Theorem seqfveq2 13930

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 13449	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (𝐾...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝐾...𝑁))
4		eleq1 2825	. . . . . 6 ⊢ (𝑥 = 𝐾 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = 𝐾 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝐾))
6		fveq2 6842	. . . . . . 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 2825	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑛 ∈ (𝐾...𝑁)))
11		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
12		fveq2 6842	. . . . . . 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 2825	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝐾...𝑁) ↔ (𝑛 + 1) ∈ (𝐾...𝑁)))
17		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
18		fveq2 6842	. . . . . . 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 2825	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
23		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
24		fveq2 6842	. . . . . . 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 12773	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ ℤ)
31		seq1 13919	. . . . . . 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 2779	. . . . 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 13454	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁)) → 𝑛 ∈ (𝐾...𝑁))
36	35	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (𝐾...𝑁))
37	36	expr 457	. . . . . 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 7364	. . . . . 6 ⊢ ((seq𝑀( + , 𝐹)‘𝑛) = (seq𝐾( + , 𝐺)‘𝑛) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) = ((seq𝐾( + , 𝐺)‘𝑛) + (𝐹‘(𝑛 + 1))))
40		simpl 483	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁)) → 𝑛 ∈ (ℤ≥‘𝐾))
41		uztrn 12781	. . . . . . . . 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 13921	. . . . . . . 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 13921	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ≥‘𝐾) → (seq𝐾( + , 𝐺)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
46	45	ad2antrl 726	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
47		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
48		fveq2 6842	. . . . . . . . . . 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 3143	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘))
52	51	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ∀𝑘 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑘) = (𝐺‘𝑘))
53		eluzp1p1 12791	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝐾) → (𝑛 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
54	53	ad2antrl 726	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
55		elfzuz3 13438	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
56	55	ad2antll 727	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
57		elfzuzb 13435	. . . . . . . . . . 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 3582	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘(𝑛 + 1)))
60	59	oveq2d 7373	. . . . . . . 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 2779	. . . . . . 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	syl5ibr 245	. . . . 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 844	. . . 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 12835	. . 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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ‘cfv 6496  (class class class)co 7357  1c1 11052   + caddc 11054  ℤcz 12499  ℤ_≥cuz 12763  ...cfz 13424  seqcseq 13906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-seq 13907

This theorem is referenced by:  seqfeq2  13931  seqfveq  13932  seqz  13956  gsumsplit1r  18542