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Theorem wlkiswwlks1 27697
 Description: The sequence of vertices in a walk is a walk as word in a pseudograph. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 9-Apr-2021.)
Assertion
Ref Expression
wlkiswwlks1 (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃𝑃 ∈ (WWalks‘𝐺)))

Proof of Theorem wlkiswwlks1
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wlkn0 27454 . 2 (𝐹(Walks‘𝐺)𝑃𝑃 ≠ ∅)
2 eqid 2798 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2798 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
42, 3upgriswlk 27474 . . 3 (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))
5 simpr 488 . . . . . 6 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ 𝑃 ≠ ∅) → 𝑃 ≠ ∅)
6 ffz0iswrd 13904 . . . . . . . 8 (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → 𝑃 ∈ Word (Vtx‘𝐺))
763ad2ant2 1131 . . . . . . 7 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → 𝑃 ∈ Word (Vtx‘𝐺))
87ad2antlr 726 . . . . . 6 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ 𝑃 ≠ ∅) → 𝑃 ∈ Word (Vtx‘𝐺))
9 upgruhgr 26939 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
103uhgrfun 26903 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
11 funfn 6362 . . . . . . . . . . . . . . . . . . 19 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1211biimpi 219 . . . . . . . . . . . . . . . . . 18 (Fun (iEdg‘𝐺) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
139, 10, 123syl 18 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1413ad2antlr 726 . . . . . . . . . . . . . . . 16 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
15 wrdsymbcl 13890 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (𝐹𝑖) ∈ dom (iEdg‘𝐺))
1615ad4ant14 751 . . . . . . . . . . . . . . . 16 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (𝐹𝑖) ∈ dom (iEdg‘𝐺))
17 fnfvelrn 6835 . . . . . . . . . . . . . . . 16 (((iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ (𝐹𝑖) ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(𝐹𝑖)) ∈ ran (iEdg‘𝐺))
1814, 16, 17syl2anc 587 . . . . . . . . . . . . . . 15 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹𝑖)) ∈ ran (iEdg‘𝐺))
19 edgval 26886 . . . . . . . . . . . . . . 15 (Edg‘𝐺) = ran (iEdg‘𝐺)
2018, 19eleqtrrdi 2901 . . . . . . . . . . . . . 14 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹𝑖)) ∈ (Edg‘𝐺))
21 eleq1 2877 . . . . . . . . . . . . . . 15 ({(𝑃𝑖), (𝑃‘(𝑖 + 1))} = ((iEdg‘𝐺)‘(𝐹𝑖)) → ({(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ((iEdg‘𝐺)‘(𝐹𝑖)) ∈ (Edg‘𝐺)))
2221eqcoms 2806 . . . . . . . . . . . . . 14 (((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} → ({(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ((iEdg‘𝐺)‘(𝐹𝑖)) ∈ (Edg‘𝐺)))
2320, 22syl5ibrcom 250 . . . . . . . . . . . . 13 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → (((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} → {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
2423ralimdva 3144 . . . . . . . . . . . 12 (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) → (∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} → ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
2524ex 416 . . . . . . . . . . 11 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (𝐺 ∈ UPGraph → (∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} → ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
2625com23 86 . . . . . . . . . 10 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} → (𝐺 ∈ UPGraph → ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
27263impia 1114 . . . . . . . . 9 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → (𝐺 ∈ UPGraph → ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
2827impcom 411 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) → ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))
29 lencl 13896 . . . . . . . . . . . . . 14 (𝐹 ∈ Word dom (iEdg‘𝐺) → (♯‘𝐹) ∈ ℕ0)
30 ffz0hash 13821 . . . . . . . . . . . . . . . 16 (((♯‘𝐹) ∈ ℕ0𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (♯‘𝑃) = ((♯‘𝐹) + 1))
3130ex 416 . . . . . . . . . . . . . . 15 ((♯‘𝐹) ∈ ℕ0 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (♯‘𝑃) = ((♯‘𝐹) + 1)))
32 oveq1 7152 . . . . . . . . . . . . . . . . 17 ((♯‘𝑃) = ((♯‘𝐹) + 1) → ((♯‘𝑃) − 1) = (((♯‘𝐹) + 1) − 1))
33 nn0cn 11913 . . . . . . . . . . . . . . . . . 18 ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℂ)
34 pncan1 11071 . . . . . . . . . . . . . . . . . 18 ((♯‘𝐹) ∈ ℂ → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
3533, 34syl 17 . . . . . . . . . . . . . . . . 17 ((♯‘𝐹) ∈ ℕ0 → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
3632, 35sylan9eqr 2855 . . . . . . . . . . . . . . . 16 (((♯‘𝐹) ∈ ℕ0 ∧ (♯‘𝑃) = ((♯‘𝐹) + 1)) → ((♯‘𝑃) − 1) = (♯‘𝐹))
3736ex 416 . . . . . . . . . . . . . . 15 ((♯‘𝐹) ∈ ℕ0 → ((♯‘𝑃) = ((♯‘𝐹) + 1) → ((♯‘𝑃) − 1) = (♯‘𝐹)))
3831, 37syld 47 . . . . . . . . . . . . . 14 ((♯‘𝐹) ∈ ℕ0 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((♯‘𝑃) − 1) = (♯‘𝐹)))
3929, 38syl 17 . . . . . . . . . . . . 13 (𝐹 ∈ Word dom (iEdg‘𝐺) → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((♯‘𝑃) − 1) = (♯‘𝐹)))
4039imp 410 . . . . . . . . . . . 12 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → ((♯‘𝑃) − 1) = (♯‘𝐹))
4140oveq2d 7161 . . . . . . . . . . 11 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (0..^((♯‘𝑃) − 1)) = (0..^(♯‘𝐹)))
4241raleqdv 3365 . . . . . . . . . 10 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
43423adant3 1129 . . . . . . . . 9 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
4443adantl 485 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(♯‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
4528, 44mpbird 260 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))
4645adantr 484 . . . . . 6 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ 𝑃 ≠ ∅) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))
47 eqid 2798 . . . . . . 7 (Edg‘𝐺) = (Edg‘𝐺)
482, 47iswwlks 27666 . . . . . 6 (𝑃 ∈ (WWalks‘𝐺) ↔ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
495, 8, 46, 48syl3anbrc 1340 . . . . 5 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ 𝑃 ≠ ∅) → 𝑃 ∈ (WWalks‘𝐺))
5049ex 416 . . . 4 ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) → (𝑃 ≠ ∅ → 𝑃 ∈ (WWalks‘𝐺)))
5150ex 416 . . 3 (𝐺 ∈ UPGraph → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃 ≠ ∅ → 𝑃 ∈ (WWalks‘𝐺))))
524, 51sylbid 243 . 2 (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 → (𝑃 ≠ ∅ → 𝑃 ∈ (WWalks‘𝐺))))
531, 52mpdi 45 1 (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃𝑃 ∈ (WWalks‘𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∅c0 4246  {cpr 4530   class class class wbr 5034  dom cdm 5523  ran crn 5524  Fun wfun 6326   Fn wfn 6327  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145  ℂcc 10542  0cc0 10544  1c1 10545   + caddc 10547   − cmin 10877  ℕ0cn0 11903  ...cfz 12905  ..^cfzo 13048  ♯chash 13706  Word cword 13877  Vtxcvtx 26833  iEdgciedg 26834  Edgcedg 26884  UHGraphcuhgr 26893  UPGraphcupgr 26917  Walkscwlks 27430  WWalkscwwlks 27655 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-2o 8104  df-oadd 8107  df-er 8290  df-map 8409  df-pm 8410  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-dju 9332  df-card 9370  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-2 11706  df-n0 11904  df-xnn0 11976  df-z 11990  df-uz 12252  df-fz 12906  df-fzo 13049  df-hash 13707  df-word 13878  df-edg 26885  df-uhgr 26895  df-upgr 26919  df-wlks 27433  df-wwlks 27660 This theorem is referenced by:  wlklnwwlkln1  27698  wlkiswwlks  27706  wlkiswwlkupgr  27708  elwspths2spth  27797
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