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Theorem seqcoll2 14371
Description: The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 13-Dec-2014.)
Hypotheses
Ref Expression
seqcoll2.1 ((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)
seqcoll2.1b ((𝜑𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)
seqcoll2.c ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
seqcoll2.a (𝜑𝑍𝑆)
seqcoll2.2 (𝜑𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))
seqcoll2.3 (𝜑𝐴 ≠ ∅)
seqcoll2.5 (𝜑𝐴 ⊆ (𝑀...𝑁))
seqcoll2.6 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)
seqcoll2.7 ((𝜑𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)
seqcoll2.8 ((𝜑𝑛 ∈ (1...(♯‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))
Assertion
Ref Expression
seqcoll2 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘𝐴)))
Distinct variable groups:   𝑘,𝑛,𝐴   𝑘,𝐹,𝑛   𝑘,𝐺,𝑛   𝑛,𝐻   𝑘,𝑀,𝑛   𝜑,𝑘,𝑛   𝑘,𝑁   + ,𝑘,𝑛   𝑆,𝑘,𝑛   𝑘,𝑍
Allowed substitution hints:   𝐻(𝑘)   𝑁(𝑛)   𝑍(𝑛)

Proof of Theorem seqcoll2
StepHypRef Expression
1 seqcoll2.1b . . 3 ((𝜑𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)
2 fzssuz 13489 . . . 4 (𝑀...𝑁) ⊆ (ℤ𝑀)
3 seqcoll2.5 . . . . 5 (𝜑𝐴 ⊆ (𝑀...𝑁))
4 seqcoll2.2 . . . . . . . 8 (𝜑𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))
5 isof1o 7273 . . . . . . . 8 (𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴) → 𝐺:(1...(♯‘𝐴))–1-1-onto𝐴)
64, 5syl 17 . . . . . . 7 (𝜑𝐺:(1...(♯‘𝐴))–1-1-onto𝐴)
7 f1of 6789 . . . . . . 7 (𝐺:(1...(♯‘𝐴))–1-1-onto𝐴𝐺:(1...(♯‘𝐴))⟶𝐴)
86, 7syl 17 . . . . . 6 (𝜑𝐺:(1...(♯‘𝐴))⟶𝐴)
9 seqcoll2.3 . . . . . . . . . 10 (𝜑𝐴 ≠ ∅)
10 fzfi 13884 . . . . . . . . . . . . 13 (𝑀...𝑁) ∈ Fin
11 ssfi 9124 . . . . . . . . . . . . 13 (((𝑀...𝑁) ∈ Fin ∧ 𝐴 ⊆ (𝑀...𝑁)) → 𝐴 ∈ Fin)
1210, 3, 11sylancr 588 . . . . . . . . . . . 12 (𝜑𝐴 ∈ Fin)
13 hasheq0 14270 . . . . . . . . . . . 12 (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
1412, 13syl 17 . . . . . . . . . . 11 (𝜑 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
1514necon3bbid 2982 . . . . . . . . . 10 (𝜑 → (¬ (♯‘𝐴) = 0 ↔ 𝐴 ≠ ∅))
169, 15mpbird 257 . . . . . . . . 9 (𝜑 → ¬ (♯‘𝐴) = 0)
17 hashcl 14263 . . . . . . . . . . . 12 (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
1812, 17syl 17 . . . . . . . . . . 11 (𝜑 → (♯‘𝐴) ∈ ℕ0)
19 elnn0 12422 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 ↔ ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0))
2018, 19sylib 217 . . . . . . . . . 10 (𝜑 → ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0))
2120ord 863 . . . . . . . . 9 (𝜑 → (¬ (♯‘𝐴) ∈ ℕ → (♯‘𝐴) = 0))
2216, 21mt3d 148 . . . . . . . 8 (𝜑 → (♯‘𝐴) ∈ ℕ)
23 nnuz 12813 . . . . . . . 8 ℕ = (ℤ‘1)
2422, 23eleqtrdi 2848 . . . . . . 7 (𝜑 → (♯‘𝐴) ∈ (ℤ‘1))
25 eluzfz2 13456 . . . . . . 7 ((♯‘𝐴) ∈ (ℤ‘1) → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
2624, 25syl 17 . . . . . 6 (𝜑 → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
278, 26ffvelcdmd 7041 . . . . 5 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ 𝐴)
283, 27sseldd 3950 . . . 4 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (𝑀...𝑁))
292, 28sselid 3947 . . 3 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (ℤ𝑀))
30 elfzuz3 13445 . . . 4 ((𝐺‘(♯‘𝐴)) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ‘(𝐺‘(♯‘𝐴))))
3128, 30syl 17 . . 3 (𝜑𝑁 ∈ (ℤ‘(𝐺‘(♯‘𝐴))))
32 fzss2 13488 . . . . . . 7 (𝑁 ∈ (ℤ‘(𝐺‘(♯‘𝐴))) → (𝑀...(𝐺‘(♯‘𝐴))) ⊆ (𝑀...𝑁))
3331, 32syl 17 . . . . . 6 (𝜑 → (𝑀...(𝐺‘(♯‘𝐴))) ⊆ (𝑀...𝑁))
3433sselda 3949 . . . . 5 ((𝜑𝑘 ∈ (𝑀...(𝐺‘(♯‘𝐴)))) → 𝑘 ∈ (𝑀...𝑁))
35 seqcoll2.6 . . . . 5 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)
3634, 35syldan 592 . . . 4 ((𝜑𝑘 ∈ (𝑀...(𝐺‘(♯‘𝐴)))) → (𝐹𝑘) ∈ 𝑆)
37 seqcoll2.c . . . 4 ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
3829, 36, 37seqcl 13935 . . 3 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) ∈ 𝑆)
39 peano2uz 12833 . . . . . . . 8 ((𝐺‘(♯‘𝐴)) ∈ (ℤ𝑀) → ((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ𝑀))
4029, 39syl 17 . . . . . . 7 (𝜑 → ((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ𝑀))
41 fzss1 13487 . . . . . . 7 (((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ𝑀) → (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
4240, 41syl 17 . . . . . 6 (𝜑 → (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
4342sselda 3949 . . . . 5 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
44 eluzelre 12781 . . . . . . . . 9 ((𝐺‘(♯‘𝐴)) ∈ (ℤ𝑀) → (𝐺‘(♯‘𝐴)) ∈ ℝ)
4529, 44syl 17 . . . . . . . 8 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ ℝ)
4645adantr 482 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) ∈ ℝ)
47 peano2re 11335 . . . . . . . 8 ((𝐺‘(♯‘𝐴)) ∈ ℝ → ((𝐺‘(♯‘𝐴)) + 1) ∈ ℝ)
4846, 47syl 17 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → ((𝐺‘(♯‘𝐴)) + 1) ∈ ℝ)
49 elfzelz 13448 . . . . . . . . 9 (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℤ)
5049zred 12614 . . . . . . . 8 (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℝ)
5150adantl 483 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ℝ)
5246ltp1d 12092 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) < ((𝐺‘(♯‘𝐴)) + 1))
53 elfzle1 13451 . . . . . . . 8 (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → ((𝐺‘(♯‘𝐴)) + 1) ≤ 𝑘)
5453adantl 483 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → ((𝐺‘(♯‘𝐴)) + 1) ≤ 𝑘)
5546, 48, 51, 52, 54ltletrd 11322 . . . . . 6 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) < 𝑘)
566adantr 482 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺:(1...(♯‘𝐴))–1-1-onto𝐴)
57 f1ocnv 6801 . . . . . . . . . . . . . 14 (𝐺:(1...(♯‘𝐴))–1-1-onto𝐴𝐺:𝐴1-1-onto→(1...(♯‘𝐴)))
5856, 57syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺:𝐴1-1-onto→(1...(♯‘𝐴)))
59 f1of 6789 . . . . . . . . . . . . 13 (𝐺:𝐴1-1-onto→(1...(♯‘𝐴)) → 𝐺:𝐴⟶(1...(♯‘𝐴)))
6058, 59syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺:𝐴⟶(1...(♯‘𝐴)))
61 simprr 772 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝑘𝐴)
6260, 61ffvelcdmd 7041 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ (1...(♯‘𝐴)))
6362elfzelzd 13449 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℤ)
6463zred 12614 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℝ)
6518adantr 482 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (♯‘𝐴) ∈ ℕ0)
6665nn0red 12481 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (♯‘𝐴) ∈ ℝ)
67 elfzle2 13452 . . . . . . . . . 10 ((𝐺𝑘) ∈ (1...(♯‘𝐴)) → (𝐺𝑘) ≤ (♯‘𝐴))
6862, 67syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ≤ (♯‘𝐴))
6964, 66, 68lensymd 11313 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ¬ (♯‘𝐴) < (𝐺𝑘))
704adantr 482 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))
7126adantr 482 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
72 isorel 7276 . . . . . . . . . 10 ((𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴) ∧ ((♯‘𝐴) ∈ (1...(♯‘𝐴)) ∧ (𝐺𝑘) ∈ (1...(♯‘𝐴)))) → ((♯‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(♯‘𝐴)) < (𝐺‘(𝐺𝑘))))
7370, 71, 62, 72syl12anc 836 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((♯‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(♯‘𝐴)) < (𝐺‘(𝐺𝑘))))
74 f1ocnvfv2 7228 . . . . . . . . . . 11 ((𝐺:(1...(♯‘𝐴))–1-1-onto𝐴𝑘𝐴) → (𝐺‘(𝐺𝑘)) = 𝑘)
7556, 61, 74syl2anc 585 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺‘(𝐺𝑘)) = 𝑘)
7675breq2d 5122 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((𝐺‘(♯‘𝐴)) < (𝐺‘(𝐺𝑘)) ↔ (𝐺‘(♯‘𝐴)) < 𝑘))
7773, 76bitrd 279 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((♯‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(♯‘𝐴)) < 𝑘))
7869, 77mtbid 324 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ¬ (𝐺‘(♯‘𝐴)) < 𝑘)
7978expr 458 . . . . . 6 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝑘𝐴 → ¬ (𝐺‘(♯‘𝐴)) < 𝑘))
8055, 79mt2d 136 . . . . 5 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → ¬ 𝑘𝐴)
8143, 80eldifd 3926 . . . 4 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
82 seqcoll2.7 . . . 4 ((𝜑𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)
8381, 82syldan 592 . . 3 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐹𝑘) = 𝑍)
841, 29, 31, 38, 83seqid2 13961 . 2 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) = (seq𝑀( + , 𝐹)‘𝑁))
85 seqcoll2.1 . . 3 ((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)
86 seqcoll2.a . . 3 (𝜑𝑍𝑆)
873, 2sstrdi 3961 . . 3 (𝜑𝐴 ⊆ (ℤ𝑀))
8833ssdifd 4105 . . . . 5 (𝜑 → ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴) ⊆ ((𝑀...𝑁) ∖ 𝐴))
8988sselda 3949 . . . 4 ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
9089, 82syldan 592 . . 3 ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)
91 seqcoll2.8 . . 3 ((𝜑𝑛 ∈ (1...(♯‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))
9285, 1, 37, 86, 4, 26, 87, 36, 90, 91seqcoll 14370 . 2 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) = (seq1( + , 𝐻)‘(♯‘𝐴)))
9384, 92eqtr3d 2779 1 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2944  cdif 3912  wss 3915  c0 4287   class class class wbr 5110  ccnv 5637  wf 6497  1-1-ontowf1o 6500  cfv 6501   Isom wiso 6502  (class class class)co 7362  Fincfn 8890  cr 11057  0cc0 11058  1c1 11059   + caddc 11061   < clt 11196  cle 11197  cn 12160  0cn0 12420  cuz 12770  ...cfz 13431  seqcseq 13913  chash 14237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-seq 13914  df-hash 14238
This theorem is referenced by:  isercolllem3  15558  gsumval3  19691
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