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Theorem smupp1 16360
Description: The initial element of the partial sum sequence. (Contributed by Mario Carneiro, 9-Sep-2016.)
Hypotheses
Ref Expression
smuval.a (𝜑𝐴 ⊆ ℕ0)
smuval.b (𝜑𝐵 ⊆ ℕ0)
smuval.p 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
smuval.n (𝜑𝑁 ∈ ℕ0)
Assertion
Ref Expression
smupp1 (𝜑 → (𝑃‘(𝑁 + 1)) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
Distinct variable groups:   𝑚,𝑛,𝑝,𝐴   𝑛,𝑁   𝜑,𝑛   𝐵,𝑚,𝑛,𝑝
Allowed substitution hints:   𝜑(𝑚,𝑝)   𝑃(𝑚,𝑛,𝑝)   𝑁(𝑚,𝑝)

Proof of Theorem smupp1
Dummy variables 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smuval.n . . . . 5 (𝜑𝑁 ∈ ℕ0)
2 nn0uz 12805 . . . . 5 0 = (ℤ‘0)
31, 2eleqtrdi 2848 . . . 4 (𝜑𝑁 ∈ (ℤ‘0))
4 seqp1 13921 . . . 4 (𝑁 ∈ (ℤ‘0) → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑁 + 1)) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))))
53, 4syl 17 . . 3 (𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑁 + 1)) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))))
6 smuval.p . . . 4 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
76fveq1i 6843 . . 3 (𝑃‘(𝑁 + 1)) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑁 + 1))
86fveq1i 6843 . . . 4 (𝑃𝑁) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)
98oveq1i 7367 . . 3 ((𝑃𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)))
105, 7, 93eqtr4g 2801 . 2 (𝜑 → (𝑃‘(𝑁 + 1)) = ((𝑃𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))))
11 1nn0 12429 . . . . . . 7 1 ∈ ℕ0
1211a1i 11 . . . . . 6 (𝜑 → 1 ∈ ℕ0)
131, 12nn0addcld 12477 . . . . 5 (𝜑 → (𝑁 + 1) ∈ ℕ0)
14 eqeq1 2740 . . . . . . 7 (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
15 oveq1 7364 . . . . . . 7 (𝑛 = (𝑁 + 1) → (𝑛 − 1) = ((𝑁 + 1) − 1))
1614, 15ifbieq2d 4512 . . . . . 6 (𝑛 = (𝑁 + 1) → if(𝑛 = 0, ∅, (𝑛 − 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
17 eqid 2736 . . . . . 6 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))
18 0ex 5264 . . . . . . 7 ∅ ∈ V
19 ovex 7390 . . . . . . 7 ((𝑁 + 1) − 1) ∈ V
2018, 19ifex 4536 . . . . . 6 if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) ∈ V
2116, 17, 20fvmpt 6948 . . . . 5 ((𝑁 + 1) ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
2213, 21syl 17 . . . 4 (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
23 nn0p1nn 12452 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
241, 23syl 17 . . . . . 6 (𝜑 → (𝑁 + 1) ∈ ℕ)
2524nnne0d 12203 . . . . 5 (𝜑 → (𝑁 + 1) ≠ 0)
26 ifnefalse 4498 . . . . 5 ((𝑁 + 1) ≠ 0 → if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) = ((𝑁 + 1) − 1))
2725, 26syl 17 . . . 4 (𝜑 → if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) = ((𝑁 + 1) − 1))
281nn0cnd 12475 . . . . 5 (𝜑𝑁 ∈ ℂ)
2912nn0cnd 12475 . . . . 5 (𝜑 → 1 ∈ ℂ)
3028, 29pncand 11513 . . . 4 (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
3122, 27, 303eqtrd 2780 . . 3 (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = 𝑁)
3231oveq2d 7373 . 2 (𝜑 → ((𝑃𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))) = ((𝑃𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))𝑁))
33 smuval.a . . . . 5 (𝜑𝐴 ⊆ ℕ0)
34 smuval.b . . . . 5 (𝜑𝐵 ⊆ ℕ0)
3533, 34, 6smupf 16358 . . . 4 (𝜑𝑃:ℕ0⟶𝒫 ℕ0)
3635, 1ffvelcdmd 7036 . . 3 (𝜑 → (𝑃𝑁) ∈ 𝒫 ℕ0)
37 simpl 483 . . . . 5 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → 𝑥 = (𝑃𝑁))
38 simpr 485 . . . . . . . . 9 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
3938eleq1d 2822 . . . . . . . 8 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → (𝑦𝐴𝑁𝐴))
4038oveq2d 7373 . . . . . . . . 9 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → (𝑘𝑦) = (𝑘𝑁))
4140eleq1d 2822 . . . . . . . 8 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → ((𝑘𝑦) ∈ 𝐵 ↔ (𝑘𝑁) ∈ 𝐵))
4239, 41anbi12d 631 . . . . . . 7 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → ((𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵) ↔ (𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵)))
4342rabbidv 3415 . . . . . 6 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵)})
44 oveq1 7364 . . . . . . . . 9 (𝑘 = 𝑛 → (𝑘𝑁) = (𝑛𝑁))
4544eleq1d 2822 . . . . . . . 8 (𝑘 = 𝑛 → ((𝑘𝑁) ∈ 𝐵 ↔ (𝑛𝑁) ∈ 𝐵))
4645anbi2d 629 . . . . . . 7 (𝑘 = 𝑛 → ((𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵) ↔ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)))
4746cbvrabv 3417 . . . . . 6 {𝑘 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}
4843, 47eqtrdi 2792 . . . . 5 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)})
4937, 48oveq12d 7375 . . . 4 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → (𝑥 sadd {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)}) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
50 oveq1 7364 . . . . 5 (𝑝 = 𝑥 → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}) = (𝑥 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))
51 eleq1w 2820 . . . . . . . . 9 (𝑚 = 𝑦 → (𝑚𝐴𝑦𝐴))
52 oveq2 7365 . . . . . . . . . 10 (𝑚 = 𝑦 → (𝑛𝑚) = (𝑛𝑦))
5352eleq1d 2822 . . . . . . . . 9 (𝑚 = 𝑦 → ((𝑛𝑚) ∈ 𝐵 ↔ (𝑛𝑦) ∈ 𝐵))
5451, 53anbi12d 631 . . . . . . . 8 (𝑚 = 𝑦 → ((𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵) ↔ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)))
5554rabbidv 3415 . . . . . . 7 (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)})
56 oveq1 7364 . . . . . . . . . 10 (𝑘 = 𝑛 → (𝑘𝑦) = (𝑛𝑦))
5756eleq1d 2822 . . . . . . . . 9 (𝑘 = 𝑛 → ((𝑘𝑦) ∈ 𝐵 ↔ (𝑛𝑦) ∈ 𝐵))
5857anbi2d 629 . . . . . . . 8 (𝑘 = 𝑛 → ((𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵) ↔ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)))
5958cbvrabv 3417 . . . . . . 7 {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)}
6055, 59eqtr4di 2794 . . . . . 6 (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)})
6160oveq2d 7373 . . . . 5 (𝑚 = 𝑦 → (𝑥 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}) = (𝑥 sadd {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)}))
6250, 61cbvmpov 7452 . . . 4 (𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})) = (𝑥 ∈ 𝒫 ℕ0, 𝑦 ∈ ℕ0 ↦ (𝑥 sadd {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)}))
63 ovex 7390 . . . 4 ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}) ∈ V
6449, 62, 63ovmpoa 7510 . . 3 (((𝑃𝑁) ∈ 𝒫 ℕ0𝑁 ∈ ℕ0) → ((𝑃𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))𝑁) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
6536, 1, 64syl2anc 584 . 2 (𝜑 → ((𝑃𝑁)(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))𝑁) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
6610, 32, 653eqtrd 2780 1 (𝜑 → (𝑃‘(𝑁 + 1)) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2943  {crab 3407  wss 3910  c0 4282  ifcif 4486  𝒫 cpw 4560  cmpt 5188  cfv 6496  (class class class)co 7357  cmpo 7359  0cc0 11051  1c1 11052   + caddc 11054  cmin 11385  cn 12153  0cn0 12413  cuz 12763  seqcseq 13906   sadd csad 16300
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-xor 1510  df-tru 1544  df-fal 1554  df-had 1595  df-cad 1608  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-seq 13907  df-sad 16331
This theorem is referenced by:  smuval2  16362  smupvallem  16363  smu01lem  16365  smupval  16368  smup1  16369  smueqlem  16370
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