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Theorem smupp1 15821
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 12272 . . . . 5 0 = (ℤ‘0)
31, 2eleqtrdi 2921 . . . 4 (𝜑𝑁 ∈ (ℤ‘0))
4 seqp1 13376 . . . 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 6664 . . 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 6664 . . . 4 (𝑃𝑁) = (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)
98oveq1i 7158 . . 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 2879 . 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 11905 . . . . . . 7 1 ∈ ℕ0
1211a1i 11 . . . . . 6 (𝜑 → 1 ∈ ℕ0)
131, 12nn0addcld 11951 . . . . 5 (𝜑 → (𝑁 + 1) ∈ ℕ0)
14 eqeq1 2823 . . . . . . 7 (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
15 oveq1 7155 . . . . . . 7 (𝑛 = (𝑁 + 1) → (𝑛 − 1) = ((𝑁 + 1) − 1))
1614, 15ifbieq2d 4490 . . . . . 6 (𝑛 = (𝑁 + 1) → if(𝑛 = 0, ∅, (𝑛 − 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
17 eqid 2819 . . . . . 6 (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))
18 0ex 5202 . . . . . . 7 ∅ ∈ V
19 ovex 7181 . . . . . . 7 ((𝑁 + 1) − 1) ∈ V
2018, 19ifex 4513 . . . . . 6 if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) ∈ V
2116, 17, 20fvmpt 6761 . . . . 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 11928 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
241, 23syl 17 . . . . . 6 (𝜑 → (𝑁 + 1) ∈ ℕ)
2524nnne0d 11679 . . . . 5 (𝜑 → (𝑁 + 1) ≠ 0)
26 ifnefalse 4477 . . . . 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 11949 . . . . 5 (𝜑𝑁 ∈ ℂ)
2912nn0cnd 11949 . . . . 5 (𝜑 → 1 ∈ ℂ)
3028, 29pncand 10990 . . . 4 (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
3122, 27, 303eqtrd 2858 . . 3 (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = 𝑁)
3231oveq2d 7164 . 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 15819 . . . 4 (𝜑𝑃:ℕ0⟶𝒫 ℕ0)
3635, 1ffvelrnd 6845 . . 3 (𝜑 → (𝑃𝑁) ∈ 𝒫 ℕ0)
37 simpl 485 . . . . 5 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → 𝑥 = (𝑃𝑁))
38 simpr 487 . . . . . . . . 9 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
3938eleq1d 2895 . . . . . . . 8 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → (𝑦𝐴𝑁𝐴))
4038oveq2d 7164 . . . . . . . . 9 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → (𝑘𝑦) = (𝑘𝑁))
4140eleq1d 2895 . . . . . . . 8 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → ((𝑘𝑦) ∈ 𝐵 ↔ (𝑘𝑁) ∈ 𝐵))
4239, 41anbi12d 632 . . . . . . 7 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → ((𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵) ↔ (𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵)))
4342rabbidv 3479 . . . . . 6 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵)})
44 oveq1 7155 . . . . . . . . 9 (𝑘 = 𝑛 → (𝑘𝑁) = (𝑛𝑁))
4544eleq1d 2895 . . . . . . . 8 (𝑘 = 𝑛 → ((𝑘𝑁) ∈ 𝐵 ↔ (𝑛𝑁) ∈ 𝐵))
4645anbi2d 630 . . . . . . 7 (𝑘 = 𝑛 → ((𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵) ↔ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)))
4746cbvrabv 3490 . . . . . 6 {𝑘 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑘𝑁) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}
4843, 47syl6eq 2870 . . . . 5 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)})
4937, 48oveq12d 7166 . . . 4 ((𝑥 = (𝑃𝑁) ∧ 𝑦 = 𝑁) → (𝑥 sadd {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)}) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
50 oveq1 7155 . . . . 5 (𝑝 = 𝑥 → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}) = (𝑥 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}))
51 eleq1w 2893 . . . . . . . . 9 (𝑚 = 𝑦 → (𝑚𝐴𝑦𝐴))
52 oveq2 7156 . . . . . . . . . 10 (𝑚 = 𝑦 → (𝑛𝑚) = (𝑛𝑦))
5352eleq1d 2895 . . . . . . . . 9 (𝑚 = 𝑦 → ((𝑛𝑚) ∈ 𝐵 ↔ (𝑛𝑦) ∈ 𝐵))
5451, 53anbi12d 632 . . . . . . . 8 (𝑚 = 𝑦 → ((𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵) ↔ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)))
5554rabbidv 3479 . . . . . . 7 (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)})
56 oveq1 7155 . . . . . . . . . 10 (𝑘 = 𝑛 → (𝑘𝑦) = (𝑛𝑦))
5756eleq1d 2895 . . . . . . . . 9 (𝑘 = 𝑛 → ((𝑘𝑦) ∈ 𝐵 ↔ (𝑛𝑦) ∈ 𝐵))
5857anbi2d 630 . . . . . . . 8 (𝑘 = 𝑛 → ((𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵) ↔ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)))
5958cbvrabv 3490 . . . . . . 7 {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑛𝑦) ∈ 𝐵)}
6055, 59syl6eqr 2872 . . . . . 6 (𝑚 = 𝑦 → {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)} = {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)})
6160oveq2d 7164 . . . . 5 (𝑚 = 𝑦 → (𝑥 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)}) = (𝑥 sadd {𝑘 ∈ ℕ0 ∣ (𝑦𝐴 ∧ (𝑘𝑦) ∈ 𝐵)}))
6250, 61cbvmpov 7241 . . . 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 7181 . . . 4 ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}) ∈ V
6449, 62, 63ovmpoa 7297 . . 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 586 . 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 2858 1 (𝜑 → (𝑃‘(𝑁 + 1)) = ((𝑃𝑁) sadd {𝑛 ∈ ℕ0 ∣ (𝑁𝐴 ∧ (𝑛𝑁) ∈ 𝐵)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  wcel 2107  wne 3014  {crab 3140  wss 3934  c0 4289  ifcif 4465  𝒫 cpw 4537  cmpt 5137  cfv 6348  (class class class)co 7148  cmpo 7150  0cc0 10529  1c1 10530   + caddc 10532  cmin 10862  cn 11630  0cn0 11889  cuz 12235  seqcseq 13361   sadd csad 15761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-xor 1498  df-tru 1533  df-had 1587  df-cad 1601  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885  df-seq 13362  df-sad 15792
This theorem is referenced by:  smuval2  15823  smupvallem  15824  smu01lem  15826  smupval  15829  smup1  15830  smueqlem  15831
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