Theorem seqsplit 14008

Description: Split a sequence into two sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Hypotheses

Ref	Expression
seqsplit.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seqsplit.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
seqsplit.3	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))
seqsplit.4	⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐾))
seqsplit.5	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)

Assertion

Ref	Expression
seqsplit	⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐹   𝑥,𝐾,𝑦,𝑧   𝑥,𝑀,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝑁,𝑦,𝑧   𝑥, + ,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem seqsplit

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqsplit.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))
2		eluzfz2 13516	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → 𝑁 ∈ ((𝑀 + 1)...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ ((𝑀 + 1)...𝑁))
4		eleq1 2820	. . . . . 6 ⊢ (𝑥 = (𝑀 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑀 + 1) ∈ ((𝑀 + 1)...𝑁)))
5		fveq2 6891	. . . . . . 7 ⊢ (𝑥 = (𝑀 + 1) → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘(𝑀 + 1)))
6		fveq2 6891	. . . . . . . 8 ⊢ (𝑥 = (𝑀 + 1) → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)))
7	6	oveq2d 7428	. . . . . . 7 ⊢ (𝑥 = (𝑀 + 1) → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))))
8	5, 7	eqeq12d 2747	. . . . . 6 ⊢ (𝑥 = (𝑀 + 1) → ((seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) ↔ (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)))))
9	4, 8	imbi12d 344	. . . . 5 ⊢ (𝑥 = (𝑀 + 1) → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥))) ↔ ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))))))
10	9	imbi2d 340	. . . 4 ⊢ (𝑥 = (𝑀 + 1) → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)))) ↔ (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)))))))
11		eleq1 2820	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑛 ∈ ((𝑀 + 1)...𝑁)))
12		fveq2 6891	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘𝑛))
13		fveq2 6891	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘𝑛))
14	13	oveq2d 7428	. . . . . . 7 ⊢ (𝑥 = 𝑛 → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)))
15	12, 14	eqeq12d 2747	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) ↔ (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛))))
16	11, 15	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥))) ↔ (𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)))))
17	16	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)))) ↔ (𝜑 → (𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛))))))
18		eleq1 2820	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)))
19		fveq2 6891	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘(𝑛 + 1)))
20		fveq2 6891	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))
21	20	oveq2d 7428	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))))
22	19, 21	eqeq12d 2747	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) ↔ (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))))
23	18, 22	imbi12d 344	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥))) ↔ ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))))))
24	23	imbi2d 340	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)))) ↔ (𝜑 → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))))))
25		eleq1 2820	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑁 ∈ ((𝑀 + 1)...𝑁)))
26		fveq2 6891	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘𝑁))
27		fveq2 6891	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘𝑁))
28	27	oveq2d 7428	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))
29	26, 28	eqeq12d 2747	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) ↔ (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁))))
30	25, 29	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥))) ↔ (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))))
31	30	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑥) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)))) ↔ (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁))))))
32		seqsplit.4	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐾))
33		seqp1 13988	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ≥‘𝐾) → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1))))
34	32, 33	syl 17	. . . . . 6 ⊢ (𝜑 → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1))))
35		eluzel2 12834	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → (𝑀 + 1) ∈ ℤ)
36		seq1 13986	. . . . . . . 8 ⊢ ((𝑀 + 1) ∈ ℤ → (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)) = (𝐹‘(𝑀 + 1)))
37	1, 35, 36	3syl 18	. . . . . . 7 ⊢ (𝜑 → (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)) = (𝐹‘(𝑀 + 1)))
38	37	oveq2d 7428	. . . . . 6 ⊢ (𝜑 → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))) = ((seq𝐾( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1))))
39	34, 38	eqtr4d 2774	. . . . 5 ⊢ (𝜑 → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))))
40	39	a1i13 27	. . . 4 ⊢ ((𝑀 + 1) ∈ ℤ → (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))))))
41		peano2fzr 13521	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
42	41	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
43	42	expr 456	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → 𝑛 ∈ ((𝑀 + 1)...𝑁)))
44	43	imim1d 82	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)))))
45		oveq1 7419	. . . . . 6 ⊢ ((seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) → ((seq𝐾( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) = (((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) + (𝐹‘(𝑛 + 1))))
46		simprl 768	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ≥‘(𝑀 + 1)))
47		peano2uz 12892	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ≥‘𝐾) → (𝑀 + 1) ∈ (ℤ≥‘𝐾))
48	32, 47	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 1) ∈ (ℤ≥‘𝐾))
49	48	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑀 + 1) ∈ (ℤ≥‘𝐾))
50		uztrn 12847	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (ℤ≥‘𝐾)) → 𝑛 ∈ (ℤ≥‘𝐾))
51	46, 49, 50	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ≥‘𝐾))
52		seqp1 13988	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ≥‘𝐾) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
53	51, 52	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
54		seqp1 13988	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) → (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)) = ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
55	46, 54	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)) = ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
56	55	oveq2d 7428	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))) = ((seq𝐾( + , 𝐹)‘𝑀) + ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
57		simpl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝜑)
58		eluzelz 12839	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ≥‘𝐾) → 𝑀 ∈ ℤ)
59	32, 58	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℤ)
60		peano2uzr 12894	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀))
61	59, 1, 60	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
62		fzss2 13548	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾...𝑀) ⊆ (𝐾...𝑁))
63	61, 62	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾...𝑀) ⊆ (𝐾...𝑁))
64	63	sselda 3982	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑀)) → 𝑥 ∈ (𝐾...𝑁))
65		seqsplit.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
66	64, 65	syldan 590	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
67		seqsplit.1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
68	32, 66, 67	seqcl 13995	. . . . . . . . . 10 ⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝑀) ∈ 𝑆)
69	68	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq𝐾( + , 𝐹)‘𝑀) ∈ 𝑆)
70		elfzuz3 13505	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ((𝑀 + 1)...𝑁) → 𝑁 ∈ (ℤ≥‘𝑛))
71		fzss2 13548	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘𝑛) → ((𝑀 + 1)...𝑛) ⊆ ((𝑀 + 1)...𝑁))
72	42, 70, 71	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((𝑀 + 1)...𝑛) ⊆ ((𝑀 + 1)...𝑁))
73		fzss1 13547	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘𝐾) → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁))
74	32, 47, 73	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁))
75	74	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁))
76	72, 75	sstrd 3992	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((𝑀 + 1)...𝑛) ⊆ (𝐾...𝑁))
77	76	sselda 3982	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑛)) → 𝑥 ∈ (𝐾...𝑁))
78	65	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
79	77, 78	syldan 590	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑛)) → (𝐹‘𝑥) ∈ 𝑆)
80	67	adantlr 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
81	46, 79, 80	seqcl 13995	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq(𝑀 + 1)( + , 𝐹)‘𝑛) ∈ 𝑆)
82		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
83	82	eleq1d 2817	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
84	65	ralrimiva 3145	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (𝐾...𝑁)(𝐹‘𝑥) ∈ 𝑆)
85	84	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ∀𝑥 ∈ (𝐾...𝑁)(𝐹‘𝑥) ∈ 𝑆)
86		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))
87		ssel2 3977	. . . . . . . . . . 11 ⊢ ((((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝑛 + 1) ∈ (𝐾...𝑁))
88	74, 86, 87	syl2an 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑛 + 1) ∈ (𝐾...𝑁))
89	83, 85, 88	rspcdva 3613	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
90		seqsplit.2	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
91	90	caovassg 7609	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((seq𝐾( + , 𝐹)‘𝑀) ∈ 𝑆 ∧ (seq(𝑀 + 1)( + , 𝐹)‘𝑛) ∈ 𝑆 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑆)) → (((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) + (𝐹‘(𝑛 + 1))) = ((seq𝐾( + , 𝐹)‘𝑀) + ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
92	57, 69, 81, 89, 91	syl13anc 1371	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) + (𝐹‘(𝑛 + 1))) = ((seq𝐾( + , 𝐹)‘𝑀) + ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
93	56, 92	eqtr4d 2774	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))) = (((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) + (𝐹‘(𝑛 + 1))))
94	53, 93	eqeq12d 2747	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1))) ↔ ((seq𝐾( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) = (((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) + (𝐹‘(𝑛 + 1)))))
95	45, 94	imbitrrid 245	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))))
96	44, 95	animpimp2impd 843	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) → ((𝜑 → (𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑛) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)))) → (𝜑 → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))))))
97	10, 17, 24, 31, 40, 96	uzind4 12897	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))))
98	1, 97	mpcom 38	. 2 ⊢ (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁))))
99	3, 98	mpd 15	1 ⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3060   ⊆ wss 3948  ‘cfv 6543  (class class class)co 7412  1c1 11117   + caddc 11119  ℤcz 12565  ℤ_≥cuz 12829  ...cfz 13491  seqcseq 13973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-n0 12480  df-z 12566  df-uz 12830  df-fz 13492  df-seq 13974

This theorem is referenced by:  seq1p  14009  seqf1olem2  14015  bcval5  14285  clim2ser  15608  clim2ser2  15609  isumsplit  15793  clim2div  15842  gsumsgrpccat  18763  mulgnndir  19026  mblfinlem2  36992  fmul01lt1lem1  44761  fmul01lt1lem2  44762