Theorem clim2prod 11792

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

Step	Hyp	Ref	Expression
1		eqid 2204	. 2 ⊢ (ℤ≥‘(𝑁 + 1)) = (ℤ≥‘(𝑁 + 1))
2		clim2prod.1	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
3		uzssz 9667	. . . . 5 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
4	2, 3	eqsstri 3224	. . . 4 ⊢ 𝑍 ⊆ ℤ
5		clim2prod.2	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
6	4, 5	sselid 3190	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7	6	peano2zd 9497	. 2 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
8		clim2prod.4	. 2 ⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹) ⇝ 𝐴)
9	5, 2	eleqtrdi 2297	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
10		eluzel2 9652	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
11	9, 10	syl 14	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
12		clim2prod.3	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
13	2, 11, 12	prodf 11791	. . 3 ⊢ (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ)
14	13, 5	ffvelcdmd 5715	. 2 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ∈ ℂ)
15		seqex 10592	. . 3 ⊢ seq𝑀( · , 𝐹) ∈ V
16	15	a1i 9	. 2 ⊢ (𝜑 → seq𝑀( · , 𝐹) ∈ V)
17		peano2uz 9703	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀))
18		uzss 9668	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘𝑀) → (ℤ≥‘(𝑁 + 1)) ⊆ (ℤ≥‘𝑀))
19	9, 17, 18	3syl 17	. . . . . . 7 ⊢ (𝜑 → (ℤ≥‘(𝑁 + 1)) ⊆ (ℤ≥‘𝑀))
20	19, 2	sseqtrrdi 3241	. . . . . 6 ⊢ (𝜑 → (ℤ≥‘(𝑁 + 1)) ⊆ 𝑍)
21	20	sselda 3192	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍)
22	21, 12	syldan 282	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → (𝐹‘𝑘) ∈ ℂ)
23	1, 7, 22	prodf 11791	. . 3 ⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹):(ℤ≥‘(𝑁 + 1))⟶ℂ)
24	23	ffvelcdmda 5714	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘𝑘) ∈ ℂ)
25		fveq2 5575	. . . . . 6 ⊢ (𝑥 = (𝑁 + 1) → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘(𝑁 + 1)))
26		fveq2 5575	. . . . . . 7 ⊢ (𝑥 = (𝑁 + 1) → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1)))
27	26	oveq2d 5959	. . . . . 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 2219	. . . . 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 230	. . . 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 5575	. . . . . 6 ⊢ (𝑥 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘𝑛))
31		fveq2 5575	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘𝑛))
32	31	oveq2d 5959	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛)))
33	30, 32	eqeq12d 2219	. . . . 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 230	. . . 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 5575	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘(𝑛 + 1)))
36		fveq2 5575	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1)))
37	36	oveq2d 5959	. . . . . 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 2219	. . . . 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 230	. . . 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 5575	. . . . . 6 ⊢ (𝑥 = 𝑘 → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘𝑘))
41		fveq2 5575	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘𝑘))
42	41	oveq2d 5959	. . . . . 6 ⊢ (𝑥 = 𝑘 → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑘)))
43	40, 42	eqeq12d 2219	. . . . 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 230	. . . 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	2	eleq2i 2271	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ≥‘𝑀))
46	45, 12	sylan2br 288	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
47		mulcl 8051	. . . . . . . 8 ⊢ ((𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑘 · 𝑣) ∈ ℂ)
48	47	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑘 · 𝑣) ∈ ℂ)
49	9, 46, 48	seq3p1 10608	. . . . . 6 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (𝐹‘(𝑁 + 1))))
50	7, 22, 48	seq3-1 10605	. . . . . . 7 ⊢ (𝜑 → (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1)) = (𝐹‘(𝑁 + 1)))
51	50	oveq2d 5959	. . . . . 6 ⊢ (𝜑 → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1))) = ((seq𝑀( · , 𝐹)‘𝑁) · (𝐹‘(𝑁 + 1))))
52	49, 51	eqtr4d 2240	. . . . 5 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1))))
53	52	a1i 9	. . . 4 ⊢ ((𝑁 + 1) ∈ ℤ → (𝜑 → (seq𝑀( · , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1)))))
54	19	sselda 3192	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑛 ∈ (ℤ≥‘𝑀))
55	46	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
56	47	adantl 277	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑘 · 𝑣) ∈ ℂ)
57	54, 55, 56	seq3p1 10608	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
58	57	adantr 276	. . . . . . . 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))))
59		oveq1 5950	. . . . . . . . 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))))
60	59	adantl 277	. . . . . . . 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))))
61	14	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( · , 𝐹)‘𝑁) ∈ ℂ)
62	23	ffvelcdmda 5714	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘𝑛) ∈ ℂ)
63		peano2uz 9703	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ (ℤ≥‘𝑀))
64	63, 2	eleqtrrdi 2298	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ 𝑍)
65	54, 64	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (𝑛 + 1) ∈ 𝑍)
66	12	ralrimiva 2578	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
67		fveq2 5575	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
68	67	eleq1d 2273	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
69	68	rspcv 2872	. . . . . . . . . . . . 13 ⊢ ((𝑛 + 1) ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → (𝐹‘(𝑛 + 1)) ∈ ℂ))
70	66, 69	mpan9 281	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
71	65, 70	syldan 282	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
72	61, 62, 71	mulassd 8095	. . . . . . . . . 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)))))
73	72	adantr 276	. . . . . . . . 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)))))
74		simpr 110	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))
75	22	adantlr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → (𝐹‘𝑘) ∈ ℂ)
76	74, 75, 56	seq3p1 10608	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1)) = ((seq(𝑁 + 1)( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
77	76	oveq2d 5959	. . . . . . . . . 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)))))
78	77	adantr 276	. . . . . . . . 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)))))
79	73, 78	eqtr4d 2240	. . . . . . . 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))))
80	58, 60, 79	3eqtrd 2241	. . . . . . 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))))
81	80	exp31 364	. . . . . 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))))))
82	81	com12 30	. . . . 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))))))
83	82	a2d 26	. . . 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))))))
84	29, 34, 39, 44, 53, 83	uzind4 9708	. . 3 ⊢ (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝜑 → (seq𝑀( · , 𝐹)‘𝑘) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑘))))
85	84	impcom 125	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( · , 𝐹)‘𝑘) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑘)))
86	1, 7, 8, 14, 16, 24, 85	climmulc2 11584	1 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ ((seq𝑀( · , 𝐹)‘𝑁) · 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1372   ∈ wcel 2175  ∀wral 2483  Vcvv 2771   ⊆ wss 3165   class class class wbr 4043  ‘cfv 5270  (class class class)co 5943  ℂcc 7922  1c1 7925   + caddc 7927   · cmul 7929  ℤcz 9371  ℤ_≥cuz 9647  seqcseq 10590   ⇝ cli 11531

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-rp 9775  df-seqfrec 10591  df-exp 10682  df-cj 11095  df-re 11096  df-im 11097  df-rsqrt 11251  df-abs 11252  df-clim 11532

This theorem is referenced by:  ntrivcvgap  11801