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Theorem prodfrec 15928

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 13569	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		fveq2 6907	. . . . 5 ⊢ (𝑚 = 𝑀 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑀))
5		fveq2 6907	. . . . . 6 ⊢ (𝑚 = 𝑀 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑀))
6	5	oveq2d 7447	. . . . 5 ⊢ (𝑚 = 𝑀 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑀)))
7	4, 6	eqeq12d 2751	. . . 4 ⊢ (𝑚 = 𝑀 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀))))
8	7	imbi2d 340	. . 3 ⊢ (𝑚 = 𝑀 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀)))))
9		fveq2 6907	. . . . 5 ⊢ (𝑚 = 𝑛 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑛))
10		fveq2 6907	. . . . . 6 ⊢ (𝑚 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑛))
11	10	oveq2d 7447	. . . . 5 ⊢ (𝑚 = 𝑛 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑛)))
12	9, 11	eqeq12d 2751	. . . 4 ⊢ (𝑚 = 𝑛 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))))
13	12	imbi2d 340	. . 3 ⊢ (𝑚 = 𝑛 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)))))
14		fveq2 6907	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘(𝑛 + 1)))
15		fveq2 6907	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘(𝑛 + 1)))
16	15	oveq2d 7447	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))
17	14, 16	eqeq12d 2751	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1)))))
18	17	imbi2d 340	. . 3 ⊢ (𝑚 = (𝑛 + 1) → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
19		fveq2 6907	. . . . 5 ⊢ (𝑚 = 𝑁 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑁))
20		fveq2 6907	. . . . . 6 ⊢ (𝑚 = 𝑁 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑁))
21	20	oveq2d 7447	. . . . 5 ⊢ (𝑚 = 𝑁 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))
22	19, 21	eqeq12d 2751	. . . 4 ⊢ (𝑚 = 𝑁 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁))))
23	22	imbi2d 340	. . 3 ⊢ (𝑚 = 𝑁 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))))
24		eluzfz1 13568	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
25	1, 24	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
26		fveq2 6907	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝐺‘𝑘) = (𝐺‘𝑀))
27		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
28	27	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (1 / (𝐹‘𝑘)) = (1 / (𝐹‘𝑀)))
29	26, 28	eqeq12d 2751	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → ((𝐺‘𝑘) = (1 / (𝐹‘𝑘)) ↔ (𝐺‘𝑀) = (1 / (𝐹‘𝑀))))
30	29	imbi2d 340	. . . . . . 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 413	. . . . . . 7 ⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐺‘𝑘) = (1 / (𝐹‘𝑘))))
33	30, 32	vtoclga 3577	. . . . . 6 ⊢ (𝑀 ∈ (𝑀...𝑁) → (𝜑 → (𝐺‘𝑀) = (1 / (𝐹‘𝑀))))
34	25, 33	mpcom 38	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑀) = (1 / (𝐹‘𝑀)))
35		eluzel2 12881	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
36	1, 35	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
37		seq1 14052	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (seq𝑀( · , 𝐺)‘𝑀) = (𝐺‘𝑀))
38	36, 37	syl 17	. . . . 5 ⊢ (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (𝐺‘𝑀))
39		seq1 14052	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹‘𝑀))
40	36, 39	syl 17	. . . . . 6 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹‘𝑀))
41	40	oveq2d 7447	. . . . 5 ⊢ (𝜑 → (1 / (seq𝑀( · , 𝐹)‘𝑀)) = (1 / (𝐹‘𝑀)))
42	34, 38, 41	3eqtr4d 2785	. . . 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 7438	. . . . . . . . 9 ⊢ ((seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)) → ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))) = ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))))
45	44	3ad2ant3 1134	. . . . . . . 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 13800	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
47		fveq2 6907	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑛 + 1)))
48		fveq2 6907	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
49	48	oveq2d 7447	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → (1 / (𝐹‘𝑘)) = (1 / (𝐹‘(𝑛 + 1))))
50	47, 49	eqeq12d 2751	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → ((𝐺‘𝑘) = (1 / (𝐹‘𝑘)) ↔ (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1)))))
51	50	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐺‘𝑘) = (1 / (𝐹‘𝑘))) ↔ (𝜑 → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1))))))
52	51, 32	vtoclga 3577	. . . . . . . . . . . . 13 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1)))))
53	46, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1)))))
54	53	impcom 407	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1))))
55	54	oveq2d 7447	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))) = ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))))
56		1cnd 11254	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ)
57		elfzouz 13700	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ_≥‘𝑀))
58	57	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ_≥‘𝑀))
59		elfzouz2 13711	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ_≥‘𝑛))
60		fzss2 13601	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
61	59, 60	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
62	61	sselda 3995	. . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑀..^𝑁) ∧ 𝑘 ∈ (𝑀...𝑛))) → (𝐹‘𝑘) ∈ ℂ)
65	64	anassrs 467	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℂ)
66		mulcl 11237	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 · 𝑥) ∈ ℂ)
67	66	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 · 𝑥) ∈ ℂ)
68	58, 65, 67	seqcl 14060	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ)
69	48	eleq1d 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
70	69	imbi2d 340	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹‘𝑘) ∈ ℂ) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ)))
71	63	expcom 413	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑘) ∈ ℂ))
72	70, 71	vtoclga 3577	. . . . . . . . . . . . . 14 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))
73	46, 72	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))
74	73	impcom 407	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
75		prodfn0.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ≠ 0)
76	62, 75	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑀..^𝑁) ∧ 𝑘 ∈ (𝑀...𝑛))) → (𝐹‘𝑘) ≠ 0)
77	76	anassrs 467	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ≠ 0)
78	58, 65, 77	prodfn0 15927	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ≠ 0)
79	48	neeq1d 2998	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ≠ 0 ↔ (𝐹‘(𝑛 + 1)) ≠ 0))
80	79	imbi2d 340	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹‘𝑘) ≠ 0) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) ≠ 0)))
81	75	expcom 413	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑘) ≠ 0))
82	80, 81	vtoclga 3577	. . . . . . . . . . . . . 14 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ≠ 0))
83	46, 82	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ≠ 0))
84	83	impcom 407	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ≠ 0)
85	56, 68, 56, 74, 78, 84	divmuldivd 12082	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))) = ((1 · 1) / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
86		1t1e1 12426	. . . . . . . . . . . 12 ⊢ (1 · 1) = 1
87	86	oveq1i 7441	. . . . . . . . . . 11 ⊢ ((1 · 1) / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
88	85, 87	eqtrdi 2791	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
89	55, 88	eqtrd 2775	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
90	89	3adant3 1131	. . . . . . . 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 2775	. . . . . . 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 14054	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))))
93	57, 92	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))))
94	93	3ad2ant2 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐺)‘𝑛) · (𝐺‘(𝑛 + 1))))
95		seqp1 14054	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
96	57, 95	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
97	96	oveq2d 7447	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))))
98	97	3ad2ant2 1133	. . . . . . 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 2785	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))
100	99	3exp 1118	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (𝑀..^𝑁) → ((seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
101	100	com12 32	. . . 4 ⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)) → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))))
102	101	a2d 29	. . 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 13821	. 2 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁))))
104	3, 103	mpcom 38	1 ⊢ (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 / cdiv 11918 ℤcz 12611 ℤ_≥cuz 12876 ...cfz 13544 ..^cfzo 13691 seqcseq 14039

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040

This theorem is referenced by: prodfdiv 15929

W3C validator