3297	. . . . . . . . . . . . . . . 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 294	. . . . . . . . . . . . . . 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 689	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → ((𝑛 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
32	31	expd 415	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → (𝑛 ∈ (0...𝑁) → (𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))))
33	32	rexlimdv 3137	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
34	33	ex 412	. . . . . . . . . . 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 1118	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((♯‘𝑥) = (♯‘𝑦) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚)))
37	36	imp 406	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))
38		oveq2 7369	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
39	38	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
40	39	cbvrexvw 3217	. . . . . . . 8 ⊢ (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑚))
41	37, 40	sylibr 234	. . . . . . 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 1129	. . . . . 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 30155	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
44	43	ancoms 458	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
45	42, 44	imbitrrid 246	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∼ 𝑥))
46	45	expd 415	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((♯‘𝑥) = (♯‘𝑦) → 𝑦 ∼ 𝑥)))
47	4, 46	sylbid 240	. . 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 3437	1 ⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  Vcvv 3430   class class class wbr 5086  {copab 5148  ‘cfv 6493  (class class class)co 7361  0cc0 11032  ...cfz 13455  ♯chash 14286  Word cword 14469   cyclShift ccsh 14744  Vtxcvtx 29082   ClWWalksN cclwwlkn 30112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  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 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-inf 9350  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-div 11802  df-nn 12169  df-2 12238  df-n0 12432  df-z 12519  df-uz 12783  df-rp 12937  df-fz 13456  df-fzo 13603  df-fl 13745  df-mod 13823  df-hash 14287  df-word 14470  df-concat 14527  df-substr 14598  df-pfx 14628  df-csh 14745  df-clwwlk 30070  df-clwwlkn 30113

This theorem is referenced by:  erclwwlkn  30160  eclclwwlkn1  30163