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Theorem seqcoll2 13106
 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 12253 . . . 4 (𝑀...𝑁) ⊆ (ℤ𝑀)
3 seqcoll2.5 . . . . 5 (𝜑𝐴 ⊆ (𝑀...𝑁))
4 seqcoll2.2 . . . . . . . 8 (𝜑𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))
5 isof1o 6473 . . . . . . . 8 (𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) → 𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
64, 5syl 17 . . . . . . 7 (𝜑𝐺:(1...(#‘𝐴))–1-1-onto𝐴)
7 f1of 6050 . . . . . . 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 12633 . . . . . . . . . . . . 13 (𝑀...𝑁) ∈ Fin
11 ssfi 8065 . . . . . . . . . . . . 13 (((𝑀...𝑁) ∈ Fin ∧ 𝐴 ⊆ (𝑀...𝑁)) → 𝐴 ∈ Fin)
1210, 3, 11sylancr 694 . . . . . . . . . . . 12 (𝜑𝐴 ∈ Fin)
13 hasheq0 13015 . . . . . . . . . . . 12 (𝐴 ∈ Fin → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))
1412, 13syl 17 . . . . . . . . . . 11 (𝜑 → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))
1514necon3bbid 2819 . . . . . . . . . 10 (𝜑 → (¬ (#‘𝐴) = 0 ↔ 𝐴 ≠ ∅))
169, 15mpbird 246 . . . . . . . . 9 (𝜑 → ¬ (#‘𝐴) = 0)
17 hashcl 13009 . . . . . . . . . . . 12 (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0)
1812, 17syl 17 . . . . . . . . . . 11 (𝜑 → (#‘𝐴) ∈ ℕ0)
19 elnn0 11171 . . . . . . . . . . 11 ((#‘𝐴) ∈ ℕ0 ↔ ((#‘𝐴) ∈ ℕ ∨ (#‘𝐴) = 0))
2018, 19sylib 207 . . . . . . . . . 10 (𝜑 → ((#‘𝐴) ∈ ℕ ∨ (#‘𝐴) = 0))
2120ord 391 . . . . . . . . 9 (𝜑 → (¬ (#‘𝐴) ∈ ℕ → (#‘𝐴) = 0))
2216, 21mt3d 139 . . . . . . . 8 (𝜑 → (#‘𝐴) ∈ ℕ)
23 nnuz 11599 . . . . . . . 8 ℕ = (ℤ‘1)
2422, 23syl6eleq 2698 . . . . . . 7 (𝜑 → (#‘𝐴) ∈ (ℤ‘1))
25 eluzfz2 12220 . . . . . . 7 ((#‘𝐴) ∈ (ℤ‘1) → (#‘𝐴) ∈ (1...(#‘𝐴)))
2624, 25syl 17 . . . . . 6 (𝜑 → (#‘𝐴) ∈ (1...(#‘𝐴)))
278, 26ffvelrnd 6268 . . . . 5 (𝜑 → (𝐺‘(#‘𝐴)) ∈ 𝐴)
283, 27sseldd 3569 . . . 4 (𝜑 → (𝐺‘(#‘𝐴)) ∈ (𝑀...𝑁))
292, 28sseldi 3566 . . 3 (𝜑 → (𝐺‘(#‘𝐴)) ∈ (ℤ𝑀))
30 elfzuz3 12210 . . . 4 ((𝐺‘(#‘𝐴)) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ‘(𝐺‘(#‘𝐴))))
3128, 30syl 17 . . 3 (𝜑𝑁 ∈ (ℤ‘(𝐺‘(#‘𝐴))))
32 fzss2 12252 . . . . . . 7 (𝑁 ∈ (ℤ‘(𝐺‘(#‘𝐴))) → (𝑀...(𝐺‘(#‘𝐴))) ⊆ (𝑀...𝑁))
3331, 32syl 17 . . . . . 6 (𝜑 → (𝑀...(𝐺‘(#‘𝐴))) ⊆ (𝑀...𝑁))
3433sselda 3568 . . . . 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 12683 . . 3 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(#‘𝐴))) ∈ 𝑆)
39 peano2uz 11617 . . . . . . . 8 ((𝐺‘(#‘𝐴)) ∈ (ℤ𝑀) → ((𝐺‘(#‘𝐴)) + 1) ∈ (ℤ𝑀))
4029, 39syl 17 . . . . . . 7 (𝜑 → ((𝐺‘(#‘𝐴)) + 1) ∈ (ℤ𝑀))
41 fzss1 12251 . . . . . . 7 (((𝐺‘(#‘𝐴)) + 1) ∈ (ℤ𝑀) → (((𝐺‘(#‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
4240, 41syl 17 . . . . . 6 (𝜑 → (((𝐺‘(#‘𝐴)) + 1)...𝑁) ⊆ (𝑀...𝑁))
4342sselda 3568 . . . . 5 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
44 eluzelre 11574 . . . . . . . . 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 10088 . . . . . . . 8 ((𝐺‘(#‘𝐴)) ∈ ℝ → ((𝐺‘(#‘𝐴)) + 1) ∈ ℝ)
4846, 47syl 17 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → ((𝐺‘(#‘𝐴)) + 1) ∈ ℝ)
49 elfzelz 12213 . . . . . . . . 9 (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℤ)
5049zred 11358 . . . . . . . 8 (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) → 𝑘 ∈ ℝ)
5150adantl 481 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → 𝑘 ∈ ℝ)
5246ltp1d 10833 . . . . . . 7 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝐺‘(#‘𝐴)) < ((𝐺‘(#‘𝐴)) + 1))
53 elfzle1 12215 . . . . . . . 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 10076 . . . . . 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 6062 . . . . . . . . . . . . 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 6050 . . . . . . . . . . . 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 792 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → 𝑘𝐴)
6260, 61ffvelrnd 6268 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ (1...(#‘𝐴)))
63 elfzle2 12216 . . . . . . . . . 10 ((𝐺𝑘) ∈ (1...(#‘𝐴)) → (𝐺𝑘) ≤ (#‘𝐴))
6462, 63syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ≤ (#‘𝐴))
65 elfzelz 12213 . . . . . . . . . . . 12 ((𝐺𝑘) ∈ (1...(#‘𝐴)) → (𝐺𝑘) ∈ ℤ)
6662, 65syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℤ)
6766zred 11358 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺𝑘) ∈ ℝ)
6818adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (#‘𝐴) ∈ ℕ0)
6968nn0red 11229 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (#‘𝐴) ∈ ℝ)
7067, 69lenltd 10062 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((𝐺𝑘) ≤ (#‘𝐴) ↔ ¬ (#‘𝐴) < (𝐺𝑘)))
7164, 70mpbid 221 . . . . . . . 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 6476 . . . . . . . . . 10 ((𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴) ∧ ((#‘𝐴) ∈ (1...(#‘𝐴)) ∧ (𝐺𝑘) ∈ (1...(#‘𝐴)))) → ((#‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(#‘𝐴)) < (𝐺‘(𝐺𝑘))))
7572, 73, 62, 74syl12anc 1316 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((#‘𝐴) < (𝐺𝑘) ↔ (𝐺‘(#‘𝐴)) < (𝐺‘(𝐺𝑘))))
76 f1ocnvfv2 6433 . . . . . . . . . . 11 ((𝐺:(1...(#‘𝐴))–1-1-onto𝐴𝑘𝐴) → (𝐺‘(𝐺𝑘)) = 𝑘)
7756, 61, 76syl2anc 691 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → (𝐺‘(𝐺𝑘)) = 𝑘)
7877breq2d 4595 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁) ∧ 𝑘𝐴)) → ((𝐺‘(#‘𝐴)) < (𝐺‘(𝐺𝑘)) ↔ (𝐺‘(#‘𝐴)) < 𝑘))
7975, 78bitrd 267 . . . . . . . 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 641 . . . . . 6 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → (𝑘𝐴 → ¬ (𝐺‘(#‘𝐴)) < 𝑘))
8255, 81mt2d 130 . . . . 5 ((𝜑𝑘 ∈ (((𝐺‘(#‘𝐴)) + 1)...𝑁)) → ¬ 𝑘𝐴)
8343, 82eldifd 3551 . . . 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 12709 . 2 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(#‘𝐴))) = (seq𝑀( + , 𝐹)‘𝑁))
87 seqcoll2.1 . . 3 ((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)
88 seqcoll2.a . . 3 (𝜑𝑍𝑆)
893, 2syl6ss 3580 . . 3 (𝜑𝐴 ⊆ (ℤ𝑀))
9033ssdifd 3708 . . . . 5 (𝜑 → ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴) ⊆ ((𝑀...𝑁) ∖ 𝐴))
9190sselda 3568 . . . 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 13105 . 2 (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘(#‘𝐴))) = (seq1( + , 𝐻)‘(#‘𝐴)))
9586, 94eqtr3d 2646 1 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(#‘𝐴)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583  ◡ccnv 5037  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805  (class class class)co 6549  Fincfn 7841  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  seqcseq 12663  #chash 12979 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-hash 12980 This theorem is referenced by:  isercolllem3  14245  gsumval3  18131
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