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Theorem prodfrec 15937

Description: The reciprocal of an infinite product. (Contributed by Scott Fenton, 15-Jan-2018.)

**Hypotheses**
Ref	Expression
prodfn0.1	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
prodfn0.2	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℂ)
prodfn0.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ≠ 0)
prodfrec.4	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) = (1 / (𝐹‘𝑘)))

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

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

Proof of Theorem prodfrec Dummy variables 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prodfn0.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		eluzfz2 13548	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 18	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		fveq2 6871	. . . . 5 ⊢ (𝑚 = 𝑀 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑀))
5		fveq2 6871	. . . . . 6 ⊢ (𝑚 = 𝑀 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑀))
6	5	oveq2d 7416	. . . . 5 ⊢ (𝑚 = 𝑀 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑀)))
7	4, 6	eqeq12d 2781	. . . 4 ⊢ (𝑚 = 𝑀 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀))))
8	7	imbi2d 343	. . 3 ⊢ (𝑚 = 𝑀 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀)))))
9		fveq2 6871	. . . . 5 ⊢ (𝑚 = 𝑛 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑛))
10		fveq2 6871	. . . . . 6 ⊢ (𝑚 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑛))
11	10	oveq2d 7416	. . . . 5 ⊢ (𝑚 = 𝑛 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑛)))
12	9, 11	eqeq12d 2781	. . . 4 ⊢ (𝑚 = 𝑛 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))))
13	12	imbi2d 343	. . 3 ⊢ (𝑚 = 𝑛 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)))))
14		fveq2 6871	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘(𝑛 + 1)))
15		fveq2 6871	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘(𝑛 + 1)))
16	15	oveq2d 7416	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))
17	14, 16	eqeq12d 2781	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1)))))
18	17	imbi2d 343	. . 3 ⊢ (𝑚 = (𝑛 + 1) → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
19		fveq2 6871	. . . . 5 ⊢ (𝑚 = 𝑁 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑁))
20		fveq2 6871	. . . . . 6 ⊢ (𝑚 = 𝑁 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑁))
21	20	oveq2d 7416	. . . . 5 ⊢ (𝑚 = 𝑁 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))
22	19, 21	eqeq12d 2781	. . . 4 ⊢ (𝑚 = 𝑁 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁))))
23	22	imbi2d 343	. . 3 ⊢ (𝑚 = 𝑁 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))))
24		eluzfz1 13547	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
25	1, 24	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
26		fveq2 6871	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝐺‘𝑘) = (𝐺‘𝑀))
27		fveq2 6871	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
28	27	oveq2d 7416	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (1 / (𝐹‘𝑘)) = (1 / (𝐹‘𝑀)))
29	26, 28	eqeq12d 2781	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → ((𝐺‘𝑘) = (1 / (𝐹‘𝑘)) ↔ (𝐺‘𝑀) = (1 / (𝐹‘𝑀))))
30	29	imbi2d 343	. . . . . . 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 418	. . . . . . 7 ⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐺‘𝑘) = (1 / (𝐹‘𝑘))))
33	30, 32	vtoclga 3544	. . . . . 6 ⊢ (𝑀 ∈ (𝑀...𝑁) → (𝜑 → (𝐺‘𝑀) = (1 / (𝐹‘𝑀))))
34	25, 33	mpcom 39	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑀) = (1 / (𝐹‘𝑀)))
35		eluzel2 12855	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
36	1, 35	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
37		seq1 14038	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (seq𝑀( · , 𝐺)‘𝑀) = (𝐺‘𝑀))
38	36, 37	syl 18	. . . . 5 ⊢ (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (𝐺‘𝑀))
39		seq1 14038	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹‘𝑀))
40	36, 39	syl 18	. . . . . 6 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹‘𝑀))
41	40	oveq2d 7416	. . . . 5 ⊢ (𝜑 → (1 / (seq𝑀( · , 𝐹)‘𝑀)) = (1 / (𝐹‘𝑀)))
42	34, 38, 41	3eqtr4d 2810	. . . 4 ⊢ (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀)))
43	42	a1i 11	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀))))
44		oveq1 7407	. . . . . . . . 9 ⊢ ((seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)) → ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))) = ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))))
45	44	3ad2ant3 1151	. . . . . . . 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))))
46		fzofzp1 13781	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
47		fveq2 6871	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑛 + 1)))
48		fveq2 6871	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
49	48	oveq2d 7416	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → (1 / (𝐹‘𝑘)) = (1 / (𝐹‘(𝑛 + 1))))
50	47, 49	eqeq12d 2781	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → ((𝐺‘𝑘) = (1 / (𝐹‘𝑘)) ↔ (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1)))))
51	50	imbi2d 343	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐺‘𝑘) = (1 / (𝐹‘𝑘))) ↔ (𝜑 → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1))))))
52	51, 32	vtoclga 3544	. . . . . . . . . . . . 13 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1)))))
53	46, 52	syl 18	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1)))))
54	53	impcom 412	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1))))
55	54	oveq2d 7416	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))) = ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))))
56		1cnd 11190	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ)
57		elfzouz 13680	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ_≥‘𝑀))
58	57	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ_≥‘𝑀))
59		elfzouz2 13691	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ_≥‘𝑛))
60		fzss2 13580	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
61	59, 60	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
62	61	sselda 3939	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (𝑀..^𝑁) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ (𝑀...𝑁))
63		prodfn0.2	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℂ)
64	62, 63	sylan2 604	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑀..^𝑁) ∧ 𝑘 ∈ (𝑀...𝑛))) → (𝐹‘𝑘) ∈ ℂ)
65	64	anassrs 472	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℂ)
66		mulcl 11172	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 · 𝑥) ∈ ℂ)
67	66	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 · 𝑥) ∈ ℂ)
68	58, 65, 67	seqcl 14046	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ)
69	48	eleq1d 2850	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
70	69	imbi2d 343	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹‘𝑘) ∈ ℂ) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ)))
71	63	expcom 418	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑘) ∈ ℂ))
72	70, 71	vtoclga 3544	. . . . . . . . . . . . . 14 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))
73	46, 72	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))
74	73	impcom 412	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
75		prodfn0.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ≠ 0)
76	62, 75	sylan2 604	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑀..^𝑁) ∧ 𝑘 ∈ (𝑀...𝑛))) → (𝐹‘𝑘) ≠ 0)
77	76	anassrs 472	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ≠ 0)
78	58, 65, 77	prodfn0 15936	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ≠ 0)
79	48	neeq1d 3019	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ≠ 0 ↔ (𝐹‘(𝑛 + 1)) ≠ 0))
80	79	imbi2d 343	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹‘𝑘) ≠ 0) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) ≠ 0)))
81	75	expcom 418	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑘) ≠ 0))
82	80, 81	vtoclga 3544	. . . . . . . . . . . . . 14 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ≠ 0))
83	46, 82	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ≠ 0))
84	83	impcom 412	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ≠ 0)
85	56, 68, 56, 74, 78, 84	divmuldivd 12020	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))) = ((1 · 1) / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
86		1t1e1 12390	. . . . . . . . . . . 12 ⊢ (1 · 1) = 1
87	86	oveq1i 7410	. . . . . . . . . . 11 ⊢ ((1 · 1) / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
88	85, 87	eqtrdi 2816	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
89	55, 88	eqtrd 2800	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
90	89	3adant3 1148	. . . . . . . 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)))))
91	45, 90	eqtrd 2800	. . . . . . 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)))))
92		seqp1 14040	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))))
93	57, 92	syl 18	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))))
94	93	3ad2ant2 1150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))))
95		seqp1 14040	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
96	57, 95	syl 18	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
97	96	oveq2d 7416	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
98	97	3ad2ant2 1150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
99	91, 94, 98	3eqtr4d 2810	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))
100	99	3exp 1135	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (𝑀..^𝑁) → ((seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
101	100	com12 33	. . . 4 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
102	101	a2d 30	. . 3 ⊢ (𝑛 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (𝜑 → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
103	8, 13, 18, 23, 43, 102	fzind2 13805	. 2 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁))))
104	3, 103	mpcom 39	1 ⊢ (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 ⊆ wss 3907 ‘cfv 6525 (class class class)co 7400 ℂcc 11086 0cc0 11088 1c1 11089 + caddc 11091 · cmul 11093 / cdiv 11859 ℤcz 12579 ℤ_≥cuz 12850 ...cfz 13523 ..^cfzo 13670 seqcseq 14025

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12222 df-n0 12493 df-z 12580 df-uz 12851 df-fz 13524 df-fzo 13671 df-seq 14026

This theorem is referenced by: prodfdiv 15938

W3C validator