Theorem wlkl1loop 30152

Description: A walk of length 1 from a vertex to itself is a loop. (Contributed by AV, 23-Apr-2021.)

Assertion

Ref	Expression
wlkl1loop	⊢ (((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))

Proof of Theorem wlkl1loop

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkv 30127	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
2		simp3l 1220	. . . . . . . . 9 ⊢ ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → Fun (iEdg‘𝐺))
3		simp2 1155	. . . . . . . . 9 ⊢ ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → 𝐹(Walks‘𝐺)𝑃)
4		c0ex 11272	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
5	4	snid 4622	. . . . . . . . . . . 12 ⊢ 0 ∈ {0}
6		oveq2 7416	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 1 → (0..^(♯‘𝐹)) = (0..^1))
7		fzo01 13851	. . . . . . . . . . . . 13 ⊢ (0..^1) = {0}
8	6, 7	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 1 → (0..^(♯‘𝐹)) = {0})
9	5, 8	eleqtrrid 2867	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 1 → 0 ∈ (0..^(♯‘𝐹)))
10	9	ad2antrl 741	. . . . . . . . . 10 ⊢ ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → 0 ∈ (0..^(♯‘𝐹)))
11	10	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → 0 ∈ (0..^(♯‘𝐹)))
12		eqid 2760	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
13	12	iedginwlk 30151	. . . . . . . . 9 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃 ∧ 0 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘0)) ∈ ran (iEdg‘𝐺))
14	2, 3, 11, 13	syl3anc 1398	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → ((iEdg‘𝐺)‘(𝐹‘0)) ∈ ran (iEdg‘𝐺))
15		eqid 2760	. . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
16	15, 12	iswlkg 30128	. . . . . . . . . 10 ⊢ (𝐺 ∈ V → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))))))
17	8	raleqdv 3319	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 1 → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ↔ ∀𝑘 ∈ {0}if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))
18		oveq1 7415	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
19		0p1e1 12433	. . . . . . . . . . . . . . . . . 18 ⊢ (0 + 1) = 1
20	18, 19	eqtrdi 2811	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → (𝑘 + 1) = 1)
21		wkslem2 30123	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 = 0 ∧ (𝑘 + 1) = 1) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ↔ if-((𝑃‘0) = (𝑃‘1), ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)}, {(𝑃‘0), (𝑃‘1)} ⊆ ((iEdg‘𝐺)‘(𝐹‘0)))))
22	20, 21	mpdan 700	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ↔ if-((𝑃‘0) = (𝑃‘1), ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)}, {(𝑃‘0), (𝑃‘1)} ⊆ ((iEdg‘𝐺)‘(𝐹‘0)))))
23	4, 22	ralsn 4641	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ {0}if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ↔ if-((𝑃‘0) = (𝑃‘1), ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)}, {(𝑃‘0), (𝑃‘1)} ⊆ ((iEdg‘𝐺)‘(𝐹‘0))))
24	17, 23	bitrdi 290	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) = 1 → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ↔ if-((𝑃‘0) = (𝑃‘1), ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)}, {(𝑃‘0), (𝑃‘1)} ⊆ ((iEdg‘𝐺)‘(𝐹‘0)))))
25	24	ad2antrl 741	. . . . . . . . . . . . 13 ⊢ ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ↔ if-((𝑃‘0) = (𝑃‘1), ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)}, {(𝑃‘0), (𝑃‘1)} ⊆ ((iEdg‘𝐺)‘(𝐹‘0)))))
26		ifptru 1091	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃‘0) = (𝑃‘1) → (if-((𝑃‘0) = (𝑃‘1), ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)}, {(𝑃‘0), (𝑃‘1)} ⊆ ((iEdg‘𝐺)‘(𝐹‘0))) ↔ ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)}))
27	26	biimpa 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃‘0) = (𝑃‘1) ∧ if-((𝑃‘0) = (𝑃‘1), ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)}, {(𝑃‘0), (𝑃‘1)} ⊆ ((iEdg‘𝐺)‘(𝐹‘0)))) → ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)})
28	27	eqcomd 2766	. . . . . . . . . . . . . . 15 ⊢ (((𝑃‘0) = (𝑃‘1) ∧ if-((𝑃‘0) = (𝑃‘1), ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)}, {(𝑃‘0), (𝑃‘1)} ⊆ ((iEdg‘𝐺)‘(𝐹‘0)))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0)))
29	28	ex 418	. . . . . . . . . . . . . 14 ⊢ ((𝑃‘0) = (𝑃‘1) → (if-((𝑃‘0) = (𝑃‘1), ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)}, {(𝑃‘0), (𝑃‘1)} ⊆ ((iEdg‘𝐺)‘(𝐹‘0))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0))))
30	29	ad2antll 742	. . . . . . . . . . . . 13 ⊢ ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → (if-((𝑃‘0) = (𝑃‘1), ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)}, {(𝑃‘0), (𝑃‘1)} ⊆ ((iEdg‘𝐺)‘(𝐹‘0))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0))))
31	25, 30	sylbid 243	. . . . . . . . . . . 12 ⊢ ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0))))
32	31	com12 33	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0))))
33	32	3ad2ant3 1153	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0))))
34	16, 33	biimtrdi 256	. . . . . . . . 9 ⊢ (𝐺 ∈ V → (𝐹(Walks‘𝐺)𝑃 → ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0)))))
35	34	3imp 1128	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0)))
36		edgval 29561	. . . . . . . . 9 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
37	36	a1i 11	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → (Edg‘𝐺) = ran (iEdg‘𝐺))
38	14, 35, 37	3eltr4d 2875	. . . . . . 7 ⊢ ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → {(𝑃‘0)} ∈ (Edg‘𝐺))
39	38	3exp 1137	. . . . . 6 ⊢ (𝐺 ∈ V → (𝐹(Walks‘𝐺)𝑃 → ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))))
40	39	3ad2ant1 1151	. . . . 5 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(Walks‘𝐺)𝑃 → ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))))
41	1, 40	mpcom 39	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺)))
42	41	expd 421	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (Fun (iEdg‘𝐺) → (((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)) → {(𝑃‘0)} ∈ (Edg‘𝐺))))
43	42	impcom 413	. 2 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃) → (((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)) → {(𝑃‘0)} ∈ (Edg‘𝐺)))
44	43	imp 412	1 ⊢ (((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  if-wif 1078   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ⊆ wss 3898  {csn 4583  {cpr 4585   class class class wbr 5102  dom cdm 5647  ran crn 5648  Fun wfun 6521  ⟶wf 6523  ‘cfv 6527  (class class class)co 7408  0cc0 11172  1c1 11173   + caddc 11175  ...cfz 13609  ..^cfzo 13757  ♯chash 14442  Word cword 14626  Vtxcvtx 29508  iEdgciedg 29509  Edgcedg 29559  Walkscwlks 30111

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11228  ax-resscn 11229  ax-1cn 11230  ax-icn 11231  ax-addcl 11232  ax-addrcl 11233  ax-mulcl 11234  ax-mulrcl 11235  ax-mulcom 11236  ax-addass 11237  ax-mulass 11238  ax-distr 11239  ax-i2m1 11240  ax-1ne0 11241  ax-1rid 11242  ax-rnegex 11243  ax-rrecex 11244  ax-cnre 11245  ax-pre-lttri 11246  ax-pre-lttrn 11247  ax-pre-ltadd 11248  ax-pre-mulgt0 11249

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  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 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-er 8695  df-map 8827  df-pm 8828  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-card 9992  df-pnf 11317  df-mnf 11318  df-xr 11319  df-ltxr 11320  df-le 11321  df-sub 11515  df-neg 11516  df-nn 12306  df-n0 12577  df-z 12664  df-uz 12936  df-fz 13610  df-fzo 13758  df-hash 14443  df-word 14627  df-edg 29560  df-wlks 30114

This theorem is used by:  clwlkl1loop  30304  loop1cycl  30678