Theorem erclwwlknsym 30140

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 30138	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
4	1, 2	erclwwlkneq 30137	. . . 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 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
8	7	clwwlknbp 30105	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
9		eqcom 2743	. . . . . . . . . . . . . . . . . . . 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 2854	. . . . . . . . . . . . . . . . 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 30104	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) → 𝑦 ∈ Word (Vtx‘𝐺))
16	15, 1	eleq2s 2854	. . . . . . . . . . . . . . . . . . . 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 14789	. . . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (♯‘𝑥) → (0...𝑁) = (0...(♯‘𝑥)))
23		oveq2 7375	. . . . . . . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = (♯‘𝑥) ∧ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ (♯‘𝑥) = (♯‘𝑦))) → (0...𝑁) = (0...(♯‘𝑦)))
26	25	eleq2d 2822	. . . . . . . . . . . . . . . . 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 3296	. . . . . . . . . . . . . . . 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 3136	. . . . . . . . . . . 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 7375	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
39	38	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
40	39	cbvrexvw 3216	. . . . . . . 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 30137	. . . . . . 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 3436	1 ⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  Vcvv 3429   class class class wbr 5085  {copab 5147  ‘cfv 6498  (class class class)co 7367  0cc0 11038  ...cfz 13461  ♯chash 14292  Word cword 14475   cyclShift ccsh 14750  Vtxcvtx 29065   ClWWalksN cclwwlkn 30094

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-inf 9356  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-n0 12438  df-z 12525  df-uz 12789  df-rp 12943  df-fz 13462  df-fzo 13609  df-fl 13751  df-mod 13829  df-hash 14293  df-word 14476  df-concat 14533  df-substr 14604  df-pfx 14634  df-csh 14751  df-clwwlk 30052  df-clwwlkn 30095

This theorem is referenced by:  erclwwlkn  30142  eclclwwlkn1  30145