Theorem erclwwlknsym 30006

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 30004	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
4	1, 2	erclwwlkneq 30003	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))))
5		simpl2 1193	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∈ 𝑊)
6		simpl1 1192	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑥 ∈ 𝑊)
7		eqid 2730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
8	7	clwwlknbp 29971	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
9		eqcom 2737	. . . . . . . . . . . . . . . . . . . 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 2847	. . . . . . . . . . . . . . . . 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 29970	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → 𝑦 ∈ Word (Vtx‘𝐺))
16	15, 1	eleq2s 2847	. . . . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (♯‘𝑥) = (♯‘𝑦))
21	19, 20	cshwcshid 14800	. . . . . . . . . . . . . . . 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 7398	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (♯‘𝑥) → (0...𝑁) = (0...(♯‘𝑥)))
23		oveq2 7398	. . . . . . . . . . . . . . . . . . . 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 2785	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (0...𝑁) = (0...(♯‘𝑦)))
26	25	eleq2d 2815	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(♯‘𝑦))))
27	26	anbi1d 631	. . . . . . . . . . . . . . . 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 3302	. . . . . . . . . . . . . . . 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 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 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 3133	. . . . . . . . . . . 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 1117	. . . . . . . . 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 7398	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
39	38	eqeq2d 2741	. . . . . . . . 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 1128	. . . . . 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 30003	. . . . . . 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 3457	1 ⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3054  Vcvv 3450   class class class wbr 5110  {copab 5172  ‘cfv 6514  (class class class)co 7390  0cc0 11075  ...cfz 13475  ♯chash 14302  Word cword 14485   cyclShift ccsh 14760  Vtxcvtx 28930   ClWWalksN cclwwlkn 29960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-n0 12450  df-z 12537  df-uz 12801  df-rp 12959  df-fz 13476  df-fzo 13623  df-fl 13761  df-mod 13839  df-hash 14303  df-word 14486  df-concat 14543  df-substr 14613  df-pfx 14643  df-csh 14761  df-clwwlk 29918  df-clwwlkn 29961

This theorem is referenced by:  erclwwlkn  30008  eclclwwlkn1  30011