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Theorem seqcoll2 13639
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 12767 . . . 4 (𝑀...𝑁) ⊆ (ℤ𝑀)
3 seqcoll2.5 . . . . 5 (𝜑𝐴 ⊆ (𝑀...𝑁))
4 seqcoll2.2 . . . . . . . 8 (𝜑𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))
5 isof1o 6901 . . . . . . . 8 (𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴) → 𝐺:(1...(♯‘𝐴))–1-1-onto𝐴)
64, 5syl 17 . . . . . . 7 (𝜑𝐺:(1...(♯‘𝐴))–1-1-onto𝐴)
7 f1of 6446 . . . . . . 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 13158 . . . . . . . . . . . . 13 (𝑀...𝑁) ∈ Fin
11 ssfi 8535 . . . . . . . . . . . . 13 (((𝑀...𝑁) ∈ Fin ∧ 𝐴 ⊆ (𝑀...𝑁)) → 𝐴 ∈ Fin)
1210, 3, 11sylancr 578 . . . . . . . . . . . 12 (𝜑𝐴 ∈ Fin)
13 hasheq0 13542 . . . . . . . . . . . 12 (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
1412, 13syl 17 . . . . . . . . . . 11 (𝜑 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
1514necon3bbid 3004 . . . . . . . . . 10 (𝜑 → (¬ (♯‘𝐴) = 0 ↔ 𝐴 ≠ ∅))
169, 15mpbird 249 . . . . . . . . 9 (𝜑 → ¬ (♯‘𝐴) = 0)
17 hashcl 13535 . . . . . . . . . . . 12 (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
1812, 17syl 17 . . . . . . . . . . 11 (𝜑 → (♯‘𝐴) ∈ ℕ0)
19 elnn0 11712 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 ↔ ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0))
2018, 19sylib 210 . . . . . . . . . 10 (𝜑 → ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0))
2120ord 850 . . . . . . . . 9 (𝜑 → (¬ (♯‘𝐴) ∈ ℕ → (♯‘𝐴) = 0))
2216, 21mt3d 143 . . . . . . . 8 (𝜑 → (♯‘𝐴) ∈ ℕ)
23 nnuz 12098 . . . . . . . 8 ℕ = (ℤ‘1)
2422, 23syl6eleq 2876 . . . . . . 7 (𝜑 → (♯‘𝐴) ∈ (ℤ‘1))
25 eluzfz2 12734 . . . . . . 7 ((♯‘𝐴) ∈ (ℤ‘1) → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
2624, 25syl 17 . . . . . 6 (𝜑 → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
278, 26ffvelrnd 6679 . . . . 5 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ 𝐴)
283, 27sseldd 3861 . . . 4 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (𝑀...𝑁))
292, 28sseldi 3858 . . 3 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (ℤ𝑀))
30 elfzuz3 12724 . . . 4 ((𝐺‘(♯‘𝐴)) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ‘(𝐺‘(♯‘𝐴))))
3128, 30syl 17 . . 3 (𝜑𝑁 ∈ (ℤ‘(𝐺‘(♯‘𝐴))))
32 fzss2 12766 . . . . . . 7 (𝑁 ∈ (ℤ‘(𝐺‘(♯‘𝐴))) → (𝑀...(𝐺‘(♯‘𝐴))) ⊆ (𝑀...𝑁))
3331, 32syl 17 . . . . . 6 (𝜑 → (𝑀...(𝐺‘(♯‘𝐴))) ⊆ (𝑀...𝑁))
3433sselda 3860 . . . . 5 ((𝜑𝑘 ∈ (𝑀...(𝐺‘(♯‘𝐴)))) → 𝑘 ∈ (𝑀...𝑁))
35 seqcoll2.6 . . . . 5 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)
3634, 35syldan 582 . . . 4 ((𝜑𝑘 ∈ (𝑀...(𝐺‘(♯‘𝐴)))) → (𝐹𝑘) ∈ 𝑆)
37 seqcoll2.c . . . 4 ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
3829, 36, 37seqcl 13208 . . 3 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) ∈ 𝑆)
39 peano2uz 12118 . . . . . . . 8 ((𝐺‘(♯‘𝐴)) ∈ (ℤ𝑀) → ((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ𝑀))
4029, 39syl 17 . . . . . . 7 (𝜑 → ((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ𝑀))
41 fzss1 12765 . . . . . . 7 (((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ𝑀) → (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
4240, 41syl 17 . . . . . 6 (𝜑 → (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
4342sselda 3860 . . . . 5 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
44 eluzelre 12072 . . . . . . . . 9 ((𝐺‘(♯‘𝐴)) ∈ (ℤ𝑀) → (𝐺‘(♯‘𝐴)) ∈ ℝ)
4529, 44syl 17 . . . . . . . 8 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ ℝ)
4645adantr 473 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) ∈ ℝ)
47 peano2re 10615 . . . . . . . 8 ((𝐺‘(♯‘𝐴)) ∈ ℝ → ((𝐺‘(♯‘𝐴)) + 1) ∈ ℝ)
4846, 47syl 17 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → ((𝐺‘(♯‘𝐴)) + 1) ∈ ℝ)
49 elfzelz 12727 . . . . . . . . 9 (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℤ)
5049zred 11903 . . . . . . . 8 (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℝ)
5150adantl 474 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ℝ)
5246ltp1d 11373 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) < ((𝐺‘(♯‘𝐴)) + 1))
53 elfzle1 12729 . . . . . . . 8 (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → ((𝐺‘(♯‘𝐴)) + 1) ≤ 𝑘)
5453adantl 474 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → ((𝐺‘(♯‘𝐴)) + 1) ≤ 𝑘)
5546, 48, 51, 52, 54ltletrd 10602 . . . . . 6 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) < 𝑘)
566adantr 473 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺:(1...(♯‘𝐴))–1-1-onto𝐴)
57 f1ocnv 6458 . . . . . . . . . . . . . 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 6446 . . . . . . . . . . . . 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 760 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝑘𝐴)
6260, 61ffvelrnd 6679 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ (1...(♯‘𝐴)))
63 elfzelz 12727 . . . . . . . . . . 11 ((𝐺𝑘) ∈ (1...(♯‘𝐴)) → (𝐺𝑘) ∈ ℤ)
6462, 63syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℤ)
6564zred 11903 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℝ)
6618adantr 473 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (♯‘𝐴) ∈ ℕ0)
6766nn0red 11771 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (♯‘𝐴) ∈ ℝ)
68 elfzle2 12730 . . . . . . . . . 10 ((𝐺𝑘) ∈ (1...(♯‘𝐴)) → (𝐺𝑘) ≤ (♯‘𝐴))
6962, 68syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ≤ (♯‘𝐴))
7065, 67, 69lensymd 10593 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ¬ (♯‘𝐴) < (𝐺𝑘))
714adantr 473 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))
7226adantr 473 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
73 isorel 6904 . . . . . . . . . 10 ((𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴) ∧ ((♯‘𝐴) ∈ (1...(♯‘𝐴)) ∧ (𝐺𝑘) ∈ (1...(♯‘𝐴)))) → ((♯‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(♯‘𝐴)) < (𝐺‘(𝐺𝑘))))
7471, 72, 62, 73syl12anc 824 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((♯‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(♯‘𝐴)) < (𝐺‘(𝐺𝑘))))
75 f1ocnvfv2 6861 . . . . . . . . . . 11 ((𝐺:(1...(♯‘𝐴))–1-1-onto𝐴𝑘𝐴) → (𝐺‘(𝐺𝑘)) = 𝑘)
7656, 61, 75syl2anc 576 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺‘(𝐺𝑘)) = 𝑘)
7776breq2d 4942 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((𝐺‘(♯‘𝐴)) < (𝐺‘(𝐺𝑘)) ↔ (𝐺‘(♯‘𝐴)) < 𝑘))
7874, 77bitrd 271 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((♯‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(♯‘𝐴)) < 𝑘))
7970, 78mtbid 316 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ¬ (𝐺‘(♯‘𝐴)) < 𝑘)
8079expr 449 . . . . . 6 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝑘𝐴 → ¬ (𝐺‘(♯‘𝐴)) < 𝑘))
8155, 80mt2d 134 . . . . 5 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → ¬ 𝑘𝐴)
8243, 81eldifd 3842 . . . 4 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
83 seqcoll2.7 . . . 4 ((𝜑𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)
8482, 83syldan 582 . . 3 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐹𝑘) = 𝑍)
851, 29, 31, 38, 84seqid2 13234 . 2 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) = (seq𝑀( + , 𝐹)‘𝑁))
86 seqcoll2.1 . . 3 ((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)
87 seqcoll2.a . . 3 (𝜑𝑍𝑆)
883, 2syl6ss 3872 . . 3 (𝜑𝐴 ⊆ (ℤ𝑀))
8933ssdifd 4009 . . . . 5 (𝜑 → ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴) ⊆ ((𝑀...𝑁) ∖ 𝐴))
9089sselda 3860 . . . 4 ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
9190, 83syldan 582 . . 3 ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)
92 seqcoll2.8 . . 3 ((𝜑𝑛 ∈ (1...(♯‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))
9386, 1, 37, 87, 4, 26, 88, 36, 91, 92seqcoll 13638 . 2 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) = (seq1( + , 𝐻)‘(♯‘𝐴)))
9485, 93eqtr3d 2816 1 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wo 833   = wceq 1507  wcel 2050  wne 2967  cdif 3828  wss 3831  c0 4180   class class class wbr 4930  ccnv 5407  wf 6186  1-1-ontowf1o 6189  cfv 6190   Isom wiso 6191  (class class class)co 6978  Fincfn 8308  cr 10336  0cc0 10337  1c1 10338   + caddc 10340   < clt 10476  cle 10477  cn 11441  0cn0 11710  cz 11796  cuz 12061  ...cfz 12711  seqcseq 13187  chash 13508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-cnex 10393  ax-resscn 10394  ax-1cn 10395  ax-icn 10396  ax-addcl 10397  ax-addrcl 10398  ax-mulcl 10399  ax-mulrcl 10400  ax-mulcom 10401  ax-addass 10402  ax-mulass 10403  ax-distr 10404  ax-i2m1 10405  ax-1ne0 10406  ax-1rid 10407  ax-rnegex 10408  ax-rrecex 10409  ax-cnre 10410  ax-pre-lttri 10411  ax-pre-lttrn 10412  ax-pre-ltadd 10413  ax-pre-mulgt0 10414
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-isom 6199  df-riota 6939  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-1st 7503  df-2nd 7504  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-1o 7907  df-er 8091  df-en 8309  df-dom 8310  df-sdom 8311  df-fin 8312  df-card 9164  df-pnf 10478  df-mnf 10479  df-xr 10480  df-ltxr 10481  df-le 10482  df-sub 10674  df-neg 10675  df-nn 11442  df-n0 11711  df-z 11797  df-uz 12062  df-fz 12712  df-seq 13188  df-hash 13509
This theorem is referenced by:  isercolllem3  14887  gsumval3  18784
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