Theorem seqfveq2 14047

Description: Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Hypotheses

Ref	Expression
seqfveq2.1	⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))
seqfveq2.2	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺‘𝐾))
seqfveq2.3	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))
seqfveq2.4	⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))

Assertion

Ref	Expression
seqfveq2	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))

Distinct variable groups:   𝑘,𝐹   𝑘,𝐺   𝑘,𝐾   𝑘,𝑁   𝜑,𝑘

Allowed substitution hints:   + (𝑘)   𝑀(𝑘)

Proof of Theorem seqfveq2

Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqfveq2.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))
2		eluzfz2 13554	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (𝐾...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝐾...𝑁))
4		eleq1 2823	. . . . . 6 ⊢ (𝑥 = 𝐾 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5		fveq2 6881	. . . . . . 7 ⊢ (𝑥 = 𝐾 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝐾))
6		fveq2 6881	. . . . . . 7 ⊢ (𝑥 = 𝐾 → (seq𝐾( + , 𝐺)‘𝑥) = (seq𝐾( + , 𝐺)‘𝐾))
7	5, 6	eqeq12d 2752	. . . . . 6 ⊢ (𝑥 = 𝐾 → ((seq𝑀( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾)))
8	4, 7	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝐾 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑥)) ↔ (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾))))
9	8	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝐾 → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑥))) ↔ (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾)))))
10		eleq1 2823	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑛 ∈ (𝐾...𝑁)))
11		fveq2 6881	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
12		fveq2 6881	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝐾( + , 𝐺)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑛))
13	11, 12	eqeq12d 2752	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝑛) = (seq𝐾( + , 𝐺)‘𝑛)))
14	10, 13	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑥)) ↔ (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝐾( + , 𝐺)‘𝑛))))
15	14	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑥))) ↔ (𝜑 → (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝐾( + , 𝐺)‘𝑛)))))
16		eleq1 2823	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝐾...𝑁) ↔ (𝑛 + 1) ∈ (𝐾...𝑁)))
17		fveq2 6881	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
18		fveq2 6881	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝐾( + , 𝐺)‘𝑥) = (seq𝐾( + , 𝐺)‘(𝑛 + 1)))
19	17, 18	eqeq12d 2752	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq𝐾( + , 𝐺)‘(𝑛 + 1))))
20	16, 19	imbi12d 344	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑥)) ↔ ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq𝐾( + , 𝐺)‘(𝑛 + 1)))))
21	20	imbi2d 340	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑥))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq𝐾( + , 𝐺)‘(𝑛 + 1))))))
22		eleq1 2823	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
23		fveq2 6881	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
24		fveq2 6881	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝐾( + , 𝐺)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑁))
25	23, 24	eqeq12d 2752	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁)))
26	22, 25	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑥)) ↔ (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))))
27	26	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑥))) ↔ (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁)))))
28		seqfveq2.2	. . . . . 6 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺‘𝐾))
29		seqfveq2.1	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))
30		eluzelz 12867	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ ℤ)
31		seq1 14037	. . . . . . 7 ⊢ (𝐾 ∈ ℤ → (seq𝐾( + , 𝐺)‘𝐾) = (𝐺‘𝐾))
32	29, 30, 31	3syl 18	. . . . . 6 ⊢ (𝜑 → (seq𝐾( + , 𝐺)‘𝐾) = (𝐺‘𝐾))
33	28, 32	eqtr4d 2774	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾))
34	33	a1d 25	. . . 4 ⊢ (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝐾( + , 𝐺)‘𝐾)))
35		peano2fzr 13559	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁)) → 𝑛 ∈ (𝐾...𝑁))
36	35	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (𝐾...𝑁))
37	36	expr 456	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → 𝑛 ∈ (𝐾...𝑁)))
38	37	imim1d 82	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → ((𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝐾( + , 𝐺)‘𝑛)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝐾( + , 𝐺)‘𝑛))))
39		oveq1 7417	. . . . . 6 ⊢ ((seq𝑀( + , 𝐹)‘𝑛) = (seq𝐾( + , 𝐺)‘𝑛) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) = ((seq𝐾( + , 𝐺)‘𝑛) + (𝐹‘(𝑛 + 1))))
40		simpl 482	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁)) → 𝑛 ∈ (ℤ≥‘𝐾))
41		uztrn 12875	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ (ℤ≥‘𝑀))
42	40, 29, 41	syl2anr 597	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (ℤ≥‘𝑀))
43		seqp1 14039	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
44	42, 43	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
45		seqp1 14039	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ≥‘𝐾) → (seq𝐾( + , 𝐺)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
46	45	ad2antrl 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
47		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
48		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑛 + 1)))
49	47, 48	eqeq12d 2752	. . . . . . . . . 10 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘(𝑛 + 1)) = (𝐺‘(𝑛 + 1))))
50		seqfveq2.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))
51	50	ralrimiva 3133	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘))
52	51	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘))
53		eluzp1p1 12885	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝐾) → (𝑛 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
54	53	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
55		elfzuz3 13543	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
56	55	ad2antll 729	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
57		elfzuzb 13540	. . . . . . . . . . 11 ⊢ ((𝑛 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑛 + 1) ∈ (ℤ≥‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑛 + 1))))
58	54, 56, 57	sylanbrc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ ((𝐾 + 1)...𝑁))
59	49, 52, 58	rspcdva 3607	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘(𝑛 + 1)))
60	59	oveq2d 7426	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝐾( + , 𝐺)‘𝑛) + (𝐹‘(𝑛 + 1))) = ((seq𝐾( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
61	46, 60	eqtr4d 2774	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐺)‘𝑛) + (𝐹‘(𝑛 + 1))))
62	44, 61	eqeq12d 2752	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq𝐾( + , 𝐺)‘(𝑛 + 1)) ↔ ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) = ((seq𝐾( + , 𝐺)‘𝑛) + (𝐹‘(𝑛 + 1)))))
63	39, 62	imbitrrid 246	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝑛) = (seq𝐾( + , 𝐺)‘𝑛) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq𝐾( + , 𝐺)‘(𝑛 + 1))))
64	38, 63	animpimp2impd 846	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘𝐾) → ((𝜑 → (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝐾( + , 𝐺)‘𝑛))) → (𝜑 → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = (seq𝐾( + , 𝐺)‘(𝑛 + 1))))))
65	9, 15, 21, 27, 34, 64	uzind4i 12931	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))))
66	1, 65	mpcom 38	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁)))
67	3, 66	mpd 15	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ‘cfv 6536  (class class class)co 7410  1c1 11135   + caddc 11137  ℤcz 12593  ℤ_≥cuz 12857  ...cfz 13529  seqcseq 14024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-n0 12507  df-z 12594  df-uz 12858  df-fz 13530  df-seq 14025

This theorem is referenced by:  seqfeq2  14048  seqfveq  14049  seqz  14073  gsumsplit1r  18670