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Theorem iseqsplit 9458

Description: Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.)

**Hypotheses**
Ref	Expression
iseqsplit.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqsplit.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
iseqsplit.3	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑀 + 1)))
iseqsplit.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
iseqsplit.4	⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝐾))
iseqsplit.5	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → (𝐹‘𝑥) ∈ 𝑆)

**Assertion**
Ref	Expression
iseqsplit	⊢ (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘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)

Proof of Theorem iseqsplit Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqsplit.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑀 + 1)))
2		eluzfz2 9051	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑀 + 1)) → 𝑁 ∈ ((𝑀 + 1)...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ ((𝑀 + 1)...𝑁))
4		eleq1 2141	. . . . . 6 ⊢ (𝑥 = (𝑀 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑀 + 1) ∈ ((𝑀 + 1)...𝑁)))
5		fveq2 5198	. . . . . . 7 ⊢ (𝑥 = (𝑀 + 1) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)))
6		fveq2 5198	. . . . . . . 8 ⊢ (𝑥 = (𝑀 + 1) → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1)))
7	6	oveq2d 5548	. . . . . . 7 ⊢ (𝑥 = (𝑀 + 1) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1))))
8	5, 7	eqeq12d 2095	. . . . . 6 ⊢ (𝑥 = (𝑀 + 1) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) ↔ (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1)))))
9	4, 8	imbi12d 232	. . . . 5 ⊢ (𝑥 = (𝑀 + 1) → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥))) ↔ ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1))))))
10	9	imbi2d 228	. . . 4 ⊢ (𝑥 = (𝑀 + 1) → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)))) ↔ (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1)))))))
11		eleq1 2141	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑛 ∈ ((𝑀 + 1)...𝑁)))
12		fveq2 5198	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘𝑛))
13		fveq2 5198	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))
14	13	oveq2d 5548	. . . . . . 7 ⊢ (𝑥 = 𝑛 → ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)))
15	12, 14	eqeq12d 2095	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) ↔ (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))))
16	11, 15	imbi12d 232	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥))) ↔ (𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)))))
17	16	imbi2d 228	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)))) ↔ (𝜑 → (𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))))))
18		eleq1 2141	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)))
19		fveq2 5198	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)))
20		fveq2 5198	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)))
21	20	oveq2d 5548	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))))
22	19, 21	eqeq12d 2095	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) ↔ (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)))))
23	18, 22	imbi12d 232	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥))) ↔ ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))))))
24	23	imbi2d 228	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)))) ↔ (𝜑 → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)))))))
25		eleq1 2141	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ ((𝑀 + 1)...𝑁) ↔ 𝑁 ∈ ((𝑀 + 1)...𝑁)))
26		fveq2 5198	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = (seq𝐾( + , 𝐹, 𝑆)‘𝑁))
27		fveq2 5198	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥) = (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁))
28	27	oveq2d 5548	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))
29	26, 28	eqeq12d 2095	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)) ↔ (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁))))
30	25, 29	imbi12d 232	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥))) ↔ (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))))
31	30	imbi2d 228	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑥) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑥)))) ↔ (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁))))))
32		iseqsplit.4	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝐾))
33		iseqsplit.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
34		iseqsplit.5	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → (𝐹‘𝑥) ∈ 𝑆)
35		iseqsplit.1	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
36	32, 33, 34, 35	iseqp1 9445	. . . . . . 7 ⊢ (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (𝐹‘(𝑀 + 1))))
37		eluzel2 8624	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑀 + 1)) → (𝑀 + 1) ∈ ℤ)
38	1, 37	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
39		eluzelz 8628	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (ℤ_≥‘𝐾) → 𝑀 ∈ ℤ)
40	32, 39	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
41		peano2uzr 8673	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ (ℤ_≥‘(𝑀 + 1))) → 𝑥 ∈ (ℤ_≥‘𝑀))
42	40, 41	sylan 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝑀 + 1))) → 𝑥 ∈ (ℤ_≥‘𝑀))
43	32	adantr 270	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝑀 + 1))) → 𝑀 ∈ (ℤ_≥‘𝐾))
44		uztrn 8635	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (ℤ_≥‘𝑀) ∧ 𝑀 ∈ (ℤ_≥‘𝐾)) → 𝑥 ∈ (ℤ_≥‘𝐾))
45	42, 43, 44	syl2anc 403	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝑀 + 1))) → 𝑥 ∈ (ℤ_≥‘𝐾))
46	45, 34	syldan 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝑆)
47	38, 33, 46, 35	iseq1 9442	. . . . . . . 8 ⊢ (𝜑 → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1)) = (𝐹‘(𝑀 + 1)))
48	47	oveq2d 5548	. . . . . . 7 ⊢ (𝜑 → ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1))) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (𝐹‘(𝑀 + 1))))
49	36, 48	eqtr4d 2116	. . . . . 6 ⊢ (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1))))
50	49	a1d 22	. . . . 5 ⊢ (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1)))))
51	50	a1i 9	. . . 4 ⊢ ((𝑀 + 1) ∈ ℤ → (𝜑 → ((𝑀 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑀 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑀 + 1))))))
52		peano2fzr 9056	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
53	52	adantl 271	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ ((𝑀 + 1)...𝑁))
54	53	expr 367	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑀 + 1))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → 𝑛 ∈ ((𝑀 + 1)...𝑁)))
55	54	imim1d 74	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑀 + 1))) → ((𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)))))
56		oveq1 5539	. . . . . . . . . 10 ⊢ ((seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))) = (((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) + (𝐹‘(𝑛 + 1))))
57		simprl 497	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ_≥‘(𝑀 + 1)))
58		peano2uz 8671	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ_≥‘𝐾) → (𝑀 + 1) ∈ (ℤ_≥‘𝐾))
59	32, 58	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 + 1) ∈ (ℤ_≥‘𝐾))
60	59	adantr 270	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑀 + 1) ∈ (ℤ_≥‘𝐾))
61		uztrn 8635	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (ℤ_≥‘𝐾)) → 𝑛 ∈ (ℤ_≥‘𝐾))
62	57, 60, 61	syl2anc 403	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑛 ∈ (ℤ_≥‘𝐾))
63	33	adantr 270	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝑆 ∈ 𝑉)
64	34	adantlr 460	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → (𝐹‘𝑥) ∈ 𝑆)
65	35	adantlr 460	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
66	62, 63, 64, 65	iseqp1 9445	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
67	46	adantlr 460	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) ∧ 𝑥 ∈ (ℤ_≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝑆)
68	57, 63, 67, 65	iseqp1 9445	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))))
69	68	oveq2d 5548	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + ((seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))))
70		simpl 107	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → 𝜑)
71	32, 33, 34, 35	iseqcl 9443	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘𝑀) ∈ 𝑆)
72	71	adantr 270	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq𝐾( + , 𝐹, 𝑆)‘𝑀) ∈ 𝑆)
73	57, 63, 67, 65	iseqcl 9443	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛) ∈ 𝑆)
74		fzss1 9081	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 + 1) ∈ (ℤ_≥‘𝐾) → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁))
75	32, 58, 74	3syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁))
76		simpr 108	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))
77		ssel2 2994	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 + 1)...𝑁) ⊆ (𝐾...𝑁) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁)) → (𝑛 + 1) ∈ (𝐾...𝑁))
78	75, 76, 77	syl2an 283	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝑛 + 1) ∈ (𝐾...𝑁))
79		elfzuz 9041	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐾...𝑁) → 𝑥 ∈ (ℤ_≥‘𝐾))
80	79, 34	sylan2 280	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)
81	80	ralrimiva 2434	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ (𝐾...𝑁)(𝐹‘𝑥) ∈ 𝑆)
82	81	adantr 270	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ∀𝑥 ∈ (𝐾...𝑁)(𝐹‘𝑥) ∈ 𝑆)
83		fveq2 5198	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
84	83	eleq1d 2147	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑆))
85	84	rspcv 2697	. . . . . . . . . . . . . 14 ⊢ ((𝑛 + 1) ∈ (𝐾...𝑁) → (∀𝑥 ∈ (𝐾...𝑁)(𝐹‘𝑥) ∈ 𝑆 → (𝐹‘(𝑛 + 1)) ∈ 𝑆))
86	78, 82, 85	sylc 61	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ 𝑆)
87		iseqsplit.2	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
88	87	caovassg 5679	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) ∈ 𝑆 ∧ (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛) ∈ 𝑆 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑆)) → (((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) + (𝐹‘(𝑛 + 1))) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + ((seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))))
89	70, 72, 73, 86, 88	syl13anc 1171	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → (((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) + (𝐹‘(𝑛 + 1))) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + ((seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))))
90	69, 89	eqtr4d 2116	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))) = (((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) + (𝐹‘(𝑛 + 1))))
91	66, 90	eqeq12d 2095	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))) ↔ ((seq𝐾( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1))) = (((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) + (𝐹‘(𝑛 + 1)))))
92	56, 91	syl5ibr 154	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) ∧ (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)))))
93	92	expr 367	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑀 + 1))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → ((seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))))))
94	93	a2d 26	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑀 + 1))) → (((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))))))
95	55, 94	syld 44	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(𝑀 + 1))) → ((𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1))))))
96	95	expcom 114	. . . . 5 ⊢ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) → (𝜑 → ((𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛))) → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)))))))
97	96	a2d 26	. . . 4 ⊢ (𝑛 ∈ (ℤ_≥‘(𝑀 + 1)) → ((𝜑 → (𝑛 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑛) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑛)))) → (𝜑 → ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘(𝑛 + 1)) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘(𝑛 + 1)))))))
98	10, 17, 24, 31, 51, 97	uzind4 8676	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑀 + 1)) → (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))))
99	1, 98	mpcom 36	. 2 ⊢ (𝜑 → (𝑁 ∈ ((𝑀 + 1)...𝑁) → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁))))
100	3, 99	mpd 13	1 ⊢ (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 919 = wceq 1284 ∈ wcel 1433 ∀wral 2348 ⊆ wss 2973 ‘cfv 4922 (class class class)co 5532 1c1 6982 + caddc 6984 ℤcz 8351 ℤ_≥cuz 8619 ...cfz 9029 seqcseq 9431

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092

This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-fz 9030 df-iseq 9432

This theorem is referenced by: iseq1p 9459 ibcval5 9690 clim2iser 10175 clim2iser2 10176

W3C validator