Theorem erclwwlksym 30612

Description: ∼ is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by AV, 29-Apr-2021.)

Hypothesis

Ref	Expression
erclwwlk.r	⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}

Assertion

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

Distinct variable groups:   𝑛,𝐺,𝑢,𝑤   𝑥,𝑛,𝑢,𝑤,𝑦

Allowed substitution hints:   ∼ (𝑥, 𝑦, 𝑤, 𝑢, 𝑛)   𝐺(𝑥, 𝑦)

Proof of Theorem erclwwlksym

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		erclwwlk.r	. . . 4 ⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
2	1	erclwwlkeqlen 30610	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
3	1	erclwwlkeq 30609	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))))
4		simpl2 1211	. . . . . . 7 ⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∈ (ClWWalks‘𝐺))
5		simpl1 1210	. . . . . . 7 ⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑥 ∈ (ClWWalks‘𝐺))
6		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
7	6	clwwlkbp 30576	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑦 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ≠ ∅))
8	7	simp2d 1161	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (ClWWalks‘𝐺) → 𝑦 ∈ Word (Vtx‘𝐺))
9	8	ad2antlr 740	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∈ Word (Vtx‘𝐺))
10		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ (♯‘𝑥) = (♯‘𝑦)) → (♯‘𝑥) = (♯‘𝑦))
11	9, 10	cshwcshid 14978	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ (♯‘𝑥) = (♯‘𝑦)) → ((𝑛 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑛)) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
12	11	expd 421	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ (♯‘𝑥) = (♯‘𝑦)) → (𝑛 ∈ (0...(♯‘𝑦)) → (𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))))
13	12	rexlimdv 3161	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
14	13	ex 418	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → ((♯‘𝑥) = (♯‘𝑦) → (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))))
15	14	com23 87	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((♯‘𝑥) = (♯‘𝑦) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))))
16	15	3impia 1135	. . . . . . . . 9 ⊢ ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((♯‘𝑥) = (♯‘𝑦) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚)))
17	16	imp 412	. . . . . . . 8 ⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))
18		oveq2 7428	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 𝑚))
19	18	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift 𝑚)))
20	19	cbvrexvw 3241	. . . . . . . 8 ⊢ (∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑚))
21	17, 20	sylibr 237	. . . . . . 7 ⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
22	4, 5, 21	3jca 1146	. . . . . 6 ⊢ (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))
23	1	erclwwlkeq 30609	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))))
24	23	ancoms 464	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑥 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))))
25	22, 24	imbitrrid 249	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) ∧ (♯‘𝑥) = (♯‘𝑦)) → 𝑦 ∼ 𝑥))
26	25	expd 421	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((♯‘𝑥) = (♯‘𝑦) → 𝑦 ∼ 𝑥)))
27	3, 26	sylbid 243	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → ((♯‘𝑥) = (♯‘𝑦) → 𝑦 ∼ 𝑥)))
28	2, 27	mpdd 44	. 2 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥))
29	28	el2v 3457	1 ⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086  Vcvv 3450  ∅c0 4278   class class class wbr 5102  {copab 5166  ‘cfv 6538  (class class class)co 7420  0cc0 11200  ...cfz 13639  ♯chash 14474  Word cword 14658   cyclShift ccsh 14939  Vtxcvtx 29574  ClWWalkscclwwlk 30572

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-inf 9435  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-n0 12607  df-z 12694  df-uz 12966  df-rp 13121  df-fz 13640  df-fzo 13789  df-fl 13932  df-mod 14010  df-hash 14475  df-word 14659  df-concat 14716  df-substr 14789  df-pfx 14821  df-csh 14940  df-clwwlk 30573

This theorem is used by:  erclwwlk  30614