Theorem prodfrecap 11509

Description: The reciprocal of a finite product. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)

Hypotheses

Ref	Expression
prodfap0.1	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
prodfap0.2	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
prodfap0.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) # 0)
prodfrec.4	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) = (1 / (𝐹‘𝑘)))
prodfrecap.g	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ)

Assertion

Ref	Expression
prodfrecap	⊢ (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))

Distinct variable groups:   𝑘,𝐹   𝑘,𝑀   𝑘,𝑁   𝜑,𝑘   𝑘,𝐺

Proof of Theorem prodfrecap

Dummy variables 𝑛 𝑣 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prodfap0.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzfz2 9988	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		fveq2 5496	. . . . 5 ⊢ (𝑚 = 𝑀 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑀))
5		fveq2 5496	. . . . . 6 ⊢ (𝑚 = 𝑀 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑀))
6	5	oveq2d 5869	. . . . 5 ⊢ (𝑚 = 𝑀 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑀)))
7	4, 6	eqeq12d 2185	. . . 4 ⊢ (𝑚 = 𝑀 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀))))
8	7	imbi2d 229	. . 3 ⊢ (𝑚 = 𝑀 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀)))))
9		fveq2 5496	. . . . 5 ⊢ (𝑚 = 𝑛 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑛))
10		fveq2 5496	. . . . . 6 ⊢ (𝑚 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑛))
11	10	oveq2d 5869	. . . . 5 ⊢ (𝑚 = 𝑛 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑛)))
12	9, 11	eqeq12d 2185	. . . 4 ⊢ (𝑚 = 𝑛 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))))
13	12	imbi2d 229	. . 3 ⊢ (𝑚 = 𝑛 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)))))
14		fveq2 5496	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘(𝑛 + 1)))
15		fveq2 5496	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘(𝑛 + 1)))
16	15	oveq2d 5869	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))
17	14, 16	eqeq12d 2185	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1)))))
18	17	imbi2d 229	. . 3 ⊢ (𝑚 = (𝑛 + 1) → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
19		fveq2 5496	. . . . 5 ⊢ (𝑚 = 𝑁 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑁))
20		fveq2 5496	. . . . . 6 ⊢ (𝑚 = 𝑁 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑁))
21	20	oveq2d 5869	. . . . 5 ⊢ (𝑚 = 𝑁 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))
22	19, 21	eqeq12d 2185	. . . 4 ⊢ (𝑚 = 𝑁 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁))))
23	22	imbi2d 229	. . 3 ⊢ (𝑚 = 𝑁 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))))
24		eluzfz1 9987	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
25	1, 24	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
26		fveq2 5496	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝐺‘𝑘) = (𝐺‘𝑀))
27		fveq2 5496	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
28	27	oveq2d 5869	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (1 / (𝐹‘𝑘)) = (1 / (𝐹‘𝑀)))
29	26, 28	eqeq12d 2185	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → ((𝐺‘𝑘) = (1 / (𝐹‘𝑘)) ↔ (𝐺‘𝑀) = (1 / (𝐹‘𝑀))))
30	29	imbi2d 229	. . . . . . 7 ⊢ (𝑘 = 𝑀 → ((𝜑 → (𝐺‘𝑘) = (1 / (𝐹‘𝑘))) ↔ (𝜑 → (𝐺‘𝑀) = (1 / (𝐹‘𝑀)))))
31		prodfrec.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) = (1 / (𝐹‘𝑘)))
32	31	expcom 115	. . . . . . 7 ⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐺‘𝑘) = (1 / (𝐹‘𝑘))))
33	30, 32	vtoclga 2796	. . . . . 6 ⊢ (𝑀 ∈ (𝑀...𝑁) → (𝜑 → (𝐺‘𝑀) = (1 / (𝐹‘𝑀))))
34	25, 33	mpcom 36	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑀) = (1 / (𝐹‘𝑀)))
35		eluzel2 9492	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
36	1, 35	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
37		prodfrecap.g	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ)
38		mulcl 7901	. . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑘 · 𝑣) ∈ ℂ)
39	38	adantl 275	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑘 · 𝑣) ∈ ℂ)
40	36, 37, 39	seq3-1 10416	. . . . 5 ⊢ (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (𝐺‘𝑀))
41		prodfap0.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
42	36, 41, 39	seq3-1 10416	. . . . . 6 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹‘𝑀))
43	42	oveq2d 5869	. . . . 5 ⊢ (𝜑 → (1 / (seq𝑀( · , 𝐹)‘𝑀)) = (1 / (𝐹‘𝑀)))
44	34, 40, 43	3eqtr4d 2213	. . . 4 ⊢ (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀)))
45	44	a1i 9	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀))))
46		oveq1 5860	. . . . . . . . 9 ⊢ ((seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)) → ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))) = ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))))
47	46	3ad2ant3 1015	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))) = ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))))
48		fzofzp1 10183	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
49		fveq2 5496	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑛 + 1)))
50		fveq2 5496	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
51	50	oveq2d 5869	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → (1 / (𝐹‘𝑘)) = (1 / (𝐹‘(𝑛 + 1))))
52	49, 51	eqeq12d 2185	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → ((𝐺‘𝑘) = (1 / (𝐹‘𝑘)) ↔ (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1)))))
53	52	imbi2d 229	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐺‘𝑘) = (1 / (𝐹‘𝑘))) ↔ (𝜑 → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1))))))
54	53, 32	vtoclga 2796	. . . . . . . . . . . . 13 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1)))))
55	48, 54	syl 14	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1)))))
56	55	impcom 124	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1))))
57	56	oveq2d 5869	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))) = ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))))
58		1cnd 7936	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ)
59		eqid 2170	. . . . . . . . . . . . . . 15 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
60	59, 36, 41	prodf 11501	. . . . . . . . . . . . . 14 ⊢ (𝜑 → seq𝑀( · , 𝐹):(ℤ≥‘𝑀)⟶ℂ)
61	60	adantr 274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → seq𝑀( · , 𝐹):(ℤ≥‘𝑀)⟶ℂ)
62		elfzouz 10107	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
63	62	adantl 275	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ≥‘𝑀))
64	61, 63	ffvelrnd 5632	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ)
65	50	eleq1d 2239	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
66	65	imbi2d 229	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹‘𝑘) ∈ ℂ) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ)))
67		elfzuz 9977	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀))
68	41	expcom 115	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜑 → (𝐹‘𝑘) ∈ ℂ))
69	67, 68	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑘) ∈ ℂ))
70	66, 69	vtoclga 2796	. . . . . . . . . . . . . 14 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))
71	48, 70	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))
72	71	impcom 124	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
73	41	adantlr 474	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
74		elfzouz2 10117	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝑛))
75		fzss2 10020	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ≥‘𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
76	74, 75	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
77	76	sselda 3147	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (𝑀..^𝑁) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ (𝑀...𝑁))
78		prodfap0.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) # 0)
79	77, 78	sylan2 284	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑀..^𝑁) ∧ 𝑘 ∈ (𝑀...𝑛))) → (𝐹‘𝑘) # 0)
80	79	anassrs 398	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) # 0)
81	63, 73, 80	prodfap0 11508	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) # 0)
82	50	breq1d 3999	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) # 0 ↔ (𝐹‘(𝑛 + 1)) # 0))
83	82	imbi2d 229	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹‘𝑘) # 0) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) # 0)))
84	78	expcom 115	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑘) # 0))
85	83, 84	vtoclga 2796	. . . . . . . . . . . . . 14 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) # 0))
86	48, 85	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) # 0))
87	86	impcom 124	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) # 0)
88	58, 64, 58, 72, 81, 87	divmuldivapd 8749	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))) = ((1 · 1) / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
89		1t1e1 9030	. . . . . . . . . . . 12 ⊢ (1 · 1) = 1
90	89	oveq1i 5863	. . . . . . . . . . 11 ⊢ ((1 · 1) / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
91	88, 90	eqtrdi 2219	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
92	57, 91	eqtrd 2203	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
93	92	3adant3 1012	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
94	47, 93	eqtrd 2203	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
95	63	3adant3 1012	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → 𝑛 ∈ (ℤ≥‘𝑀))
96	37	3ad2antl1 1154	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ)
97	38	adantl 275	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) ∧ (𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑘 · 𝑣) ∈ ℂ)
98	95, 96, 97	seq3p1 10418	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))))
99	41	3ad2antl1 1154	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
100	95, 99, 97	seq3p1 10418	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
101	100	oveq2d 5869	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
102	94, 98, 101	3eqtr4d 2213	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))
103	102	3exp 1197	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (𝑀..^𝑁) → ((seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
104	103	com12 30	. . . 4 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
105	104	a2d 26	. . 3 ⊢ (𝑛 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (𝜑 → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
106	8, 13, 18, 23, 45, 105	fzind2 10195	. 2 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁))))
107	3, 106	mpcom 36	1 ⊢ (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 973   = wceq 1348   ∈ wcel 2141   ⊆ wss 3121   class class class wbr 3989  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853  ℂcc 7772  0cc0 7774  1c1 7775   + caddc 7777   · cmul 7779   # cap 8500   / cdiv 8589  ℤcz 9212  ℤ_≥cuz 9487  ...cfz 9965  ..^cfzo 10098  seqcseq 10401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966  df-fzo 10099  df-seqfrec 10402

This theorem is referenced by:  prodfdivap  11510