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Theorem wlkl1loop 29670
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 29644 . . . . 5 (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
2 simp3l 1200 . . . . . . . . 9 ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → Fun (iEdg‘𝐺))
3 simp2 1136 . . . . . . . . 9 ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → 𝐹(Walks‘𝐺)𝑃)
4 c0ex 11252 . . . . . . . . . . . . 13 0 ∈ V
54snid 4666 . . . . . . . . . . . 12 0 ∈ {0}
6 oveq2 7438 . . . . . . . . . . . . 13 ((♯‘𝐹) = 1 → (0..^(♯‘𝐹)) = (0..^1))
7 fzo01 13782 . . . . . . . . . . . . 13 (0..^1) = {0}
86, 7eqtrdi 2790 . . . . . . . . . . . 12 ((♯‘𝐹) = 1 → (0..^(♯‘𝐹)) = {0})
95, 8eleqtrrid 2845 . . . . . . . . . . 11 ((♯‘𝐹) = 1 → 0 ∈ (0..^(♯‘𝐹)))
109ad2antrl 728 . . . . . . . . . 10 ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → 0 ∈ (0..^(♯‘𝐹)))
11103ad2ant3 1134 . . . . . . . . 9 ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → 0 ∈ (0..^(♯‘𝐹)))
12 eqid 2734 . . . . . . . . . 10 (iEdg‘𝐺) = (iEdg‘𝐺)
1312iedginwlk 29669 . . . . . . . . 9 ((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃 ∧ 0 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘0)) ∈ ran (iEdg‘𝐺))
142, 3, 11, 13syl3anc 1370 . . . . . . . 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 2734 . . . . . . . . . . 11 (Vtx‘𝐺) = (Vtx‘𝐺)
1615, 12iswlkg 29645 . . . . . . . . . 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 3323 . . . . . . . . . . . . . . 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 7437 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
19 0p1e1 12385 . . . . . . . . . . . . . . . . . 18 (0 + 1) = 1
2018, 19eqtrdi 2790 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (𝑘 + 1) = 1)
21 wkslem2 29640 . . . . . . . . . . . . . . . . 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 4685 . . . . . . . . . . . . . . 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 1074 . . . . . . . . . . . . . . . . 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 2740 . . . . . . . . . . . . . . 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 1134 . . . . . . . . . 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 1110 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → {(𝑃‘0)} = ((iEdg‘𝐺)‘(𝐹‘0)))
36 edgval 29080 . . . . . . . . 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 2853 . . . . . . 7 ((𝐺 ∈ V ∧ 𝐹(Walks‘𝐺)𝑃 ∧ (Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1)))) → {(𝑃‘0)} ∈ (Edg‘𝐺))
39383exp 1118 . . . . . 6 (𝐺 ∈ V → (𝐹(Walks‘𝐺)𝑃 → ((Fun (iEdg‘𝐺) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))))
40393ad2ant1 1132 . . . . 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 1062  w3a 1086   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  wss 3962  {csn 4630  {cpr 4632   class class class wbr 5147  dom cdm 5688  ran crn 5689  Fun wfun 6556  wf 6558  cfv 6562  (class class class)co 7430  0cc0 11152  1c1 11153   + caddc 11155  ...cfz 13543  ..^cfzo 13690  chash 14365  Word cword 14548  Vtxcvtx 29027  iEdgciedg 29028  Edgcedg 29078  Walkscwlks 29628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-hash 14366  df-word 14549  df-edg 29079  df-wlks 29631
This theorem is referenced by:  clwlkl1loop  29815  loop1cycl  35121
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