Theorem seqsplitg 10651

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)))
seqsplitg.p	⊢ (𝜑 → + ∈ 𝑉)
seqsplitg.f	⊢ (𝜑 → 𝐹 ∈ 𝑊)
seqsplit.4	⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐾))
seqsplit.5	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)

Assertion

Ref	Expression
seqsplitg	⊢ (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

Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)

Proof of Theorem seqsplitg

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqsplit.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))
2		eluzfz2 10169	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → 𝑁 ∈ ((𝑀 + 1)...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ ((𝑀 + 1)...𝑁))
4		eleq1 2269	. . . . . 6 ⊢ (𝑥 = (𝑀 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑀 + 1) ∈ ((𝑀 + 1)...𝑁)))
5		fveq2 5588	. . . . . . 7 ⊢ (𝑥 = (𝑀 + 1) → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘(𝑀 + 1)))
6		fveq2 5588	. . . . . . . 8 ⊢ (𝑥 = (𝑀 + 1) → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)))
7	6	oveq2d 5972	. . . . . . 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 2221	. . . . . 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 234	. . . . 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 230	. . . 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 2269	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑛 ∈ ((𝑀 + 1)...𝑁)))
12		fveq2 5588	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘𝑛))
13		fveq2 5588	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘𝑛))
14	13	oveq2d 5972	. . . . . . 7 ⊢ (𝑥 = 𝑛 → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑛)))
15	12, 14	eqeq12d 2221	. . . . . 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 234	. . . . 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 230	. . . 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 2269	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)))
19		fveq2 5588	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘(𝑛 + 1)))
20		fveq2 5588	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)))
21	20	oveq2d 5972	. . . . . . 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 2221	. . . . . 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 234	. . . . 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 230	. . . 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 2269	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑁 ∈ ((𝑀 + 1)...𝑁)))
26		fveq2 5588	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝐾( + , 𝐹)‘𝑥) = (seq𝐾( + , 𝐹)‘𝑁))
27		fveq2 5588	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (seq(𝑀 + 1)( + , 𝐹)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹)‘𝑁))
28	27	oveq2d 5972	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑥)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))
29	26, 28	eqeq12d 2221	. . . . . 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 234	. . . . 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 230	. . . 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		seqsplitg.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
34		seqsplitg.p	. . . . . . 7 ⊢ (𝜑 → + ∈ 𝑉)
35		seqp1g 10628	. . . . . . 7 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐹 ∈ 𝑊 ∧ + ∈ 𝑉) → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1))))
36	32, 33, 34, 35	syl3anc 1250	. . . . . 6 ⊢ (𝜑 → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1))))
37		eluzel2 9668	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → (𝑀 + 1) ∈ ℤ)
38	1, 37	syl 14	. . . . . . . 8 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
39		seq1g 10625	. . . . . . . 8 ⊢ (((𝑀 + 1) ∈ ℤ ∧ 𝐹 ∈ 𝑊 ∧ + ∈ 𝑉) → (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)) = (𝐹‘(𝑀 + 1)))
40	38, 33, 34, 39	syl3anc 1250	. . . . . . 7 ⊢ (𝜑 → (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1)) = (𝐹‘(𝑀 + 1)))
41	40	oveq2d 5972	. . . . . 6 ⊢ (𝜑 → ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))) = ((seq𝐾( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1))))
42	36, 41	eqtr4d 2242	. . . . 5 ⊢ (𝜑 → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))))
43	42	a1i13 24	. . . 4 ⊢ ((𝑀 + 1) ∈ ℤ → (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘(𝑀 + 1))))))
44		peano2fzr 10174	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
45	44	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
46	45	expr 375	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → 𝑛 ∈ ((𝑀 + 1)...𝑁)))
47	46	imim1d 75	. . . . 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)))))
48		oveq1 5963	. . . . . 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))))
49		simprl 529	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ≥‘(𝑀 + 1)))
50		peano2uz 9719	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ≥‘𝐾) → (𝑀 + 1) ∈ (ℤ≥‘𝐾))
51	32, 50	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 1) ∈ (ℤ≥‘𝐾))
52	51	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑀 + 1) ∈ (ℤ≥‘𝐾))
53		uztrn 9680	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (ℤ≥‘𝐾)) → 𝑛 ∈ (ℤ≥‘𝐾))
54	49, 52, 53	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ≥‘𝐾))
55	33	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝐹 ∈ 𝑊)
56	34	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → + ∈ 𝑉)
57		seqp1g 10628	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝐾) ∧ 𝐹 ∈ 𝑊 ∧ + ∈ 𝑉) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
58	54, 55, 56, 57	syl3anc 1250	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq𝐾( + , 𝐹)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
59		seqp1g 10628	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ 𝐹 ∈ 𝑊 ∧ + ∈ 𝑉) → (seq(𝑀 + 1)( + , 𝐹)‘(𝑛 + 1)) = ((seq(𝑀 + 1)( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
60	49, 55, 56, 59	syl3anc 1250	. . . . . . . . 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))))
61	60	oveq2d 5972	. . . . . . . 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)))))
62		simpl 109	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝜑)
63		eluzelz 9672	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ≥‘𝐾) → 𝑀 ∈ ℤ)
64	32, 63	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℤ)
65		peano2uzr 9721	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀))
66	64, 1, 65	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
67		fzss2 10201	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐾...𝑀) ⊆ (𝐾...𝑁))
68	66, 67	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾...𝑀) ⊆ (𝐾...𝑁))
69	68	sselda 3197	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑀)) → 𝑥 ∈ (𝐾...𝑁))
70		seqsplit.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
71	69, 70	syldan 282	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
72		seqsplit.1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
73	32, 71, 72, 33, 34	seqclg 10634	. . . . . . . . . 10 ⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝑀) ∈ 𝑆)
74	73	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq𝐾( + , 𝐹)‘𝑀) ∈ 𝑆)
75		elfzuz3 10159	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ((𝑀 + 1)...𝑁) → 𝑁 ∈ (ℤ≥‘𝑛))
76		fzss2 10201	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘𝑛) → ((𝑀 + 1)...𝑛) ⊆ ((𝑀 + 1)...𝑁))
77	45, 75, 76	3syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((𝑀 + 1)...𝑛) ⊆ ((𝑀 + 1)...𝑁))
78		fzss1 10200	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘𝐾) → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁))
79	32, 50, 78	3syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁))
80	79	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁))
81	77, 80	sstrd 3207	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((𝑀 + 1)...𝑛) ⊆ (𝐾...𝑁))
82	81	sselda 3197	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑛)) → 𝑥 ∈ (𝐾...𝑁))
83	70	adantlr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
84	82, 83	syldan 282	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑛)) → (𝐹‘𝑥) ∈ 𝑆)
85	72	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
86	49, 84, 85, 55, 56	seqclg 10634	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq(𝑀 + 1)( + , 𝐹)‘𝑛) ∈ 𝑆)
87		fveq2 5588	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
88	87	eleq1d 2275	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
89	70	ralrimiva 2580	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (𝐾...𝑁)(𝐹‘𝑥) ∈ 𝑆)
90	89	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ∀𝑥 ∈ (𝐾...𝑁)(𝐹‘𝑥) ∈ 𝑆)
91		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))
92		ssel2 3192	. . . . . . . . . . 11 ⊢ ((((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝑛 + 1) ∈ (𝐾...𝑁))
93	79, 91, 92	syl2an 289	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑛 + 1) ∈ (𝐾...𝑁))
94	88, 90, 93	rspcdva 2886	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
95		seqsplit.2	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
96	95	caovassg 6117	. . . . . . . . 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)))))
97	62, 74, 86, 94, 96	syl13anc 1252	. . . . . . . 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)))))
98	61, 97	eqtr4d 2242	. . . . . . 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))))
99	58, 98	eqeq12d 2221	. . . . . 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)))))
100	48, 99	imbitrrid 156	. . . . 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)))))
101	47, 100	animpimp2impd 559	. . . 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)))))))
102	10, 17, 24, 31, 43, 101	uzind4 9724	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) → (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))))
103	1, 102	mpcom 36	. 2 ⊢ (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁))))
104	3, 103	mpd 13	1 ⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ∀wral 2485   ⊆ wss 3170  ‘cfv 5279  (class class class)co 5956  1c1 7941   + caddc 7943  ℤcz 9387  ℤ_≥cuz 9663  ...cfz 10145  seqcseq 10609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643  ax-cnex 8031  ax-resscn 8032  ax-1cn 8033  ax-1re 8034  ax-icn 8035  ax-addcl 8036  ax-addrcl 8037  ax-mulcl 8038  ax-addcom 8040  ax-addass 8042  ax-distr 8044  ax-i2m1 8045  ax-0lt1 8046  ax-0id 8048  ax-rnegex 8049  ax-cnre 8051  ax-pre-ltirr 8052  ax-pre-ltwlin 8053  ax-pre-lttrn 8054  ax-pre-ltadd 8056

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-iord 4420  df-on 4422  df-ilim 4423  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-riota 5911  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1st 6238  df-2nd 6239  df-recs 6403  df-frec 6489  df-pnf 8124  df-mnf 8125  df-xr 8126  df-ltxr 8127  df-le 8128  df-sub 8260  df-neg 8261  df-inn 9052  df-n0 9311  df-z 9388  df-uz 9664  df-fz 10146  df-fzo 10280  df-seqfrec 10610

This theorem is referenced by:  seqf1oglem2  10682