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Theorem seqcoll2 13203
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 12340 . . . 4 (𝑀...𝑁) ⊆ (ℤ𝑀)
3 seqcoll2.5 . . . . 5 (𝜑𝐴 ⊆ (𝑀...𝑁))
4 seqcoll2.2 . . . . . . . 8 (𝜑𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
5 isof1o 6538 . . . . . . . 8 (𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) → 𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
64, 5syl 17 . . . . . . 7 (𝜑𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
7 f1of 6104 . . . . . . 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 12727 . . . . . . . . . . . . 13 (𝑀...𝑁) ∈ Fin
11 ssfi 8140 . . . . . . . . . . . . 13 (((𝑀...𝑁) ∈ Fin ∧ 𝐴 ⊆ (𝑀...𝑁)) → 𝐴 ∈ Fin)
1210, 3, 11sylancr 694 . . . . . . . . . . . 12 (𝜑𝐴 ∈ Fin)
13 hasheq0 13110 . . . . . . . . . . . 12 (𝐴 ∈ Fin → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))
1412, 13syl 17 . . . . . . . . . . 11 (𝜑 → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))
1514necon3bbid 2827 . . . . . . . . . 10 (𝜑 → (¬ (#‘𝐴) = 0 ↔ 𝐴 ≠ ∅))
169, 15mpbird 247 . . . . . . . . 9 (𝜑 → ¬ (#‘𝐴) = 0)
17 hashcl 13103 . . . . . . . . . . . 12 (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0)
1812, 17syl 17 . . . . . . . . . . 11 (𝜑 → (#‘𝐴) ∈ ℕ0)
19 elnn0 11254 . . . . . . . . . . 11 ((#‘𝐴) ∈ ℕ0 ↔ ((#‘𝐴) ∈ ℕ ∨ (#‘𝐴) = 0))
2018, 19sylib 208 . . . . . . . . . 10 (𝜑 → ((#‘𝐴) ∈ ℕ ∨ (#‘𝐴) = 0))
2120ord 392 . . . . . . . . 9 (𝜑 → (¬ (#‘𝐴) ∈ ℕ → (#‘𝐴) = 0))
2216, 21mt3d 140 . . . . . . . 8 (𝜑 → (#‘𝐴) ∈ ℕ)
23 nnuz 11683 . . . . . . . 8 ℕ = (ℤ‘1)
2422, 23syl6eleq 2708 . . . . . . 7 (𝜑 → (#‘𝐴) ∈ (ℤ‘1))
25 eluzfz2 12307 . . . . . . 7 ((#‘𝐴) ∈ (ℤ‘1) → (#‘𝐴) ∈ (1...(#‘𝐴)))
2624, 25syl 17 . . . . . 6 (𝜑 → (#‘𝐴) ∈ (1...(#‘𝐴)))
278, 26ffvelrnd 6326 . . . . 5 (𝜑 → (𝐺‘(#‘𝐴)) ∈ 𝐴)
283, 27sseldd 3589 . . . 4 (𝜑 → (𝐺‘(#‘𝐴)) ∈ (𝑀...𝑁))
292, 28sseldi 3586 . . 3 (𝜑 → (𝐺‘(#‘𝐴)) ∈ (ℤ𝑀))
30 elfzuz3 12297 . . . 4 ((𝐺‘(#‘𝐴)) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ‘(𝐺‘(#‘𝐴))))
3128, 30syl 17 . . 3 (𝜑𝑁 ∈ (ℤ‘(𝐺‘(#‘𝐴))))
32 fzss2 12339 . . . . . . 7 (𝑁 ∈ (ℤ‘(𝐺‘(#‘𝐴))) → (𝑀...(𝐺‘(#‘𝐴))) ⊆ (𝑀...𝑁))
3331, 32syl 17 . . . . . 6 (𝜑 → (𝑀...(𝐺‘(#‘𝐴))) ⊆ (𝑀...𝑁))
3433sselda 3588 . . . . 5 ((𝜑𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → 𝑘 ∈ (𝑀...𝑁))
35 seqcoll2.6 . . . . 5 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)
3634, 35syldan 487 . . . 4 ((𝜑𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → (𝐹𝑘) ∈ 𝑆)
37 seqcoll2.c . . . 4 ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)
3829, 36, 37seqcl 12777 . . 3 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(#‘𝐴))) ∈ 𝑆)
39 peano2uz 11701 . . . . . . . 8 ((𝐺‘(#‘𝐴)) ∈ (ℤ𝑀) → ((𝐺‘(#‘𝐴)) + 1) ∈ (ℤ𝑀))
4029, 39syl 17 . . . . . . 7 (𝜑 → ((𝐺‘(#‘𝐴)) + 1) ∈ (ℤ𝑀))
41 fzss1 12338 . . . . . . 7 (((𝐺‘(#‘𝐴)) + 1) ∈ (ℤ𝑀) → (((𝐺‘(#‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
4240, 41syl 17 . . . . . 6 (𝜑 → (((𝐺‘(#‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
4342sselda 3588 . . . . 5 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
44 eluzelre 11658 . . . . . . . . 9 ((𝐺‘(#‘𝐴)) ∈ (ℤ𝑀) → (𝐺‘(#‘𝐴)) ∈ ℝ)
4529, 44syl 17 . . . . . . . 8 (𝜑 → (𝐺‘(#‘𝐴)) ∈ ℝ)
4645adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐺‘(#‘𝐴)) ∈ ℝ)
47 peano2re 10169 . . . . . . . 8 ((𝐺‘(#‘𝐴)) ∈ ℝ → ((𝐺‘(#‘𝐴)) + 1) ∈ ℝ)
4846, 47syl 17 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → ((𝐺‘(#‘𝐴)) + 1) ∈ ℝ)
49 elfzelz 12300 . . . . . . . . 9 (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℤ)
5049zred 11442 . . . . . . . 8 (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℝ)
5150adantl 482 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ℝ)
5246ltp1d 10914 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐺‘(#‘𝐴)) < ((𝐺‘(#‘𝐴)) + 1))
53 elfzle1 12302 . . . . . . . 8 (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) → ((𝐺‘(#‘𝐴)) + 1) ≤ 𝑘)
5453adantl 482 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → ((𝐺‘(#‘𝐴)) + 1) ≤ 𝑘)
5546, 48, 51, 52, 54ltletrd 10157 . . . . . 6 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐺‘(#‘𝐴)) < 𝑘)
566adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
57 f1ocnv 6116 . . . . . . . . . . . . 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 6104 . . . . . . . . . . . 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 795 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝑘𝐴)
6260, 61ffvelrnd 6326 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ (1...(#‘𝐴)))
63 elfzle2 12303 . . . . . . . . . 10 ((𝐺𝑘) ∈ (1...(#‘𝐴)) → (𝐺𝑘) ≤ (#‘𝐴))
6462, 63syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ≤ (#‘𝐴))
65 elfzelz 12300 . . . . . . . . . . . 12 ((𝐺𝑘) ∈ (1...(#‘𝐴)) → (𝐺𝑘) ∈ ℤ)
6662, 65syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℤ)
6766zred 11442 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℝ)
6818adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (#‘𝐴) ∈ ℕ0)
6968nn0red 11312 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (#‘𝐴) ∈ ℝ)
7067, 69lenltd 10143 . . . . . . . . 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 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
7326adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (#‘𝐴) ∈ (1...(#‘𝐴)))
74 isorel 6541 . . . . . . . . . 10 ((𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) ∧ ((#‘𝐴) ∈ (1...(#‘𝐴)) ∧ (𝐺𝑘) ∈ (1...(#‘𝐴)))) → ((#‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(#‘𝐴)) < (𝐺‘(𝐺𝑘))))
7572, 73, 62, 74syl12anc 1321 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((#‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(#‘𝐴)) < (𝐺‘(𝐺𝑘))))
76 f1ocnvfv2 6498 . . . . . . . . . . 11 ((𝐺:(1...(#‘𝐴))–1-1-onto𝐴𝑘𝐴) → (𝐺‘(𝐺𝑘)) = 𝑘)
7756, 61, 76syl2anc 692 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺‘(𝐺𝑘)) = 𝑘)
7877breq2d 4635 . . . . . . . . 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 314 . . . . . . 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 3571 . . . 4 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
84 seqcoll2.7 . . . 4 ((𝜑𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)
8583, 84syldan 487 . . 3 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐹𝑘) = 𝑍)
861, 29, 31, 38, 85seqid2 12803 . 2 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(#‘𝐴))) = (seq𝑀( + , 𝐹)‘𝑁))
87 seqcoll2.1 . . 3 ((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)
88 seqcoll2.a . . 3 (𝜑𝑍𝑆)
893, 2syl6ss 3600 . . 3 (𝜑𝐴 ⊆ (ℤ𝑀))
9033ssdifd 3730 . . . . 5 (𝜑 → ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴) ⊆ ((𝑀...𝑁) ∖ 𝐴))
9190sselda 3588 . . . 4 ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴)) → 𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴))
9291, 84syldan 487 . . 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 13202 . 2 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(#‘𝐴))) = (seq1( + , 𝐻)‘(#‘𝐴)))
9586, 94eqtr3d 2657 1 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(#‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  cdif 3557  wss 3560  c0 3897   class class class wbr 4623  ccnv 5083  wf 5853  1-1-ontowf1o 5856  cfv 5857   Isom wiso 5858  (class class class)co 6615  Fincfn 7915  cr 9895  0cc0 9896  1c1 9897   + caddc 9899   < clt 10034  cle 10035  cn 10980  0cn0 11252  cz 11337  cuz 11647  ...cfz 12284  seqcseq 12757  #chash 13073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-seq 12758  df-hash 13074
This theorem is referenced by:  isercolllem3  14347  gsumval3  18248
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