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Theorem lfgrwlkprop 26565
Description: Two adjacent vertices in a walk are different in a loop-free graph. (Contributed by AV, 28-Jan-2021.)
Hypothesis
Ref Expression
lfgrwlkprop.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
lfgrwlkprop ((𝐹(Walks‘𝐺)𝑃𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))
Distinct variable groups:   𝑘,𝐹,𝑥   𝑘,𝐺   𝑘,𝐼,𝑥   𝑃,𝑘   𝑘,𝑉,𝑥
Allowed substitution hints:   𝑃(𝑥)   𝐺(𝑥)

Proof of Theorem lfgrwlkprop
StepHypRef Expression
1 wlkv 26489 . . . . 5 (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
2 eqid 2620 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
3 lfgrwlkprop.i . . . . . 6 𝐼 = (iEdg‘𝐺)
42, 3iswlk 26487 . . . . 5 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))
51, 4syl 17 . . . 4 (𝐹(Walks‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))
6 ifptru 1022 . . . . . . . . . . . 12 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ↔ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}))
76adantr 481 . . . . . . . . . . 11 (((𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ (((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) ∧ 𝑘 ∈ (0..^(#‘𝐹)))) → (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ↔ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}))
8 simplr 791 . . . . . . . . . . . . . 14 ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) ∧ 𝑘 ∈ (0..^(#‘𝐹))) → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)})
9 wrdsymbcl 13301 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word dom 𝐼𝑘 ∈ (0..^(#‘𝐹))) → (𝐹𝑘) ∈ dom 𝐼)
109ad4ant14 1291 . . . . . . . . . . . . . 14 ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) ∧ 𝑘 ∈ (0..^(#‘𝐹))) → (𝐹𝑘) ∈ dom 𝐼)
118, 10ffvelrnd 6346 . . . . . . . . . . . . 13 ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) ∧ 𝑘 ∈ (0..^(#‘𝐹))) → (𝐼‘(𝐹𝑘)) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)})
12 fveq2 6178 . . . . . . . . . . . . . . . 16 (𝑥 = (𝐼‘(𝐹𝑘)) → (#‘𝑥) = (#‘(𝐼‘(𝐹𝑘))))
1312breq2d 4656 . . . . . . . . . . . . . . 15 (𝑥 = (𝐼‘(𝐹𝑘)) → (2 ≤ (#‘𝑥) ↔ 2 ≤ (#‘(𝐼‘(𝐹𝑘)))))
1413elrab 3357 . . . . . . . . . . . . . 14 ((𝐼‘(𝐹𝑘)) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} ↔ ((𝐼‘(𝐹𝑘)) ∈ 𝒫 𝑉 ∧ 2 ≤ (#‘(𝐼‘(𝐹𝑘)))))
15 fveq2 6178 . . . . . . . . . . . . . . . . . . 19 ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘)} → (#‘(𝐼‘(𝐹𝑘))) = (#‘{(𝑃𝑘)}))
1615breq2d 4656 . . . . . . . . . . . . . . . . . 18 ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘)} → (2 ≤ (#‘(𝐼‘(𝐹𝑘))) ↔ 2 ≤ (#‘{(𝑃𝑘)})))
17 fvex 6188 . . . . . . . . . . . . . . . . . . . . 21 (𝑃𝑘) ∈ V
18 hashsng 13142 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃𝑘) ∈ V → (#‘{(𝑃𝑘)}) = 1)
1917, 18ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 (#‘{(𝑃𝑘)}) = 1
2019breq2i 4652 . . . . . . . . . . . . . . . . . . 19 (2 ≤ (#‘{(𝑃𝑘)}) ↔ 2 ≤ 1)
21 1lt2 11179 . . . . . . . . . . . . . . . . . . . 20 1 < 2
22 1re 10024 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℝ
23 2re 11075 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ ℝ
2422, 23ltnlei 10143 . . . . . . . . . . . . . . . . . . . . 21 (1 < 2 ↔ ¬ 2 ≤ 1)
25 pm2.21 120 . . . . . . . . . . . . . . . . . . . . 21 (¬ 2 ≤ 1 → (2 ≤ 1 → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
2624, 25sylbi 207 . . . . . . . . . . . . . . . . . . . 20 (1 < 2 → (2 ≤ 1 → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
2721, 26ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (2 ≤ 1 → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))
2820, 27sylbi 207 . . . . . . . . . . . . . . . . . 18 (2 ≤ (#‘{(𝑃𝑘)}) → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))
2916, 28syl6bi 243 . . . . . . . . . . . . . . . . 17 ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘)} → (2 ≤ (#‘(𝐼‘(𝐹𝑘))) → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
3029com12 32 . . . . . . . . . . . . . . . 16 (2 ≤ (#‘(𝐼‘(𝐹𝑘))) → ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘)} → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
3130adantl 482 . . . . . . . . . . . . . . 15 (((𝐼‘(𝐹𝑘)) ∈ 𝒫 𝑉 ∧ 2 ≤ (#‘(𝐼‘(𝐹𝑘)))) → ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘)} → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
3231a1i 11 . . . . . . . . . . . . . 14 ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) ∧ 𝑘 ∈ (0..^(#‘𝐹))) → (((𝐼‘(𝐹𝑘)) ∈ 𝒫 𝑉 ∧ 2 ≤ (#‘(𝐼‘(𝐹𝑘)))) → ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘)} → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))))
3314, 32syl5bi 232 . . . . . . . . . . . . 13 ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) ∧ 𝑘 ∈ (0..^(#‘𝐹))) → ((𝐼‘(𝐹𝑘)) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} → ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘)} → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))))
3411, 33mpd 15 . . . . . . . . . . . 12 ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) ∧ 𝑘 ∈ (0..^(#‘𝐹))) → ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘)} → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
3534adantl 482 . . . . . . . . . . 11 (((𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ (((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) ∧ 𝑘 ∈ (0..^(#‘𝐹)))) → ((𝐼‘(𝐹𝑘)) = {(𝑃𝑘)} → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
367, 35sylbid 230 . . . . . . . . . 10 (((𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ (((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) ∧ 𝑘 ∈ (0..^(#‘𝐹)))) → (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
3736ex 450 . . . . . . . . 9 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) ∧ 𝑘 ∈ (0..^(#‘𝐹))) → (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))))
38 neqne 2799 . . . . . . . . . 10 (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))
39382a1d 26 . . . . . . . . 9 (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) → ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) ∧ 𝑘 ∈ (0..^(#‘𝐹))) → (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))))
4037, 39pm2.61i 176 . . . . . . . 8 ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) ∧ 𝑘 ∈ (0..^(#‘𝐹))) → (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
4140ralimdva 2959 . . . . . . 7 (((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) → (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
4241ex 450 . . . . . 6 ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} → (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))))
4342com23 86 . . . . 5 ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) → (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))))
44433impia 1259 . . . 4 ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))) → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
455, 44syl6bi 243 . . 3 (𝐹(Walks‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃 → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))))
4645pm2.43i 52 . 2 (𝐹(Walks‘𝐺)𝑃 → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
4746imp 445 1 ((𝐹(Walks‘𝐺)𝑃𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  if-wif 1011  w3a 1036   = wceq 1481  wcel 1988  wne 2791  wral 2909  {crab 2913  Vcvv 3195  wss 3567  𝒫 cpw 4149  {csn 4168  {cpr 4170   class class class wbr 4644  dom cdm 5104  wf 5872  cfv 5876  (class class class)co 6635  0cc0 9921  1c1 9922   + caddc 9924   < clt 10059  cle 10060  2c2 11055  ...cfz 12311  ..^cfzo 12449  #chash 13100  Word cword 13274  Vtxcvtx 25855  iEdgciedg 25856  Walkscwlks 26473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-er 7727  df-map 7844  df-pm 7845  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-fzo 12450  df-hash 13101  df-word 13282  df-wlks 26476
This theorem is referenced by:  lfgriswlk  26566
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