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Theorem smupp1 16517
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 12920 . . . . 5 0 = (ℤ‘0)
31, 2eleqtrdi 2851 . . . 4 (𝜑𝑁 ∈ (ℤ‘0))
4 seqp1 14057 . . . 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 6907 . . 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 6907 . . . 4 (𝑃𝑁) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)
98oveq1i 7441 . . 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 2802 . 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 12542 . . . . . . 7 1 ∈ ℕ0
1211a1i 11 . . . . . 6 (𝜑 → 1 ∈ ℕ0)
131, 12nn0addcld 12591 . . . . 5 (𝜑 → (𝑁 + 1) ∈ ℕ0)
14 eqeq1 2741 . . . . . . 7 (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
15 oveq1 7438 . . . . . . 7 (𝑛 = (𝑁 + 1) → (𝑛 − 1) = ((𝑁 + 1) − 1))
1614, 15ifbieq2d 4552 . . . . . 6 (𝑛 = (𝑁 + 1) → if(𝑛 = 0, ∅, (𝑛 − 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
17 eqid 2737 . . . . . 6 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))
18 0ex 5307 . . . . . . 7 ∅ ∈ V
19 ovex 7464 . . . . . . 7 ((𝑁 + 1) − 1) ∈ V
2018, 19ifex 4576 . . . . . 6 if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) ∈ V
2116, 17, 20fvmpt 7016 . . . . 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 12565 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
241, 23syl 17 . . . . . 6 (𝜑 → (𝑁 + 1) ∈ ℕ)
2524nnne0d 12316 . . . . 5 (𝜑 → (𝑁 + 1) ≠ 0)
26 ifnefalse 4537 . . . . 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 12589 . . . . 5 (𝜑𝑁 ∈ ℂ)
2912nn0cnd 12589 . . . . 5 (𝜑 → 1 ∈ ℂ)
3028, 29pncand 11621 . . . 4 (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
3122, 27, 303eqtrd 2781 . . 3 (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = 𝑁)
3231oveq2d 7447 . 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 16515 . . . 4 (𝜑𝑃:ℕ0⟶𝒫 ℕ0)
3635, 1ffvelcdmd 7105 . . 3 (𝜑 → (𝑃𝑁) ∈ 𝒫 ℕ0)
37 simpl 482 . . . . 5 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → 𝑥 = (𝑃𝑁))
38 simpr 484 . . . . . . . . 9 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
3938eleq1d 2826 . . . . . . . 8 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → (𝑦𝐴𝑁𝐴))
4038oveq2d 7447 . . . . . . . . 9 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → (𝑘𝑦) = (𝑘𝑁))
4140eleq1d 2826 . . . . . . . 8 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → ((𝑘𝑦) ∈ 𝐵 ↔ (𝑘𝑁) ∈ 𝐵))
4239, 41anbi12d 632 . . . . . . 7 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → ((𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵) ↔ (𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵)))
4342rabbidv 3444 . . . . . 6 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵)})
44 oveq1 7438 . . . . . . . . 9 (𝑘 = 𝑛 → (𝑘𝑁) = (𝑛𝑁))
4544eleq1d 2826 . . . . . . . 8 (𝑘 = 𝑛 → ((𝑘𝑁) ∈ 𝐵 ↔ (𝑛𝑁) ∈ 𝐵))
4645anbi2d 630 . . . . . . 7 (𝑘 = 𝑛 → ((𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵) ↔ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)))
4746cbvrabv 3447 . . . . . 6 {𝑘 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}
4843, 47eqtrdi 2793 . . . . 5 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)})
4937, 48oveq12d 7449 . . . 4 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → (𝑥 sadd {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)}) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
50 oveq1 7438 . . . . 5 (𝑝 = 𝑥 → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}) = (𝑥 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))
51 eleq1w 2824 . . . . . . . . 9 (𝑚 = 𝑦 → (𝑚𝐴𝑦𝐴))
52 oveq2 7439 . . . . . . . . . 10 (𝑚 = 𝑦 → (𝑛𝑚) = (𝑛𝑦))
5352eleq1d 2826 . . . . . . . . 9 (𝑚 = 𝑦 → ((𝑛𝑚) ∈ 𝐵 ↔ (𝑛𝑦) ∈ 𝐵))
5451, 53anbi12d 632 . . . . . . . 8 (𝑚 = 𝑦 → ((𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵) ↔ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)))
5554rabbidv 3444 . . . . . . 7 (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)})
56 oveq1 7438 . . . . . . . . . 10 (𝑘 = 𝑛 → (𝑘𝑦) = (𝑛𝑦))
5756eleq1d 2826 . . . . . . . . 9 (𝑘 = 𝑛 → ((𝑘𝑦) ∈ 𝐵 ↔ (𝑛𝑦) ∈ 𝐵))
5857anbi2d 630 . . . . . . . 8 (𝑘 = 𝑛 → ((𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵) ↔ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)))
5958cbvrabv 3447 . . . . . . 7 {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)}
6055, 59eqtr4di 2795 . . . . . 6 (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)})
6160oveq2d 7447 . . . . 5 (𝑚 = 𝑦 → (𝑥 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}) = (𝑥 sadd {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)}))
6250, 61cbvmpov 7528 . . . 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 7464 . . . 4 ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}) ∈ V
6449, 62, 63ovmpoa 7588 . . 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 2781 1 (𝜑 → (𝑃‘(𝑁 + 1)) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  {crab 3436  wss 3951  c0 4333  ifcif 4525  𝒫 cpw 4600  cmpt 5225  cfv 6561  (class class class)co 7431  cmpo 7433  0cc0 11155  1c1 11156   + caddc 11158  cmin 11492  cn 12266  0cn0 12526  cuz 12878  seqcseq 14042   sadd csad 16457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1512  df-tru 1543  df-fal 1553  df-had 1594  df-cad 1607  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-seq 14043  df-sad 16488
This theorem is referenced by:  smuval2  16519  smupvallem  16520  smu01lem  16522  smupval  16525  smup1  16526  smueqlem  16527
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