Theorem clim2prod 11685

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 2193	. 2 ⊢ (ℤ≥‘(𝑁 + 1)) = (ℤ≥‘(𝑁 + 1))
2		clim2prod.1	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
3		uzssz 9615	. . . . 5 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
4	2, 3	eqsstri 3212	. . . 4 ⊢ 𝑍 ⊆ ℤ
5		clim2prod.2	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
6	4, 5	sselid 3178	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7	6	peano2zd 9445	. 2 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
8		clim2prod.4	. 2 ⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹) ⇝ 𝐴)
9	5, 2	eleqtrdi 2286	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
10		eluzel2 9600	. . . . 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 11684	. . 3 ⊢ (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ)
14	13, 5	ffvelcdmd 5695	. 2 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ∈ ℂ)
15		seqex 10523	. . 3 ⊢ seq𝑀( · , 𝐹) ∈ V
16	15	a1i 9	. 2 ⊢ (𝜑 → seq𝑀( · , 𝐹) ∈ V)
17		peano2uz 9651	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀))
18		uzss 9616	. . . . . . . 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 3229	. . . . . 6 ⊢ (𝜑 → (ℤ≥‘(𝑁 + 1)) ⊆ 𝑍)
21	20	sselda 3180	. . . . 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 11684	. . 3 ⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹):(ℤ≥‘(𝑁 + 1))⟶ℂ)
24	23	ffvelcdmda 5694	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘𝑘) ∈ ℂ)
25		fveq2 5555	. . . . . 6 ⊢ (𝑥 = (𝑁 + 1) → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘(𝑁 + 1)))
26		fveq2 5555	. . . . . . 7 ⊢ (𝑥 = (𝑁 + 1) → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1)))
27	26	oveq2d 5935	. . . . . 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 2208	. . . . 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 5555	. . . . . 6 ⊢ (𝑥 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘𝑛))
31		fveq2 5555	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘𝑛))
32	31	oveq2d 5935	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑛)))
33	30, 32	eqeq12d 2208	. . . . 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 5555	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘(𝑛 + 1)))
36		fveq2 5555	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1)))
37	36	oveq2d 5935	. . . . . 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 2208	. . . . 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 5555	. . . . . 6 ⊢ (𝑥 = 𝑘 → (seq𝑀( · , 𝐹)‘𝑥) = (seq𝑀( · , 𝐹)‘𝑘))
41		fveq2 5555	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (seq(𝑁 + 1)( · , 𝐹)‘𝑥) = (seq(𝑁 + 1)( · , 𝐹)‘𝑘))
42	41	oveq2d 5935	. . . . . 6 ⊢ (𝑥 = 𝑘 → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑥)) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑘)))
43	40, 42	eqeq12d 2208	. . . . 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 2260	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ≥‘𝑀))
46	45, 12	sylan2br 288	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
47		mulcl 8001	. . . . . . . 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 10539	. . . . . 6 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( · , 𝐹)‘𝑁) · (𝐹‘(𝑁 + 1))))
50	7, 22, 48	seq3-1 10536	. . . . . . 7 ⊢ (𝜑 → (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1)) = (𝐹‘(𝑁 + 1)))
51	50	oveq2d 5935	. . . . . 6 ⊢ (𝜑 → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘(𝑁 + 1))) = ((seq𝑀( · , 𝐹)‘𝑁) · (𝐹‘(𝑁 + 1))))
52	49, 51	eqtr4d 2229	. . . . 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 3180	. . . . . . . . . 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 10539	. . . . . . . . 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 5926	. . . . . . . . 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 5694	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘𝑛) ∈ ℂ)
63		peano2uz 9651	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ (ℤ≥‘𝑀))
64	63, 2	eleqtrrdi 2287	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ 𝑍)
65	54, 64	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (𝑛 + 1) ∈ 𝑍)
66	12	ralrimiva 2567	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
67		fveq2 5555	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
68	67	eleq1d 2262	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
69	68	rspcv 2861	. . . . . . . . . . . . 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 8045	. . . . . . . . . 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 10539	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘(𝑛 + 1)) = ((seq(𝑁 + 1)( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
77	76	oveq2d 5935	. . . . . . . . . 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 2229	. . . . . . . 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 2230	. . . . . . 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 9656	. . 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 11477	1 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ ((seq𝑀( · , 𝐹)‘𝑁) · 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∀wral 2472  Vcvv 2760   ⊆ wss 3154   class class class wbr 4030  ‘cfv 5255  (class class class)co 5919  ℂcc 7872  1c1 7875   + caddc 7877   · cmul 7879  ℤcz 9320  ℤ_≥cuz 9595  seqcseq 10521   ⇝ cli 11424

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-rp 9723  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425

This theorem is referenced by:  ntrivcvgap  11694