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Theorem seqcoll2 13287
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 12420 . . . 4 (𝑀...𝑁) ⊆ (ℤ𝑀)
3 seqcoll2.5 . . . . 5 (𝜑𝐴 ⊆ (𝑀...𝑁))
4 seqcoll2.2 . . . . . . . 8 (𝜑𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
5 isof1o 6613 . . . . . . . 8 (𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) → 𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
64, 5syl 17 . . . . . . 7 (𝜑𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
7 f1of 6175 . . . . . . 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 12811 . . . . . . . . . . . . 13 (𝑀...𝑁) ∈ Fin
11 ssfi 8221 . . . . . . . . . . . . 13 (((𝑀...𝑁) ∈ Fin ∧ 𝐴 ⊆ (𝑀...𝑁)) → 𝐴 ∈ Fin)
1210, 3, 11sylancr 696 . . . . . . . . . . . 12 (𝜑𝐴 ∈ Fin)
13 hasheq0 13192 . . . . . . . . . . . 12 (𝐴 ∈ Fin → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))
1412, 13syl 17 . . . . . . . . . . 11 (𝜑 → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))
1514necon3bbid 2860 . . . . . . . . . 10 (𝜑 → (¬ (#‘𝐴) = 0 ↔ 𝐴 ≠ ∅))
169, 15mpbird 247 . . . . . . . . 9 (𝜑 → ¬ (#‘𝐴) = 0)
17 hashcl 13185 . . . . . . . . . . . 12 (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0)
1812, 17syl 17 . . . . . . . . . . 11 (𝜑 → (#‘𝐴) ∈ ℕ0)
19 elnn0 11332 . . . . . . . . . . 11 ((#‘𝐴) ∈ ℕ0 ↔ ((#‘𝐴) ∈ ℕ ∨ (#‘𝐴) = 0))
2018, 19sylib 208 . . . . . . . . . 10 (𝜑 → ((#‘𝐴) ∈ ℕ ∨ (#‘𝐴) = 0))
2120ord 391 . . . . . . . . 9 (𝜑 → (¬ (#‘𝐴) ∈ ℕ → (#‘𝐴) = 0))
2216, 21mt3d 140 . . . . . . . 8 (𝜑 → (#‘𝐴) ∈ ℕ)
23 nnuz 11761 . . . . . . . 8 ℕ = (ℤ‘1)
2422, 23syl6eleq 2740 . . . . . . 7 (𝜑 → (#‘𝐴) ∈ (ℤ‘1))
25 eluzfz2 12387 . . . . . . 7 ((#‘𝐴) ∈ (ℤ‘1) → (#‘𝐴) ∈ (1...(#‘𝐴)))
2624, 25syl 17 . . . . . 6 (𝜑 → (#‘𝐴) ∈ (1...(#‘𝐴)))
278, 26ffvelrnd 6400 . . . . 5 (𝜑 → (𝐺‘(#‘𝐴)) ∈ 𝐴)
283, 27sseldd 3637 . . . 4 (𝜑 → (𝐺‘(#‘𝐴)) ∈ (𝑀...𝑁))
292, 28sseldi 3634 . . 3 (𝜑 → (𝐺‘(#‘𝐴)) ∈ (ℤ𝑀))
30 elfzuz3 12377 . . . 4 ((𝐺‘(#‘𝐴)) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ‘(𝐺‘(#‘𝐴))))
3128, 30syl 17 . . 3 (𝜑𝑁 ∈ (ℤ‘(𝐺‘(#‘𝐴))))
32 fzss2 12419 . . . . . . 7 (𝑁 ∈ (ℤ‘(𝐺‘(#‘𝐴))) → (𝑀...(𝐺‘(#‘𝐴))) ⊆ (𝑀...𝑁))
3331, 32syl 17 . . . . . 6 (𝜑 → (𝑀...(𝐺‘(#‘𝐴))) ⊆ (𝑀...𝑁))
3433sselda 3636 . . . . 5 ((𝜑𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → 𝑘 ∈ (𝑀...𝑁))
35 seqcoll2.6 . . . . 5 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)
3634, 35syldan 486 . . . 4 ((𝜑𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → (𝐹𝑘) ∈ 𝑆)
37 seqcoll2.c . . . 4 ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
3829, 36, 37seqcl 12861 . . 3 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(#‘𝐴))) ∈ 𝑆)
39 peano2uz 11779 . . . . . . . 8 ((𝐺‘(#‘𝐴)) ∈ (ℤ𝑀) → ((𝐺‘(#‘𝐴)) + 1) ∈ (ℤ𝑀))
4029, 39syl 17 . . . . . . 7 (𝜑 → ((𝐺‘(#‘𝐴)) + 1) ∈ (ℤ𝑀))
41 fzss1 12418 . . . . . . 7 (((𝐺‘(#‘𝐴)) + 1) ∈ (ℤ𝑀) → (((𝐺‘(#‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
4240, 41syl 17 . . . . . 6 (𝜑 → (((𝐺‘(#‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
4342sselda 3636 . . . . 5 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
44 eluzelre 11736 . . . . . . . . 9 ((𝐺‘(#‘𝐴)) ∈ (ℤ𝑀) → (𝐺‘(#‘𝐴)) ∈ ℝ)
4529, 44syl 17 . . . . . . . 8 (𝜑 → (𝐺‘(#‘𝐴)) ∈ ℝ)
4645adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐺‘(#‘𝐴)) ∈ ℝ)
47 peano2re 10247 . . . . . . . 8 ((𝐺‘(#‘𝐴)) ∈ ℝ → ((𝐺‘(#‘𝐴)) + 1) ∈ ℝ)
4846, 47syl 17 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → ((𝐺‘(#‘𝐴)) + 1) ∈ ℝ)
49 elfzelz 12380 . . . . . . . . 9 (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℤ)
5049zred 11520 . . . . . . . 8 (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℝ)
5150adantl 481 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ℝ)
5246ltp1d 10992 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐺‘(#‘𝐴)) < ((𝐺‘(#‘𝐴)) + 1))
53 elfzle1 12382 . . . . . . . 8 (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) → ((𝐺‘(#‘𝐴)) + 1) ≤ 𝑘)
5453adantl 481 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → ((𝐺‘(#‘𝐴)) + 1) ≤ 𝑘)
5546, 48, 51, 52, 54ltletrd 10235 . . . . . 6 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐺‘(#‘𝐴)) < 𝑘)
566adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
57 f1ocnv 6187 . . . . . . . . . . . . 13 (𝐺:(1...(#‘𝐴))–1-1-onto𝐴𝐺:𝐴1-1-onto→(1...(#‘𝐴)))
5856, 57syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺:𝐴1-1-onto→(1...(#‘𝐴)))
59 f1of 6175 . . . . . . . . . . . 12 (𝐺:𝐴1-1-onto→(1...(#‘𝐴)) → 𝐺:𝐴⟶(1...(#‘𝐴)))
6058, 59syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺:𝐴⟶(1...(#‘𝐴)))
61 simprr 811 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝑘𝐴)
6260, 61ffvelrnd 6400 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ (1...(#‘𝐴)))
63 elfzle2 12383 . . . . . . . . . 10 ((𝐺𝑘) ∈ (1...(#‘𝐴)) → (𝐺𝑘) ≤ (#‘𝐴))
6462, 63syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ≤ (#‘𝐴))
65 elfzelz 12380 . . . . . . . . . . . 12 ((𝐺𝑘) ∈ (1...(#‘𝐴)) → (𝐺𝑘) ∈ ℤ)
6662, 65syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℤ)
6766zred 11520 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℝ)
6818adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (#‘𝐴) ∈ ℕ0)
6968nn0red 11390 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (#‘𝐴) ∈ ℝ)
7067, 69lenltd 10221 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((𝐺𝑘) ≤ (#‘𝐴) ↔ ¬ (#‘𝐴) < (𝐺𝑘)))
7164, 70mpbid 222 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ¬ (#‘𝐴) < (𝐺𝑘))
724adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
7326adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (#‘𝐴) ∈ (1...(#‘𝐴)))
74 isorel 6616 . . . . . . . . . 10 ((𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) ∧ ((#‘𝐴) ∈ (1...(#‘𝐴)) ∧ (𝐺𝑘) ∈ (1...(#‘𝐴)))) → ((#‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(#‘𝐴)) < (𝐺‘(𝐺𝑘))))
7572, 73, 62, 74syl12anc 1364 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((#‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(#‘𝐴)) < (𝐺‘(𝐺𝑘))))
76 f1ocnvfv2 6573 . . . . . . . . . . 11 ((𝐺:(1...(#‘𝐴))–1-1-onto𝐴𝑘𝐴) → (𝐺‘(𝐺𝑘)) = 𝑘)
7756, 61, 76syl2anc 694 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺‘(𝐺𝑘)) = 𝑘)
7877breq2d 4697 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((𝐺‘(#‘𝐴)) < (𝐺‘(𝐺𝑘)) ↔ (𝐺‘(#‘𝐴)) < 𝑘))
7975, 78bitrd 268 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((#‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(#‘𝐴)) < 𝑘))
8071, 79mtbid 313 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ¬ (𝐺‘(#‘𝐴)) < 𝑘)
8180expr 642 . . . . . 6 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝑘𝐴 → ¬ (𝐺‘(#‘𝐴)) < 𝑘))
8255, 81mt2d 131 . . . . 5 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → ¬ 𝑘𝐴)
8343, 82eldifd 3618 . . . 4 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
84 seqcoll2.7 . . . 4 ((𝜑𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)
8583, 84syldan 486 . . 3 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐹𝑘) = 𝑍)
861, 29, 31, 38, 85seqid2 12887 . 2 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(#‘𝐴))) = (seq𝑀( + , 𝐹)‘𝑁))
87 seqcoll2.1 . . 3 ((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)
88 seqcoll2.a . . 3 (𝜑𝑍𝑆)
893, 2syl6ss 3648 . . 3 (𝜑𝐴 ⊆ (ℤ𝑀))
9033ssdifd 3779 . . . . 5 (𝜑 → ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴) ⊆ ((𝑀...𝑁) ∖ 𝐴))
9190sselda 3636 . . . 4 ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
9291, 84syldan 486 . . 3 ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)
93 seqcoll2.8 . . 3 ((𝜑𝑛 ∈ (1...(#‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))
9487, 1, 37, 88, 4, 26, 89, 36, 92, 93seqcoll 13286 . 2 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(#‘𝐴))) = (seq1( + , 𝐻)‘(#‘𝐴)))
9586, 94eqtr3d 2687 1 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(#‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  cdif 3604  wss 3607  c0 3948   class class class wbr 4685  ccnv 5142  wf 5922  1-1-ontowf1o 5925  cfv 5926   Isom wiso 5927  (class class class)co 6690  Fincfn 7997  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cle 10113  cn 11058  0cn0 11330  cz 11415  cuz 11725  ...cfz 12364  seqcseq 12841  #chash 13157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-seq 12842  df-hash 13158
This theorem is referenced by:  isercolllem3  14441  gsumval3  18354
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