Theorem clim2prod 15600

Description: The limit of an infinite product with an initial segment added. (Contributed by Scott Fenton, 18-Dec-2017.)

Hypotheses

Ref	Expression
clim2prod.1	⊢ 𝑍 = (ℤ≥‘𝑀)
clim2prod.2	⊢ (𝜑 → 𝑁 ∈ 𝑍)
clim2prod.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
clim2prod.4	⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹) ⇝ 𝐴)

Assertion

Ref	Expression
clim2prod	⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ ((seq𝑀( · , 𝐹)‘𝑁) · 𝐴))

Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝜑,𝑘   𝑘,𝑀   𝑘,𝑁   𝑘,𝑍

Proof of Theorem clim2prod

Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. 2 ⊢ (ℤ≥‘(𝑁 + 1)) = (ℤ≥‘(𝑁 + 1))
2		clim2prod.1	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
3		uzssz 12603	. . . . 5 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
4	2, 3	eqsstri 3955	. . . 4 ⊢ 𝑍 ⊆ ℤ
5		clim2prod.2	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
6	4, 5	sselid 3919	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7	6	peano2zd 12429	. 2 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
8		clim2prod.4	. 2 ⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹) ⇝ 𝐴)
9	5, 2	eleqtrdi 2849	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
10		eluzel2 12587	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
12		clim2prod.3	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
13	2, 11, 12	prodf 15599	. . 3 ⊢ (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ)
14	13, 5	ffvelrnd 6962	. 2 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ∈ ℂ)
15		seqex 13723	. . 3 ⊢ seq𝑀( · , 𝐹) ∈ V
16	15	a1i 11	. 2 ⊢ (𝜑 → seq𝑀( · , 𝐹) ∈ V)
17		peano2uz 12641	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀))
18		uzss 12605	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑀) → (ℤ≥‘(𝑁 + 1)) ⊆ (ℤ≥‘𝑀))
19	9, 17, 18	3syl 18	. . . . . . 7 ⊢ (𝜑 → (ℤ≥‘(𝑁 + 1)) ⊆ (ℤ≥‘𝑀))
20	19, 2	sseqtrrdi 3972	. . . . . 6 ⊢ (𝜑 → (ℤ≥‘(𝑁 + 1)) ⊆ 𝑍)
21	20	sselda 3921	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍)
22	21, 12	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → (𝐹‘𝑘) ∈ ℂ)
23	1, 7, 22	prodf 15599	. . 3 ⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹):(ℤ≥‘(𝑁 + 1))⟶ℂ)
24	23	ffvelrnda 6961	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘𝑘) ∈ ℂ)
25		fveq2 6774	. . . . . 6 ⊢ (𝑥 = (𝑁 + 1) → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘(𝑁 + 1)))
26		fveq2 6774	. . . . . . 7 ⊢ (𝑥 = (𝑁 + 1) → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1)))
27	26	oveq2d 7291	. . . . . 6 ⊢ (𝑥 = (𝑁 + 1) → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1))))
28	25, 27	eqeq12d 2754	. . . . 5 ⊢ (𝑥 = (𝑁 + 1) → ((seq𝑀( · , 𝐹)‘𝑥) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥)) ↔ (seq𝑀( · , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1)))))
29	28	imbi2d 341	. . . 4 ⊢ (𝑥 = (𝑁 + 1) → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑥) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥))) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1))))))
30		fveq2 6774	. . . . . 6 ⊢ (𝑥 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘𝑛))
31		fveq2 6774	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘𝑛))
32	31	oveq2d 7291	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛)))
33	30, 32	eqeq12d 2754	. . . . 5 ⊢ (𝑥 = 𝑛 → ((seq𝑀( · , 𝐹)‘𝑥) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥)) ↔ (seq𝑀( · , 𝐹)‘𝑛) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛))))
34	33	imbi2d 341	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑥) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥))) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘𝑛) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛)))))
35		fveq2 6774	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘(𝑛 + 1)))
36		fveq2 6774	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1)))
37	36	oveq2d 7291	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1))))
38	35, 37	eqeq12d 2754	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((seq𝑀( · , 𝐹)‘𝑥) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥)) ↔ (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1)))))
39	38	imbi2d 341	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑥) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥))) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1))))))
40		fveq2 6774	. . . . . 6 ⊢ (𝑥 = 𝑘 → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘𝑘))
41		fveq2 6774	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘𝑘))
42	41	oveq2d 7291	. . . . . 6 ⊢ (𝑥 = 𝑘 → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑘)))
43	40, 42	eqeq12d 2754	. . . . 5 ⊢ (𝑥 = 𝑘 → ((seq𝑀( · , 𝐹)‘𝑥) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥)) ↔ (seq𝑀( · , 𝐹)‘𝑘) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑘))))
44	43	imbi2d 341	. . . 4 ⊢ (𝑥 = 𝑘 → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑥) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥))) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘𝑘) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑘)))))
45	9	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁 + 1) ∈ ℤ) → 𝑁 ∈ (ℤ≥‘𝑀))
46		seqp1 13736	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (seq𝑀( · , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (𝐹‘(𝑁 + 1))))
47	45, 46	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁 + 1) ∈ ℤ) → (seq𝑀( · , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (𝐹‘(𝑁 + 1))))
48		seq1 13734	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℤ → (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1)) = (𝐹‘(𝑁 + 1)))
49	48	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁 + 1) ∈ ℤ) → (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1)) = (𝐹‘(𝑁 + 1)))
50	49	oveq2d 7291	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁 + 1) ∈ ℤ) → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1))) = ((seq𝑀( · , 𝐹)‘𝑁) · (𝐹‘(𝑁 + 1))))
51	47, 50	eqtr4d 2781	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 + 1) ∈ ℤ) → (seq𝑀( · , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1))))
52	51	expcom 414	. . . 4 ⊢ ((𝑁 + 1) ∈ ℤ → (𝜑 → (seq𝑀( · , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1)))))
53	19	sselda 3921	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑛 ∈ (ℤ≥‘𝑀))
54		seqp1 13736	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
55	53, 54	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
56	55	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (seq𝑀( · , 𝐹)‘𝑛) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛))) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
57		oveq1 7282	. . . . . . . . 9 ⊢ ((seq𝑀( · , 𝐹)‘𝑛) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛)) → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛)) · (𝐹‘(𝑛 + 1))))
58	57	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (seq𝑀( · , 𝐹)‘𝑛) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛))) → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛)) · (𝐹‘(𝑛 + 1))))
59	14	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( · , 𝐹)‘𝑁) ∈ ℂ)
60	23	ffvelrnda 6961	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘𝑛) ∈ ℂ)
61		peano2uz 12641	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ (ℤ≥‘𝑀))
62	61, 2	eleqtrrdi 2850	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ 𝑍)
63	53, 62	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (𝑛 + 1) ∈ 𝑍)
64	12	ralrimiva 3103	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
65		fveq2 6774	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
66	65	eleq1d 2823	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
67	66	rspcv 3557	. . . . . . . . . . . . 13 ⊢ ((𝑛 + 1) ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → (𝐹‘(𝑛 + 1)) ∈ ℂ))
68	64, 67	mpan9 507	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
69	63, 68	syldan 591	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
70	59, 60, 69	mulassd 10998	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛)) · (𝐹‘(𝑛 + 1))) = ((seq𝑀( · , 𝐹)‘𝑁) · ((seq(𝑁 + 1)( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
71	70	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (seq𝑀( · , 𝐹)‘𝑛) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛))) → (((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛)) · (𝐹‘(𝑛 + 1))) = ((seq𝑀( · , 𝐹)‘𝑁) · ((seq(𝑁 + 1)( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
72		seqp1 13736	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1)) = ((seq(𝑁 + 1)( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
73	72	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1)) = ((seq(𝑁 + 1)( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
74	73	oveq2d 7291	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1))) = ((seq𝑀( · , 𝐹)‘𝑁) · ((seq(𝑁 + 1)( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
75	74	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (seq𝑀( · , 𝐹)‘𝑛) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛))) → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1))) = ((seq𝑀( · , 𝐹)‘𝑁) · ((seq(𝑁 + 1)( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
76	71, 75	eqtr4d 2781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (seq𝑀( · , 𝐹)‘𝑛) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛))) → (((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛)) · (𝐹‘(𝑛 + 1))) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1))))
77	56, 58, 76	3eqtrd 2782	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (seq𝑀( · , 𝐹)‘𝑛) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛))) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1))))
78	77	exp31 420	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → ((seq𝑀( · , 𝐹)‘𝑛) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛)) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1))))))
79	78	com12 32	. . . . 5 ⊢ (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → (𝜑 → ((seq𝑀( · , 𝐹)‘𝑛) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛)) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1))))))
80	79	a2d 29	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑛) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛))) → (𝜑 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1))))))
81	29, 34, 39, 44, 52, 80	uzind4 12646	. . 3 ⊢ (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝜑 → (seq𝑀( · , 𝐹)‘𝑘) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑘))))
82	81	impcom 408	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( · , 𝐹)‘𝑘) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑘)))
83	1, 7, 8, 14, 16, 24, 82	climmulc2 15346	1 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ ((seq𝑀( · , 𝐹)‘𝑁) · 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ⊆ wss 3887   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  ℂcc 10869  1c1 10872   + caddc 10874   · cmul 10876  ℤcz 12319  ℤ_≥cuz 12582  seqcseq 13721   ⇝ cli 15193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-seq 13722  df-exp 13783  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197

This theorem is referenced by:  ntrivcvg  15609