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Theorem 1wlkl1loop 40842
 Description: A 1-walk of length 1 from a vertex to itself is a loop. (Contributed by AV, 23-Apr-2021.)
Assertion
Ref Expression
1wlkl1loop (((Fun (iEdg‘𝐺) ∧ 𝐹(1Walks‘𝐺)𝑃) ∧ ((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))

Proof of Theorem 1wlkl1loop
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 wlkv 40815 . . . . 5 (𝐹(1Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
2 simp3l 1082 . . . . . . . . 9 ((𝐺 ∈ V ∧ 𝐹(1Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → Fun (iEdg‘𝐺))
3 simp2 1055 . . . . . . . . 9 ((𝐺 ∈ V ∧ 𝐹(1Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → 𝐹(1Walks‘𝐺)𝑃)
4 c0ex 9913 . . . . . . . . . . . . 13 0 ∈ V
54snid 4155 . . . . . . . . . . . 12 0 ∈ {0}
6 oveq2 6557 . . . . . . . . . . . . 13 ((#‘𝐹) = 1 → (0..^(#‘𝐹)) = (0..^1))
7 fzo01 12417 . . . . . . . . . . . . 13 (0..^1) = {0}
86, 7syl6eq 2660 . . . . . . . . . . . 12 ((#‘𝐹) = 1 → (0..^(#‘𝐹)) = {0})
95, 8syl5eleqr 2695 . . . . . . . . . . 11 ((#‘𝐹) = 1 → 0 ∈ (0..^(#‘𝐹)))
109ad2antrl 760 . . . . . . . . . 10 ((Fun (iEdg‘𝐺) ∧ ((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → 0 ∈ (0..^(#‘𝐹)))
11103ad2ant3 1077 . . . . . . . . 9 ((𝐺 ∈ V ∧ 𝐹(1Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → 0 ∈ (0..^(#‘𝐹)))
12 eqid 2610 . . . . . . . . . 10 (iEdg‘𝐺) = (iEdg‘𝐺)
1312iedginwlk 40841 . . . . . . . . 9 ((Fun (iEdg‘𝐺) ∧ 𝐹(1Walks‘𝐺)𝑃 ∧ 0 ∈ (0..^(#‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘0)) ∈ ran (iEdg‘𝐺))
142, 3, 11, 13syl3anc 1318 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝐹(1Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → ((iEdg‘𝐺)‘(𝐹‘0)) ∈ ran (iEdg‘𝐺))
15 eqid 2610 . . . . . . . . . . 11 (Vtx‘𝐺) = (Vtx‘𝐺)
1615, 12is1wlkg 40816 . . . . . . . . . 10 (𝐺 ∈ V → (𝐹(1Walks‘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 3121 . . . . . . . . . . . . . . 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 6556 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
19 0p1e1 11009 . . . . . . . . . . . . . . . . . 18 (0 + 1) = 1
2018, 19syl6eq 2660 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (𝑘 + 1) = 1)
21 1wlkslem2 40810 . . . . . . . . . . . . . . . . 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 699 . . . . . . . . . . . . . . . 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 4169 . . . . . . . . . . . . . . 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, 23syl6bb 275 . . . . . . . . . . . . . 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 760 . . . . . . . . . . . . 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 1017 . . . . . . . . . . . . . . . . 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 500 . . . . . . . . . . . . . . . 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 2616 . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . 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 229 . . . . . . . . . . . 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 1077 . . . . . . . . . 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, 33syl6bi 242 . . . . . . . . 9 (𝐺 ∈ V → (𝐹(1Walks‘𝐺)𝑃 → ((Fun (iEdg‘𝐺) ∧ ((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0)))))
35343imp 1249 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝐹(1Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0)))
36 edgaval 25794 . . . . . . . . 9 (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
37363ad2ant1 1075 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝐹(1Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → (Edg‘𝐺) = ran (iEdg‘𝐺))
3814, 35, 373eltr4d 2703 . . . . . . 7 ((𝐺 ∈ V ∧ 𝐹(1Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → {(𝑃‘0)} ∈ (Edg‘𝐺))
39383exp 1256 . . . . . 6 (𝐺 ∈ V → (𝐹(1Walks‘𝐺)𝑃 → ((Fun (iEdg‘𝐺) ∧ ((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))))
40393ad2ant1 1075 . . . . 5 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 → ((Fun (iEdg‘𝐺) ∧ ((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))))
411, 40mpcom 37 . . . 4 (𝐹(1Walks‘𝐺)𝑃 → ((Fun (iEdg‘𝐺) ∧ ((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺)))
4241expd 451 . . 3 (𝐹(1Walks‘𝐺)𝑃 → (Fun (iEdg‘𝐺) → (((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)) → {(𝑃‘0)} ∈ (Edg‘𝐺))))
4342impcom 445 . 2 ((Fun (iEdg‘𝐺) ∧ 𝐹(1Walks‘𝐺)𝑃) → (((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)) → {(𝑃‘0)} ∈ (Edg‘𝐺)))
4443imp 444 1 (((Fun (iEdg‘𝐺) ∧ 𝐹(1Walks‘𝐺)𝑃) ∧ ((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  if-wif 1006   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  {csn 4125  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146  Vtxcvtx 25673  iEdgciedg 25674  Edgcedga 25792  1Walksc1wlks 40796 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-edga 25793  df-1wlks 40800 This theorem is referenced by:  clwlkl1loop  40989
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