Step | Hyp | Ref
| Expression |
1 | | prodfn0.1 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
2 | | eluzfz2 13264 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
3 | 1, 2 | syl 17 |
. 2
⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
4 | | fveq2 6774 |
. . . . 5
⊢ (𝑚 = 𝑀 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑀)) |
5 | | fveq2 6774 |
. . . . . 6
⊢ (𝑚 = 𝑀 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑀)) |
6 | 5 | oveq2d 7291 |
. . . . 5
⊢ (𝑚 = 𝑀 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑀))) |
7 | 4, 6 | eqeq12d 2754 |
. . . 4
⊢ (𝑚 = 𝑀 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀)))) |
8 | 7 | imbi2d 341 |
. . 3
⊢ (𝑚 = 𝑀 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (1 / (seq𝑀( · , 𝐹)‘𝑀))))) |
9 | | fveq2 6774 |
. . . . 5
⊢ (𝑚 = 𝑛 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑛)) |
10 | | fveq2 6774 |
. . . . . 6
⊢ (𝑚 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑛)) |
11 | 10 | oveq2d 7291 |
. . . . 5
⊢ (𝑚 = 𝑛 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑛))) |
12 | 9, 11 | eqeq12d 2754 |
. . . 4
⊢ (𝑚 = 𝑛 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛)))) |
13 | 12 | imbi2d 341 |
. . 3
⊢ (𝑚 = 𝑛 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑛) = (1 / (seq𝑀( · , 𝐹)‘𝑛))))) |
14 | | fveq2 6774 |
. . . . 5
⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘(𝑛 + 1))) |
15 | | fveq2 6774 |
. . . . . 6
⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘(𝑛 + 1))) |
16 | 15 | oveq2d 7291 |
. . . . 5
⊢ (𝑚 = (𝑛 + 1) → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1)))) |
17 | 14, 16 | eqeq12d 2754 |
. . . 4
⊢ (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1))))) |
18 | 17 | imbi2d 341 |
. . 3
⊢ (𝑚 = (𝑛 + 1) → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘(𝑛 + 1)) = (1 / (seq𝑀( · , 𝐹)‘(𝑛 + 1)))))) |
19 | | fveq2 6774 |
. . . . 5
⊢ (𝑚 = 𝑁 → (seq𝑀( · , 𝐺)‘𝑚) = (seq𝑀( · , 𝐺)‘𝑁)) |
20 | | fveq2 6774 |
. . . . . 6
⊢ (𝑚 = 𝑁 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑁)) |
21 | 20 | oveq2d 7291 |
. . . . 5
⊢ (𝑚 = 𝑁 → (1 / (seq𝑀( · , 𝐹)‘𝑚)) = (1 / (seq𝑀( · , 𝐹)‘𝑁))) |
22 | 19, 21 | eqeq12d 2754 |
. . . 4
⊢ (𝑚 = 𝑁 → ((seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚)) ↔ (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁)))) |
23 | 22 | imbi2d 341 |
. . 3
⊢ (𝑚 = 𝑁 → ((𝜑 → (seq𝑀( · , 𝐺)‘𝑚) = (1 / (seq𝑀( · , 𝐹)‘𝑚))) ↔ (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁))))) |
24 | | eluzfz1 13263 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
25 | 1, 24 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
26 | | fveq2 6774 |
. . . . . . . . 9
⊢ (𝑘 = 𝑀 → (𝐺‘𝑘) = (𝐺‘𝑀)) |
27 | | fveq2 6774 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀)) |
28 | 27 | oveq2d 7291 |
. . . . . . . . 9
⊢ (𝑘 = 𝑀 → (1 / (𝐹‘𝑘)) = (1 / (𝐹‘𝑀))) |
29 | 26, 28 | eqeq12d 2754 |
. . . . . . . 8
⊢ (𝑘 = 𝑀 → ((𝐺‘𝑘) = (1 / (𝐹‘𝑘)) ↔ (𝐺‘𝑀) = (1 / (𝐹‘𝑀)))) |
30 | 29 | imbi2d 341 |
. . . . . . 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 414 |
. . . . . . 7
⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐺‘𝑘) = (1 / (𝐹‘𝑘)))) |
33 | 30, 32 | vtoclga 3513 |
. . . . . 6
⊢ (𝑀 ∈ (𝑀...𝑁) → (𝜑 → (𝐺‘𝑀) = (1 / (𝐹‘𝑀)))) |
34 | 25, 33 | mpcom 38 |
. . . . 5
⊢ (𝜑 → (𝐺‘𝑀) = (1 / (𝐹‘𝑀))) |
35 | | eluzel2 12587 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
36 | 1, 35 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℤ) |
37 | | seq1 13734 |
. . . . . 6
⊢ (𝑀 ∈ ℤ → (seq𝑀( · , 𝐺)‘𝑀) = (𝐺‘𝑀)) |
38 | 36, 37 | syl 17 |
. . . . 5
⊢ (𝜑 → (seq𝑀( · , 𝐺)‘𝑀) = (𝐺‘𝑀)) |
39 | | seq1 13734 |
. . . . . . 7
⊢ (𝑀 ∈ ℤ → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
40 | 36, 39 | syl 17 |
. . . . . 6
⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
41 | 40 | oveq2d 7291 |
. . . . 5
⊢ (𝜑 → (1 / (seq𝑀( · , 𝐹)‘𝑀)) = (1 / (𝐹‘𝑀))) |
42 | 34, 38, 41 | 3eqtr4d 2788 |
. . . 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 7282 |
. . . . . . . . 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 13484 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁)) |
47 | | fveq2 6774 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (𝑛 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑛 + 1))) |
48 | | fveq2 6774 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1))) |
49 | 48 | oveq2d 7291 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (𝑛 + 1) → (1 / (𝐹‘𝑘)) = (1 / (𝐹‘(𝑛 + 1)))) |
50 | 47, 49 | eqeq12d 2754 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (𝑛 + 1) → ((𝐺‘𝑘) = (1 / (𝐹‘𝑘)) ↔ (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1))))) |
51 | 50 | imbi2d 341 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐺‘𝑘) = (1 / (𝐹‘𝑘))) ↔ (𝜑 → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1)))))) |
52 | 51, 32 | vtoclga 3513 |
. . . . . . . . . . . . 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 408 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑛 + 1)) = (1 / (𝐹‘(𝑛 + 1)))) |
55 | 54 | oveq2d 7291 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (𝐺‘(𝑛 + 1))) = ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1))))) |
56 | | 1cnd 10970 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ) |
57 | | elfzouz 13391 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ≥‘𝑀)) |
58 | 57 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → 𝑛 ∈ (ℤ≥‘𝑀)) |
59 | | elfzouz2 13402 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝑛)) |
60 | | fzss2 13296 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁)) |
61 | 59, 60 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑀...𝑛) ⊆ (𝑀...𝑁)) |
62 | 61 | sselda 3921 |
. . . . . . . . . . . . . . 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 468 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℂ) |
66 | | mulcl 10955 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 · 𝑥) ∈ ℂ) |
67 | 66 | adantl 482 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 · 𝑥) ∈ ℂ) |
68 | 58, 65, 67 | seqcl 13743 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ) |
69 | 48 | eleq1d 2823 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ)) |
70 | 69 | imbi2d 341 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹‘𝑘) ∈ ℂ) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))) |
71 | 63 | expcom 414 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑘) ∈ ℂ)) |
72 | 70, 71 | vtoclga 3513 |
. . . . . . . . . . . . . 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 408 |
. . . . . . . . . . . 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 468 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ≠ 0) |
78 | 58, 65, 77 | prodfn0 15606 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ≠ 0) |
79 | 48 | neeq1d 3003 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ≠ 0 ↔ (𝐹‘(𝑛 + 1)) ≠ 0)) |
80 | 79 | imbi2d 341 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹‘𝑘) ≠ 0) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) ≠ 0))) |
81 | 75 | expcom 414 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑘) ≠ 0)) |
82 | 80, 81 | vtoclga 3513 |
. . . . . . . . . . . . . 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 408 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ≠ 0) |
85 | 56, 68, 56, 74, 78, 84 | divmuldivd 11792 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))) = ((1 · 1) / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))) |
86 | | 1t1e1 12135 |
. . . . . . . . . . . 12
⊢ (1
· 1) = 1 |
87 | 86 | oveq1i 7285 |
. . . . . . . . . . 11
⊢ ((1
· 1) / ((seq𝑀(
· , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))) |
88 | 85, 87 | eqtrdi 2794 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → ((1 / (seq𝑀( · , 𝐹)‘𝑛)) · (1 / (𝐹‘(𝑛 + 1)))) = (1 / ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))) |
89 | 55, 88 | eqtrd 2778 |
. . . . . . . . 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 2778 |
. . . . . . 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 13736 |
. . . . . . . . 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 13736 |
. . . . . . . . . 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 7291 |
. . . . . . . 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 2788 |
. . . . . 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 13505 |
. 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))) |