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Theorem seqcoll2 13823
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 12947 . . . 4 (𝑀...𝑁) ⊆ (ℤ𝑀)
3 seqcoll2.5 . . . . 5 (𝜑𝐴 ⊆ (𝑀...𝑁))
4 seqcoll2.2 . . . . . . . 8 (𝜑𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))
5 isof1o 7059 . . . . . . . 8 (𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴) → 𝐺:(1...(♯‘𝐴))–1-1-onto𝐴)
64, 5syl 17 . . . . . . 7 (𝜑𝐺:(1...(♯‘𝐴))–1-1-onto𝐴)
7 f1of 6594 . . . . . . 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 13339 . . . . . . . . . . . . 13 (𝑀...𝑁) ∈ Fin
11 ssfi 8726 . . . . . . . . . . . . 13 (((𝑀...𝑁) ∈ Fin ∧ 𝐴 ⊆ (𝑀...𝑁)) → 𝐴 ∈ Fin)
1210, 3, 11sylancr 590 . . . . . . . . . . . 12 (𝜑𝐴 ∈ Fin)
13 hasheq0 13724 . . . . . . . . . . . 12 (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
1412, 13syl 17 . . . . . . . . . . 11 (𝜑 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
1514necon3bbid 3027 . . . . . . . . . 10 (𝜑 → (¬ (♯‘𝐴) = 0 ↔ 𝐴 ≠ ∅))
169, 15mpbird 260 . . . . . . . . 9 (𝜑 → ¬ (♯‘𝐴) = 0)
17 hashcl 13717 . . . . . . . . . . . 12 (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
1812, 17syl 17 . . . . . . . . . . 11 (𝜑 → (♯‘𝐴) ∈ ℕ0)
19 elnn0 11891 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 ↔ ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0))
2018, 19sylib 221 . . . . . . . . . 10 (𝜑 → ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0))
2120ord 861 . . . . . . . . 9 (𝜑 → (¬ (♯‘𝐴) ∈ ℕ → (♯‘𝐴) = 0))
2216, 21mt3d 150 . . . . . . . 8 (𝜑 → (♯‘𝐴) ∈ ℕ)
23 nnuz 12273 . . . . . . . 8 ℕ = (ℤ‘1)
2422, 23eleqtrdi 2903 . . . . . . 7 (𝜑 → (♯‘𝐴) ∈ (ℤ‘1))
25 eluzfz2 12914 . . . . . . 7 ((♯‘𝐴) ∈ (ℤ‘1) → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
2624, 25syl 17 . . . . . 6 (𝜑 → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
278, 26ffvelrnd 6833 . . . . 5 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ 𝐴)
283, 27sseldd 3919 . . . 4 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (𝑀...𝑁))
292, 28sseldi 3916 . . 3 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ (ℤ𝑀))
30 elfzuz3 12903 . . . 4 ((𝐺‘(♯‘𝐴)) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ‘(𝐺‘(♯‘𝐴))))
3128, 30syl 17 . . 3 (𝜑𝑁 ∈ (ℤ‘(𝐺‘(♯‘𝐴))))
32 fzss2 12946 . . . . . . 7 (𝑁 ∈ (ℤ‘(𝐺‘(♯‘𝐴))) → (𝑀...(𝐺‘(♯‘𝐴))) ⊆ (𝑀...𝑁))
3331, 32syl 17 . . . . . 6 (𝜑 → (𝑀...(𝐺‘(♯‘𝐴))) ⊆ (𝑀...𝑁))
3433sselda 3918 . . . . 5 ((𝜑𝑘 ∈ (𝑀...(𝐺‘(♯‘𝐴)))) → 𝑘 ∈ (𝑀...𝑁))
35 seqcoll2.6 . . . . 5 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)
3634, 35syldan 594 . . . 4 ((𝜑𝑘 ∈ (𝑀...(𝐺‘(♯‘𝐴)))) → (𝐹𝑘) ∈ 𝑆)
37 seqcoll2.c . . . 4 ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
3829, 36, 37seqcl 13390 . . 3 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) ∈ 𝑆)
39 peano2uz 12293 . . . . . . . 8 ((𝐺‘(♯‘𝐴)) ∈ (ℤ𝑀) → ((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ𝑀))
4029, 39syl 17 . . . . . . 7 (𝜑 → ((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ𝑀))
41 fzss1 12945 . . . . . . 7 (((𝐺‘(♯‘𝐴)) + 1) ∈ (ℤ𝑀) → (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
4240, 41syl 17 . . . . . 6 (𝜑 → (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
4342sselda 3918 . . . . 5 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
44 eluzelre 12246 . . . . . . . . 9 ((𝐺‘(♯‘𝐴)) ∈ (ℤ𝑀) → (𝐺‘(♯‘𝐴)) ∈ ℝ)
4529, 44syl 17 . . . . . . . 8 (𝜑 → (𝐺‘(♯‘𝐴)) ∈ ℝ)
4645adantr 484 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) ∈ ℝ)
47 peano2re 10806 . . . . . . . 8 ((𝐺‘(♯‘𝐴)) ∈ ℝ → ((𝐺‘(♯‘𝐴)) + 1) ∈ ℝ)
4846, 47syl 17 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → ((𝐺‘(♯‘𝐴)) + 1) ∈ ℝ)
49 elfzelz 12906 . . . . . . . . 9 (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℤ)
5049zred 12079 . . . . . . . 8 (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℝ)
5150adantl 485 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ℝ)
5246ltp1d 11563 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) < ((𝐺‘(♯‘𝐴)) + 1))
53 elfzle1 12909 . . . . . . . 8 (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) → ((𝐺‘(♯‘𝐴)) + 1) ≤ 𝑘)
5453adantl 485 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → ((𝐺‘(♯‘𝐴)) + 1) ≤ 𝑘)
5546, 48, 51, 52, 54ltletrd 10793 . . . . . 6 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐺‘(♯‘𝐴)) < 𝑘)
566adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺:(1...(♯‘𝐴))–1-1-onto𝐴)
57 f1ocnv 6606 . . . . . . . . . . . . . 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 6594 . . . . . . . . . . . . 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, 61ffvelrnd 6833 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ (1...(♯‘𝐴)))
63 elfzelz 12906 . . . . . . . . . . 11 ((𝐺𝑘) ∈ (1...(♯‘𝐴)) → (𝐺𝑘) ∈ ℤ)
6462, 63syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℤ)
6564zred 12079 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℝ)
6618adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (♯‘𝐴) ∈ ℕ0)
6766nn0red 11948 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (♯‘𝐴) ∈ ℝ)
68 elfzle2 12910 . . . . . . . . . 10 ((𝐺𝑘) ∈ (1...(♯‘𝐴)) → (𝐺𝑘) ≤ (♯‘𝐴))
6962, 68syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ≤ (♯‘𝐴))
7065, 67, 69lensymd 10784 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ¬ (♯‘𝐴) < (𝐺𝑘))
714adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))
7226adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (♯‘𝐴) ∈ (1...(♯‘𝐴)))
73 isorel 7062 . . . . . . . . . 10 ((𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴) ∧ ((♯‘𝐴) ∈ (1...(♯‘𝐴)) ∧ (𝐺𝑘) ∈ (1...(♯‘𝐴)))) → ((♯‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(♯‘𝐴)) < (𝐺‘(𝐺𝑘))))
7471, 72, 62, 73syl12anc 835 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((♯‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(♯‘𝐴)) < (𝐺‘(𝐺𝑘))))
75 f1ocnvfv2 7016 . . . . . . . . . . 11 ((𝐺:(1...(♯‘𝐴))–1-1-onto𝐴𝑘𝐴) → (𝐺‘(𝐺𝑘)) = 𝑘)
7656, 61, 75syl2anc 587 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺‘(𝐺𝑘)) = 𝑘)
7776breq2d 5045 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((𝐺‘(♯‘𝐴)) < (𝐺‘(𝐺𝑘)) ↔ (𝐺‘(♯‘𝐴)) < 𝑘))
7874, 77bitrd 282 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((♯‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(♯‘𝐴)) < 𝑘))
7970, 78mtbid 327 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ¬ (𝐺‘(♯‘𝐴)) < 𝑘)
8079expr 460 . . . . . 6 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝑘𝐴 → ¬ (𝐺‘(♯‘𝐴)) < 𝑘))
8155, 80mt2d 138 . . . . 5 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → ¬ 𝑘𝐴)
8243, 81eldifd 3895 . . . 4 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
83 seqcoll2.7 . . . 4 ((𝜑𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)
8482, 83syldan 594 . . 3 ((𝜑𝑘 ∈ (((𝐺‘(♯‘𝐴)) + 1)...𝑁)) → (𝐹𝑘) = 𝑍)
851, 29, 31, 38, 84seqid2 13416 . 2 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) = (seq𝑀( + , 𝐹)‘𝑁))
86 seqcoll2.1 . . 3 ((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)
87 seqcoll2.a . . 3 (𝜑𝑍𝑆)
883, 2sstrdi 3930 . . 3 (𝜑𝐴 ⊆ (ℤ𝑀))
8933ssdifd 4071 . . . . 5 (𝜑 → ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴) ⊆ ((𝑀...𝑁) ∖ 𝐴))
9089sselda 3918 . . . 4 ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
9190, 83syldan 594 . . 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 13822 . 2 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(♯‘𝐴))) = (seq1( + , 𝐻)‘(♯‘𝐴)))
9485, 93eqtr3d 2838 1 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2112  wne 2990  cdif 3881  wss 3884  c0 4246   class class class wbr 5033  ccnv 5522  wf 6324  1-1-ontowf1o 6327  cfv 6328   Isom wiso 6329  (class class class)co 7139  Fincfn 8496  cr 10529  0cc0 10530  1c1 10531   + caddc 10533   < clt 10668  cle 10669  cn 11629  0cn0 11889  cz 11973  cuz 12235  ...cfz 12889  seqcseq 13368  chash 13690
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-seq 13369  df-hash 13691
This theorem is referenced by:  isercolllem3  15019  gsumval3  19024
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