Theorem clim2prod 15795

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 2731	. 2 ⊢ (ℤ≥‘(𝑁 + 1)) = (ℤ≥‘(𝑁 + 1))
2		clim2prod.1	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
3		uzssz 12753	. . . . 5 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
4	2, 3	eqsstri 3976	. . . 4 ⊢ 𝑍 ⊆ ℤ
5		clim2prod.2	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
6	4, 5	sselid 3927	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7	6	peano2zd 12580	. 2 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
8		clim2prod.4	. 2 ⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹) ⇝ 𝐴)
9	5, 2	eleqtrdi 2841	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
10		eluzel2 12737	. . . . 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 15794	. . 3 ⊢ (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ)
14	13, 5	ffvelcdmd 7018	. 2 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ∈ ℂ)
15		seqex 13910	. . 3 ⊢ seq𝑀( · , 𝐹) ∈ V
16	15	a1i 11	. 2 ⊢ (𝜑 → seq𝑀( · , 𝐹) ∈ V)
17		peano2uz 12799	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀))
18		uzss 12755	. . . . . . . 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 3971	. . . . . 6 ⊢ (𝜑 → (ℤ≥‘(𝑁 + 1)) ⊆ 𝑍)
21	20	sselda 3929	. . . . 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 15794	. . 3 ⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹):(ℤ≥‘(𝑁 + 1))⟶ℂ)
24	23	ffvelcdmda 7017	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘𝑘) ∈ ℂ)
25		fveq2 6822	. . . . . 6 ⊢ (𝑥 = (𝑁 + 1) → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘(𝑁 + 1)))
26		fveq2 6822	. . . . . . 7 ⊢ (𝑥 = (𝑁 + 1) → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1)))
27	26	oveq2d 7362	. . . . . 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 2747	. . . . 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 340	. . . 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 6822	. . . . . 6 ⊢ (𝑥 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘𝑛))
31		fveq2 6822	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘𝑛))
32	31	oveq2d 7362	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛)))
33	30, 32	eqeq12d 2747	. . . . 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 340	. . . 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 6822	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘(𝑛 + 1)))
36		fveq2 6822	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1)))
37	36	oveq2d 7362	. . . . . 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 2747	. . . . 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 340	. . . 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 6822	. . . . . 6 ⊢ (𝑥 = 𝑘 → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘𝑘))
41		fveq2 6822	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘𝑘))
42	41	oveq2d 7362	. . . . . 6 ⊢ (𝑥 = 𝑘 → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑘)))
43	40, 42	eqeq12d 2747	. . . . 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 340	. . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁 + 1) ∈ ℤ) → 𝑁 ∈ (ℤ≥‘𝑀))
46		seqp1 13923	. . . . . . 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 13921	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℤ → (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1)) = (𝐹‘(𝑁 + 1)))
49	48	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁 + 1) ∈ ℤ) → (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1)) = (𝐹‘(𝑁 + 1)))
50	49	oveq2d 7362	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁 + 1) ∈ ℤ) → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1))) = ((seq𝑀( · , 𝐹)‘𝑁) · (𝐹‘(𝑁 + 1))))
51	47, 50	eqtr4d 2769	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 + 1) ∈ ℤ) → (seq𝑀( · , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1))))
52	51	expcom 413	. . . 4 ⊢ ((𝑁 + 1) ∈ ℤ → (𝜑 → (seq𝑀( · , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1)))))
53	19	sselda 3929	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑛 ∈ (ℤ≥‘𝑀))
54		seqp1 13923	. . . . . . . . . 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 480	. . . . . . . 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 7353	. . . . . . . . 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 481	. . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( · , 𝐹)‘𝑁) ∈ ℂ)
60	23	ffvelcdmda 7017	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘𝑛) ∈ ℂ)
61		peano2uz 12799	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ (ℤ≥‘𝑀))
62	61, 2	eleqtrrdi 2842	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ 𝑍)
63	53, 62	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (𝑛 + 1) ∈ 𝑍)
64	12	ralrimiva 3124	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
65		fveq2 6822	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
66	65	eleq1d 2816	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
67	66	rspcv 3568	. . . . . . . . . . . . 13 ⊢ ((𝑛 + 1) ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → (𝐹‘(𝑛 + 1)) ∈ ℂ))
68	64, 67	mpan9 506	. . . . . . . . . . . 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 11135	. . . . . . . . . 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 480	. . . . . . . . 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 13923	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1)) = ((seq(𝑁 + 1)( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
73	72	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1)) = ((seq(𝑁 + 1)( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
74	73	oveq2d 7362	. . . . . . . . . 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 480	. . . . . . . . 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 2769	. . . . . . . 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 2770	. . . . . . 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 419	. . . . . 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 12804	. . 3 ⊢ (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝜑 → (seq𝑀( · , 𝐹)‘𝑘) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑘))))
82	81	impcom 407	. 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 15544	1 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ ((seq𝑀( · , 𝐹)‘𝑁) · 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ⊆ wss 3897   class class class wbr 5089  ‘cfv 6481  (class class class)co 7346  ℂcc 11004  1c1 11007   + caddc 11009   · cmul 11011  ℤcz 12468  ℤ_≥cuz 12732  seqcseq 13908   ⇝ cli 15391

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-sup 9326  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-n0 12382  df-z 12469  df-uz 12733  df-rp 12891  df-fz 13408  df-seq 13909  df-exp 13969  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395

This theorem is referenced by:  ntrivcvg  15804