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Theorem seqcoll2 13055
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 12205 . . . 4 (𝑀...𝑁) ⊆ (ℤ𝑀)
3 seqcoll2.5 . . . . 5 (𝜑𝐴 ⊆ (𝑀...𝑁))
4 seqcoll2.2 . . . . . . . 8 (𝜑𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
5 isof1o 6448 . . . . . . . 8 (𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) → 𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
64, 5syl 17 . . . . . . 7 (𝜑𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
7 f1of 6032 . . . . . . 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 12585 . . . . . . . . . . . . 13 (𝑀...𝑁) ∈ Fin
11 ssfi 8039 . . . . . . . . . . . . 13 (((𝑀...𝑁) ∈ Fin ∧ 𝐴 ⊆ (𝑀...𝑁)) → 𝐴 ∈ Fin)
1210, 3, 11sylancr 693 . . . . . . . . . . . 12 (𝜑𝐴 ∈ Fin)
13 hasheq0 12964 . . . . . . . . . . . 12 (𝐴 ∈ Fin → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))
1412, 13syl 17 . . . . . . . . . . 11 (𝜑 → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))
1514necon3bbid 2815 . . . . . . . . . 10 (𝜑 → (¬ (#‘𝐴) = 0 ↔ 𝐴 ≠ ∅))
169, 15mpbird 245 . . . . . . . . 9 (𝜑 → ¬ (#‘𝐴) = 0)
17 hashcl 12958 . . . . . . . . . . . 12 (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0)
1812, 17syl 17 . . . . . . . . . . 11 (𝜑 → (#‘𝐴) ∈ ℕ0)
19 elnn0 11138 . . . . . . . . . . 11 ((#‘𝐴) ∈ ℕ0 ↔ ((#‘𝐴) ∈ ℕ ∨ (#‘𝐴) = 0))
2018, 19sylib 206 . . . . . . . . . 10 (𝜑 → ((#‘𝐴) ∈ ℕ ∨ (#‘𝐴) = 0))
2120ord 390 . . . . . . . . 9 (𝜑 → (¬ (#‘𝐴) ∈ ℕ → (#‘𝐴) = 0))
2216, 21mt3d 138 . . . . . . . 8 (𝜑 → (#‘𝐴) ∈ ℕ)
23 nnuz 11552 . . . . . . . 8 ℕ = (ℤ‘1)
2422, 23syl6eleq 2694 . . . . . . 7 (𝜑 → (#‘𝐴) ∈ (ℤ‘1))
25 eluzfz2 12172 . . . . . . 7 ((#‘𝐴) ∈ (ℤ‘1) → (#‘𝐴) ∈ (1...(#‘𝐴)))
2624, 25syl 17 . . . . . 6 (𝜑 → (#‘𝐴) ∈ (1...(#‘𝐴)))
278, 26ffvelrnd 6250 . . . . 5 (𝜑 → (𝐺‘(#‘𝐴)) ∈ 𝐴)
283, 27sseldd 3565 . . . 4 (𝜑 → (𝐺‘(#‘𝐴)) ∈ (𝑀...𝑁))
292, 28sseldi 3562 . . 3 (𝜑 → (𝐺‘(#‘𝐴)) ∈ (ℤ𝑀))
30 elfzuz3 12162 . . . 4 ((𝐺‘(#‘𝐴)) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ‘(𝐺‘(#‘𝐴))))
3128, 30syl 17 . . 3 (𝜑𝑁 ∈ (ℤ‘(𝐺‘(#‘𝐴))))
32 fzss2 12204 . . . . . . 7 (𝑁 ∈ (ℤ‘(𝐺‘(#‘𝐴))) → (𝑀...(𝐺‘(#‘𝐴))) ⊆ (𝑀...𝑁))
3331, 32syl 17 . . . . . 6 (𝜑 → (𝑀...(𝐺‘(#‘𝐴))) ⊆ (𝑀...𝑁))
3433sselda 3564 . . . . 5 ((𝜑𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → 𝑘 ∈ (𝑀...𝑁))
35 seqcoll2.6 . . . . 5 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)
3634, 35syldan 485 . . . 4 ((𝜑𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → (𝐹𝑘) ∈ 𝑆)
37 seqcoll2.c . . . 4 ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
3829, 36, 37seqcl 12635 . . 3 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(#‘𝐴))) ∈ 𝑆)
39 peano2uz 11570 . . . . . . . 8 ((𝐺‘(#‘𝐴)) ∈ (ℤ𝑀) → ((𝐺‘(#‘𝐴)) + 1) ∈ (ℤ𝑀))
4029, 39syl 17 . . . . . . 7 (𝜑 → ((𝐺‘(#‘𝐴)) + 1) ∈ (ℤ𝑀))
41 fzss1 12203 . . . . . . 7 (((𝐺‘(#‘𝐴)) + 1) ∈ (ℤ𝑀) → (((𝐺‘(#‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
4240, 41syl 17 . . . . . 6 (𝜑 → (((𝐺‘(#‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
4342sselda 3564 . . . . 5 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
44 eluzelre 11527 . . . . . . . . 9 ((𝐺‘(#‘𝐴)) ∈ (ℤ𝑀) → (𝐺‘(#‘𝐴)) ∈ ℝ)
4529, 44syl 17 . . . . . . . 8 (𝜑 → (𝐺‘(#‘𝐴)) ∈ ℝ)
4645adantr 479 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐺‘(#‘𝐴)) ∈ ℝ)
47 peano2re 10057 . . . . . . . 8 ((𝐺‘(#‘𝐴)) ∈ ℝ → ((𝐺‘(#‘𝐴)) + 1) ∈ ℝ)
4846, 47syl 17 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → ((𝐺‘(#‘𝐴)) + 1) ∈ ℝ)
49 elfzelz 12165 . . . . . . . . 9 (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℤ)
5049zred 11311 . . . . . . . 8 (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℝ)
5150adantl 480 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ℝ)
5246ltp1d 10800 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐺‘(#‘𝐴)) < ((𝐺‘(#‘𝐴)) + 1))
53 elfzle1 12167 . . . . . . . 8 (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) → ((𝐺‘(#‘𝐴)) + 1) ≤ 𝑘)
5453adantl 480 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → ((𝐺‘(#‘𝐴)) + 1) ≤ 𝑘)
5546, 48, 51, 52, 54ltletrd 10045 . . . . . 6 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐺‘(#‘𝐴)) < 𝑘)
566adantr 479 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
57 f1ocnv 6044 . . . . . . . . . . . . 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 6032 . . . . . . . . . . . 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 791 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝑘𝐴)
6260, 61ffvelrnd 6250 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ (1...(#‘𝐴)))
63 elfzle2 12168 . . . . . . . . . 10 ((𝐺𝑘) ∈ (1...(#‘𝐴)) → (𝐺𝑘) ≤ (#‘𝐴))
6462, 63syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ≤ (#‘𝐴))
65 elfzelz 12165 . . . . . . . . . . . 12 ((𝐺𝑘) ∈ (1...(#‘𝐴)) → (𝐺𝑘) ∈ ℤ)
6662, 65syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℤ)
6766zred 11311 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℝ)
6818adantr 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (#‘𝐴) ∈ ℕ0)
6968nn0red 11196 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (#‘𝐴) ∈ ℝ)
7067, 69lenltd 10031 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((𝐺𝑘) ≤ (#‘𝐴) ↔ ¬ (#‘𝐴) < (𝐺𝑘)))
7164, 70mpbid 220 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ¬ (#‘𝐴) < (𝐺𝑘))
724adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
7326adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (#‘𝐴) ∈ (1...(#‘𝐴)))
74 isorel 6451 . . . . . . . . . 10 ((𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) ∧ ((#‘𝐴) ∈ (1...(#‘𝐴)) ∧ (𝐺𝑘) ∈ (1...(#‘𝐴)))) → ((#‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(#‘𝐴)) < (𝐺‘(𝐺𝑘))))
7572, 73, 62, 74syl12anc 1315 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((#‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(#‘𝐴)) < (𝐺‘(𝐺𝑘))))
76 f1ocnvfv2 6408 . . . . . . . . . . 11 ((𝐺:(1...(#‘𝐴))–1-1-onto𝐴𝑘𝐴) → (𝐺‘(𝐺𝑘)) = 𝑘)
7756, 61, 76syl2anc 690 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺‘(𝐺𝑘)) = 𝑘)
7877breq2d 4586 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((𝐺‘(#‘𝐴)) < (𝐺‘(𝐺𝑘)) ↔ (𝐺‘(#‘𝐴)) < 𝑘))
7975, 78bitrd 266 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((#‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(#‘𝐴)) < 𝑘))
8071, 79mtbid 312 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ¬ (𝐺‘(#‘𝐴)) < 𝑘)
8180expr 640 . . . . . 6 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝑘𝐴 → ¬ (𝐺‘(#‘𝐴)) < 𝑘))
8255, 81mt2d 129 . . . . 5 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → ¬ 𝑘𝐴)
8343, 82eldifd 3547 . . . 4 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
84 seqcoll2.7 . . . 4 ((𝜑𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)
8583, 84syldan 485 . . 3 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐹𝑘) = 𝑍)
861, 29, 31, 38, 85seqid2 12661 . 2 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(#‘𝐴))) = (seq𝑀( + , 𝐹)‘𝑁))
87 seqcoll2.1 . . 3 ((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)
88 seqcoll2.a . . 3 (𝜑𝑍𝑆)
893, 2syl6ss 3576 . . 3 (𝜑𝐴 ⊆ (ℤ𝑀))
9033ssdifd 3704 . . . . 5 (𝜑 → ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴) ⊆ ((𝑀...𝑁) ∖ 𝐴))
9190sselda 3564 . . . 4 ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
9291, 84syldan 485 . . 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 13054 . 2 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(#‘𝐴))) = (seq1( + , 𝐻)‘(#‘𝐴)))
9586, 94eqtr3d 2642 1 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(#‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2776  cdif 3533  wss 3536  c0 3870   class class class wbr 4574  ccnv 5024  wf 5783  1-1-ontowf1o 5786  cfv 5787   Isom wiso 5788  (class class class)co 6524  Fincfn 7815  cr 9788  0cc0 9789  1c1 9790   + caddc 9792   < clt 9927  cle 9928  cn 10864  0cn0 11136  cz 11207  cuz 11516  ...cfz 12149  seqcseq 12615  #chash 12931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-isom 5796  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-card 8622  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-n0 11137  df-z 11208  df-uz 11517  df-fz 12150  df-seq 12616  df-hash 12932
This theorem is referenced by:  isercolllem3  14188  gsumval3  18074
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