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Theorem wlkl1loop 29656
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.
StepHypRef Expression
1 wlkv 29630 . . . . 5 (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
2 simp3l 1202 . . . . . . . . 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 11255 . . . . . . . . . . . . 13 0 ∈ V
54snid 4662 . . . . . . . . . . . 12 0 ∈ {0}
6 oveq2 7439 . . . . . . . . . . . . 13 ((♯‘𝐹) = 1 → (0..^(♯‘𝐹)) = (0..^1))
7 fzo01 13786 . . . . . . . . . . . . 13 (0..^1) = {0}
86, 7eqtrdi 2793 . . . . . . . . . . . 12 ((♯‘𝐹) = 1 → (0..^(♯‘𝐹)) = {0})
95, 8eleqtrrid 2848 . . . . . . . . . . 11 ((♯‘𝐹) = 1 → 0 ∈ (0..^(♯‘𝐹)))
109ad2antrl 728 . . . . . . . . . 10 ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → 0 ∈ (0..^(♯‘𝐹)))
11103ad2ant3 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)
1312iedginwlk 29655 . . . . . . . . 9 ((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃 ∧ 0 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘0)) ∈ ran (iEdg‘𝐺))
142, 3, 11, 13syl3anc 1373 . . . . . . . 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)
1615, 12iswlkg 29631 . . . . . . . . . 10 (𝐺 ∈ V → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))))))
178raleqdv 3326 . . . . . . . . . . . . . . 15 ((♯‘𝐹) = 1 → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) ↔ ∀𝑘 ∈ {0}if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘)))))
18 oveq1 7438 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
19 0p1e1 12388 . . . . . . . . . . . . . . . . . 18 (0 + 1) = 1
2018, 19eqtrdi 2793 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (𝑘 + 1) = 1)
21 wkslem2 29626 . . . . . . . . . . . . . . . . 17 ((𝑘 = 0 ∧ (𝑘 + 1) = 1) → (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) ↔ if-((𝑃‘0) = (𝑃‘1), ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)}, {(𝑃‘0), (𝑃‘1)} ⊆ ((iEdg‘𝐺)‘(𝐹‘0)))))
2220, 21mpdan 687 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) ↔ if-((𝑃‘0) = (𝑃‘1), ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)}, {(𝑃‘0), (𝑃‘1)} ⊆ ((iEdg‘𝐺)‘(𝐹‘0)))))
234, 22ralsn 4681 . . . . . . . . . . . . . . 15 (∀𝑘 ∈ {0}if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) ↔ if-((𝑃‘0) = (𝑃‘1), ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)}, {(𝑃‘0), (𝑃‘1)} ⊆ ((iEdg‘𝐺)‘(𝐹‘0))))
2417, 23bitrdi 287 . . . . . . . . . . . . . 14 ((♯‘𝐹) = 1 → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) ↔ if-((𝑃‘0) = (𝑃‘1), ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0)}, {(𝑃‘0), (𝑃‘1)} ⊆ ((iEdg‘𝐺)‘(𝐹‘0)))))
2524ad2antrl 728 . . . . . . . . . . . . 13 ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) ↔ 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)}))
2726biimpa 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)})
2827eqcomd 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)))
2928ex 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))))
3029ad2antll 729 . . . . . . . . . . . . 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))))
3125, 30sylbid 240 . . . . . . . . . . . 12 ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0))))
3231com12 32 . . . . . . . . . . 11 (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) → ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0))))
33323ad2ant3 1136 . . . . . . . . . 10 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘)))) → ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0))))
3416, 33biimtrdi 253 . . . . . . . . 9 (𝐺 ∈ V → (𝐹(Walks‘𝐺)𝑃 → ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0)))))
35343imp 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 29066 . . . . . . . . 9 (Edg‘𝐺) = ran (iEdg‘𝐺)
3736a1i 11 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → (Edg‘𝐺) = ran (iEdg‘𝐺))
3814, 35, 373eltr4d 2856 . . . . . . 7 ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → {(𝑃‘0)} ∈ (Edg‘𝐺))
39383exp 1120 . . . . . 6 (𝐺 ∈ V → (𝐹(Walks‘𝐺)𝑃 → ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))))
40393ad2ant1 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))))
411, 40mpcom 38 . . . 4 (𝐹(Walks‘𝐺)𝑃 → ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺)))
4241expd 415 . . 3 (𝐹(Walks‘𝐺)𝑃 → (Fun (iEdg‘𝐺) → (((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)) → {(𝑃‘0)} ∈ (Edg‘𝐺))))
4342impcom 407 . 2 ((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃) → (((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)) → {(𝑃‘0)} ∈ (Edg‘𝐺)))
4443imp 406 1 (((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  if-wif 1063  w3a 1087   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  wss 3951  {csn 4626  {cpr 4628   class class class wbr 5143  dom cdm 5685  ran crn 5686  Fun wfun 6555  wf 6557  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156   + caddc 11158  ...cfz 13547  ..^cfzo 13694  chash 14369  Word cword 14552  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064  Walkscwlks 29614
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-edg 29065  df-wlks 29617
This theorem is referenced by:  clwlkl1loop  29803  loop1cycl  35142
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