Theorem seqfveq2 13673

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 13193	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (𝐾...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝐾...𝑁))
4		eleq1 2826	. . . . . 6 ⊢ (𝑥 = 𝐾 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5		fveq2 6756	. . . . . . 7 ⊢ (𝑥 = 𝐾 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝐾))
6		fveq2 6756	. . . . . . 7 ⊢ (𝑥 = 𝐾 → (seq𝐾( + , 𝐺)‘𝑥) = (seq𝐾( + , 𝐺)‘𝐾))
7	5, 6	eqeq12d 2754	. . . . . 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 2826	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑛 ∈ (𝐾...𝑁)))
11		fveq2 6756	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
12		fveq2 6756	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝐾( + , 𝐺)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑛))
13	11, 12	eqeq12d 2754	. . . . . 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 2826	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝐾...𝑁) ↔ (𝑛 + 1) ∈ (𝐾...𝑁)))
17		fveq2 6756	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
18		fveq2 6756	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝐾( + , 𝐺)‘𝑥) = (seq𝐾( + , 𝐺)‘(𝑛 + 1)))
19	17, 18	eqeq12d 2754	. . . . . 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 2826	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
23		fveq2 6756	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
24		fveq2 6756	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝐾( + , 𝐺)‘𝑥) = (seq𝐾( + , 𝐺)‘𝑁))
25	23, 24	eqeq12d 2754	. . . . . 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 12521	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ ℤ)
31		seq1 13662	. . . . . . 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 2781	. . . . 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 13198	. . . . . . . 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 7262	. . . . . 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 12529	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ (ℤ≥‘𝑀))
42	40, 29, 41	syl2anr 596	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (ℤ≥‘𝑀))
43		seqp1 13664	. . . . . . . 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 13664	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ≥‘𝐾) → (seq𝐾( + , 𝐺)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
46	45	ad2antrl 724	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))
47		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
48		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑛 + 1)))
49	47, 48	eqeq12d 2754	. . . . . . . . . 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 3107	. . . . . . . . . . 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 12539	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝐾) → (𝑛 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
54	53	ad2antrl 724	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
55		elfzuz3 13182	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
56	55	ad2antll 725	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
57		elfzuzb 13179	. . . . . . . . . . 11 ⊢ ((𝑛 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑛 + 1) ∈ (ℤ≥‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑛 + 1))))
58	54, 56, 57	sylanbrc 582	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ ((𝐾 + 1)...𝑁))
59	49, 52, 58	rspcdva 3554	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑛 + 1)) = (𝐺‘(𝑛 + 1)))
60	59	oveq2d 7271	. . . . . . . 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 2781	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝐾( + , 𝐺)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐺)‘𝑛) + (𝐹‘(𝑛 + 1))))
62	44, 61	eqeq12d 2754	. . . . . 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 842	. . . 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 12579	. . 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 1539   ∈ wcel 2108  ∀wral 3063  ‘cfv 6418  (class class class)co 7255  1c1 10803   + caddc 10805  ℤcz 12249  ℤ_≥cuz 12511  ...cfz 13168  seqcseq 13649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-seq 13650

This theorem is referenced by:  seqfeq2  13674  seqfveq  13675  seqz  13699  gsumsplit1r  18286