Theorem wlkl1loop 29726

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 29701	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
2		simp3l 1203	. . . . . . . . 9 ⊢ ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → Fun (iEdg‘𝐺))
3		simp2 1138	. . . . . . . . 9 ⊢ ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → 𝐹(Walks‘𝐺)𝑃)
4		c0ex 11127	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
5	4	snid 4607	. . . . . . . . . . . 12 ⊢ 0 ∈ {0}
6		oveq2 7366	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) = 1 → (0..^(♯‘𝐹)) = (0..^1))
7		fzo01 13691	. . . . . . . . . . . . 13 ⊢ (0..^1) = {0}
8	6, 7	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) = 1 → (0..^(♯‘𝐹)) = {0})
9	5, 8	eleqtrrid 2844	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 1 → 0 ∈ (0..^(♯‘𝐹)))
10	9	ad2antrl 729	. . . . . . . . . 10 ⊢ ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → 0 ∈ (0..^(♯‘𝐹)))
11	10	3ad2ant3 1136	. . . . . . . . 9 ⊢ ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → 0 ∈ (0..^(♯‘𝐹)))
12		eqid 2737	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
13	12	iedginwlk 29725	. . . . . . . . 9 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃 ∧ 0 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘0)) ∈ ran (iEdg‘𝐺))
14	2, 3, 11, 13	syl3anc 1374	. . . . . . . 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 2737	. . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
16	15, 12	iswlkg 29702	. . . . . . . . . 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 3296	. . . . . . . . . . . . . . 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 7365	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
19		0p1e1 12287	. . . . . . . . . . . . . . . . . 18 ⊢ (0 + 1) = 1
20	18, 19	eqtrdi 2788	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → (𝑘 + 1) = 1)
21		wkslem2 29697	. . . . . . . . . . . . . . . . 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 688	. . . . . . . . . . . . . . . 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 4626	. . . . . . . . . . . . . . 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 287	. . . . . . . . . . . . . 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 729	. . . . . . . . . . . . 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 1075	. . . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . 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 730	. . . . . . . . . . . . 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 240	. . . . . . . . . . . 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 32	. . . . . . . . . . 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 1136	. . . . . . . . . 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 253	. . . . . . . . 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 1111	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0)))
36		edgval 29137	. . . . . . . . 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 2852	. . . . . . 7 ⊢ ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → {(𝑃‘0)} ∈ (Edg‘𝐺))
39	38	3exp 1120	. . . . . 6 ⊢ (𝐺 ∈ V → (𝐹(Walks‘𝐺)𝑃 → ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))))
40	39	3ad2ant1 1134	. . . . 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 38	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺)))
42	41	expd 415	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (Fun (iEdg‘𝐺) → (((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)) → {(𝑃‘0)} ∈ (Edg‘𝐺))))
43	42	impcom 407	. 2 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃) → (((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)) → {(𝑃‘0)} ∈ (Edg‘𝐺)))
44	43	imp 406	1 ⊢ (((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  if-wif 1063   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ⊆ wss 3890  {csn 4568  {cpr 4570   class class class wbr 5086  dom cdm 5622  ran crn 5623  Fun wfun 6484  ⟶wf 6486  ‘cfv 6490  (class class class)co 7358  0cc0 11027  1c1 11028   + caddc 11030  ...cfz 13450  ..^cfzo 13597  ♯chash 14281  Word cword 14464  Vtxcvtx 29084  iEdgciedg 29085  Edgcedg 29135  Walkscwlks 29685

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  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-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 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-pm 8767  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-card 9852  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12164  df-n0 12427  df-z 12514  df-uz 12778  df-fz 13451  df-fzo 13598  df-hash 14282  df-word 14465  df-edg 29136  df-wlks 29688

This theorem is referenced by:  clwlkl1loop  29871  loop1cycl  35340