Step |
Hyp |
Ref |
Expression |
1 |
|
prodfn0.1 |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
2 |
|
prodfn0.2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
3 |
|
prodfn0.3 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) |
4 |
|
prodfrec.4 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐺 ‘ 𝑘 ) = ( 1 / ( 𝐹 ‘ 𝑘 ) ) ) |
5 |
|
eluzfz2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ( 𝑀 ... 𝑁 ) ) |
6 |
1 5
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ( 𝑀 ... 𝑁 ) ) |
7 |
|
fveq2 |
⊢ ( 𝑚 = 𝑀 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑚 ) = ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑀 ) ) |
8 |
|
fveq2 |
⊢ ( 𝑚 = 𝑀 → ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) = ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑀 ) ) |
9 |
8
|
oveq2d |
⊢ ( 𝑚 = 𝑀 → ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑀 ) ) ) |
10 |
7 9
|
eqeq12d |
⊢ ( 𝑚 = 𝑀 → ( ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑚 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) ) ↔ ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑀 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑀 ) ) ) ) |
11 |
10
|
imbi2d |
⊢ ( 𝑚 = 𝑀 → ( ( 𝜑 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑚 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) ) ) ↔ ( 𝜑 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑀 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑀 ) ) ) ) ) |
12 |
|
fveq2 |
⊢ ( 𝑚 = 𝑛 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑚 ) = ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) ) |
13 |
|
fveq2 |
⊢ ( 𝑚 = 𝑛 → ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) = ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) |
14 |
13
|
oveq2d |
⊢ ( 𝑚 = 𝑛 → ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) ) |
15 |
12 14
|
eqeq12d |
⊢ ( 𝑚 = 𝑛 → ( ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑚 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) ) ↔ ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) ) ) |
16 |
15
|
imbi2d |
⊢ ( 𝑚 = 𝑛 → ( ( 𝜑 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑚 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) ) ) ↔ ( 𝜑 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) ) ) ) |
17 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑚 ) = ( seq 𝑀 ( · , 𝐺 ) ‘ ( 𝑛 + 1 ) ) ) |
18 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) = ( seq 𝑀 ( · , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) |
19 |
18
|
oveq2d |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) ) |
20 |
17 19
|
eqeq12d |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑚 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) ) ↔ ( seq 𝑀 ( · , 𝐺 ) ‘ ( 𝑛 + 1 ) ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) ) ) |
21 |
20
|
imbi2d |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( ( 𝜑 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑚 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) ) ) ↔ ( 𝜑 → ( seq 𝑀 ( · , 𝐺 ) ‘ ( 𝑛 + 1 ) ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) ) ) ) |
22 |
|
fveq2 |
⊢ ( 𝑚 = 𝑁 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑚 ) = ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑁 ) ) |
23 |
|
fveq2 |
⊢ ( 𝑚 = 𝑁 → ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) = ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑁 ) ) |
24 |
23
|
oveq2d |
⊢ ( 𝑚 = 𝑁 → ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑁 ) ) ) |
25 |
22 24
|
eqeq12d |
⊢ ( 𝑚 = 𝑁 → ( ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑚 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) ) ↔ ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑁 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑁 ) ) ) ) |
26 |
25
|
imbi2d |
⊢ ( 𝑚 = 𝑁 → ( ( 𝜑 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑚 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑚 ) ) ) ↔ ( 𝜑 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑁 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑁 ) ) ) ) ) |
27 |
|
eluzfz1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ( 𝑀 ... 𝑁 ) ) |
28 |
1 27
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ( 𝑀 ... 𝑁 ) ) |
29 |
|
fveq2 |
⊢ ( 𝑘 = 𝑀 → ( 𝐺 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑀 ) ) |
30 |
|
fveq2 |
⊢ ( 𝑘 = 𝑀 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑀 ) ) |
31 |
30
|
oveq2d |
⊢ ( 𝑘 = 𝑀 → ( 1 / ( 𝐹 ‘ 𝑘 ) ) = ( 1 / ( 𝐹 ‘ 𝑀 ) ) ) |
32 |
29 31
|
eqeq12d |
⊢ ( 𝑘 = 𝑀 → ( ( 𝐺 ‘ 𝑘 ) = ( 1 / ( 𝐹 ‘ 𝑘 ) ) ↔ ( 𝐺 ‘ 𝑀 ) = ( 1 / ( 𝐹 ‘ 𝑀 ) ) ) ) |
33 |
32
|
imbi2d |
⊢ ( 𝑘 = 𝑀 → ( ( 𝜑 → ( 𝐺 ‘ 𝑘 ) = ( 1 / ( 𝐹 ‘ 𝑘 ) ) ) ↔ ( 𝜑 → ( 𝐺 ‘ 𝑀 ) = ( 1 / ( 𝐹 ‘ 𝑀 ) ) ) ) ) |
34 |
4
|
expcom |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) → ( 𝜑 → ( 𝐺 ‘ 𝑘 ) = ( 1 / ( 𝐹 ‘ 𝑘 ) ) ) ) |
35 |
33 34
|
vtoclga |
⊢ ( 𝑀 ∈ ( 𝑀 ... 𝑁 ) → ( 𝜑 → ( 𝐺 ‘ 𝑀 ) = ( 1 / ( 𝐹 ‘ 𝑀 ) ) ) ) |
36 |
28 35
|
mpcom |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑀 ) = ( 1 / ( 𝐹 ‘ 𝑀 ) ) ) |
37 |
|
eluzel2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ ) |
38 |
1 37
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
39 |
|
seq1 |
⊢ ( 𝑀 ∈ ℤ → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑀 ) = ( 𝐺 ‘ 𝑀 ) ) |
40 |
38 39
|
syl |
⊢ ( 𝜑 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑀 ) = ( 𝐺 ‘ 𝑀 ) ) |
41 |
|
seq1 |
⊢ ( 𝑀 ∈ ℤ → ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑀 ) = ( 𝐹 ‘ 𝑀 ) ) |
42 |
38 41
|
syl |
⊢ ( 𝜑 → ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑀 ) = ( 𝐹 ‘ 𝑀 ) ) |
43 |
42
|
oveq2d |
⊢ ( 𝜑 → ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑀 ) ) = ( 1 / ( 𝐹 ‘ 𝑀 ) ) ) |
44 |
36 40 43
|
3eqtr4d |
⊢ ( 𝜑 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑀 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑀 ) ) ) |
45 |
44
|
a1i |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝜑 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑀 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑀 ) ) ) ) |
46 |
|
oveq1 |
⊢ ( ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) → ( ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) · ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) = ( ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) · ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) |
47 |
46
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ∧ ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) ) → ( ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) · ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) = ( ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) · ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) |
48 |
|
fzofzp1 |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) ) |
49 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( 𝐺 ‘ 𝑘 ) = ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) |
50 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) |
51 |
50
|
oveq2d |
⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( 1 / ( 𝐹 ‘ 𝑘 ) ) = ( 1 / ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
52 |
49 51
|
eqeq12d |
⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( ( 𝐺 ‘ 𝑘 ) = ( 1 / ( 𝐹 ‘ 𝑘 ) ) ↔ ( 𝐺 ‘ ( 𝑛 + 1 ) ) = ( 1 / ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
53 |
52
|
imbi2d |
⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( ( 𝜑 → ( 𝐺 ‘ 𝑘 ) = ( 1 / ( 𝐹 ‘ 𝑘 ) ) ) ↔ ( 𝜑 → ( 𝐺 ‘ ( 𝑛 + 1 ) ) = ( 1 / ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) ) |
54 |
53 34
|
vtoclga |
⊢ ( ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) → ( 𝜑 → ( 𝐺 ‘ ( 𝑛 + 1 ) ) = ( 1 / ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
55 |
48 54
|
syl |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝜑 → ( 𝐺 ‘ ( 𝑛 + 1 ) ) = ( 1 / ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
56 |
55
|
impcom |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐺 ‘ ( 𝑛 + 1 ) ) = ( 1 / ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
57 |
56
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) · ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) = ( ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) · ( 1 / ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
58 |
|
1cnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → 1 ∈ ℂ ) |
59 |
|
elfzouz |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
60 |
59
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
61 |
|
elfzouz2 |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑛 ) ) |
62 |
|
fzss2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑛 ) → ( 𝑀 ... 𝑛 ) ⊆ ( 𝑀 ... 𝑁 ) ) |
63 |
61 62
|
syl |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝑀 ... 𝑛 ) ⊆ ( 𝑀 ... 𝑁 ) ) |
64 |
63
|
sselda |
⊢ ( ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑛 ) ) → 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) |
65 |
64 2
|
sylan2 |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑛 ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
66 |
65
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑛 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
67 |
|
mulcl |
⊢ ( ( 𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ) → ( 𝑘 · 𝑥 ) ∈ ℂ ) |
68 |
67
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ ( 𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ) ) → ( 𝑘 · 𝑥 ) ∈ ℂ ) |
69 |
60 66 68
|
seqcl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ∈ ℂ ) |
70 |
50
|
eleq1d |
⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ↔ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ ℂ ) ) |
71 |
70
|
imbi2d |
⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( ( 𝜑 → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) ↔ ( 𝜑 → ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ ℂ ) ) ) |
72 |
2
|
expcom |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) → ( 𝜑 → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) ) |
73 |
71 72
|
vtoclga |
⊢ ( ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) → ( 𝜑 → ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ ℂ ) ) |
74 |
48 73
|
syl |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝜑 → ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ ℂ ) ) |
75 |
74
|
impcom |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐹 ‘ ( 𝑛 + 1 ) ) ∈ ℂ ) |
76 |
64 3
|
sylan2 |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑛 ) ) ) → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) |
77 |
76
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑛 ) ) → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) |
78 |
60 66 77
|
prodfn0 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ≠ 0 ) |
79 |
50
|
neeq1d |
⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( ( 𝐹 ‘ 𝑘 ) ≠ 0 ↔ ( 𝐹 ‘ ( 𝑛 + 1 ) ) ≠ 0 ) ) |
80 |
79
|
imbi2d |
⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( ( 𝜑 → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) ↔ ( 𝜑 → ( 𝐹 ‘ ( 𝑛 + 1 ) ) ≠ 0 ) ) ) |
81 |
3
|
expcom |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) → ( 𝜑 → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) ) |
82 |
80 81
|
vtoclga |
⊢ ( ( 𝑛 + 1 ) ∈ ( 𝑀 ... 𝑁 ) → ( 𝜑 → ( 𝐹 ‘ ( 𝑛 + 1 ) ) ≠ 0 ) ) |
83 |
48 82
|
syl |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝜑 → ( 𝐹 ‘ ( 𝑛 + 1 ) ) ≠ 0 ) ) |
84 |
83
|
impcom |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝐹 ‘ ( 𝑛 + 1 ) ) ≠ 0 ) |
85 |
58 69 58 75 78 84
|
divmuldivd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) · ( 1 / ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) = ( ( 1 · 1 ) / ( ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) · ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
86 |
|
1t1e1 |
⊢ ( 1 · 1 ) = 1 |
87 |
86
|
oveq1i |
⊢ ( ( 1 · 1 ) / ( ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) · ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) = ( 1 / ( ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) · ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
88 |
85 87
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) · ( 1 / ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) = ( 1 / ( ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) · ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
89 |
57 88
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) · ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) = ( 1 / ( ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) · ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
90 |
89
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ∧ ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) ) → ( ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) · ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) = ( 1 / ( ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) · ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
91 |
47 90
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ∧ ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) ) → ( ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) · ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) = ( 1 / ( ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) · ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
92 |
|
seqp1 |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) → ( seq 𝑀 ( · , 𝐺 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) · ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) |
93 |
59 92
|
syl |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → ( seq 𝑀 ( · , 𝐺 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) · ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) |
94 |
93
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ∧ ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) ) → ( seq 𝑀 ( · , 𝐺 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) · ( 𝐺 ‘ ( 𝑛 + 1 ) ) ) ) |
95 |
|
seqp1 |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) → ( seq 𝑀 ( · , 𝐹 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) · ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
96 |
59 95
|
syl |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → ( seq 𝑀 ( · , 𝐹 ) ‘ ( 𝑛 + 1 ) ) = ( ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) · ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) |
97 |
96
|
oveq2d |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) = ( 1 / ( ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) · ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
98 |
97
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ∧ ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) ) → ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) = ( 1 / ( ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) · ( 𝐹 ‘ ( 𝑛 + 1 ) ) ) ) ) |
99 |
91 94 98
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) ∧ ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) ) → ( seq 𝑀 ( · , 𝐺 ) ‘ ( 𝑛 + 1 ) ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) ) |
100 |
99
|
3exp |
⊢ ( 𝜑 → ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → ( ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) → ( seq 𝑀 ( · , 𝐺 ) ‘ ( 𝑛 + 1 ) ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) ) ) ) |
101 |
100
|
com12 |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝜑 → ( ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) → ( seq 𝑀 ( · , 𝐺 ) ‘ ( 𝑛 + 1 ) ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) ) ) ) |
102 |
101
|
a2d |
⊢ ( 𝑛 ∈ ( 𝑀 ..^ 𝑁 ) → ( ( 𝜑 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑛 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑛 ) ) ) → ( 𝜑 → ( seq 𝑀 ( · , 𝐺 ) ‘ ( 𝑛 + 1 ) ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ ( 𝑛 + 1 ) ) ) ) ) ) |
103 |
11 16 21 26 45 102
|
fzind2 |
⊢ ( 𝑁 ∈ ( 𝑀 ... 𝑁 ) → ( 𝜑 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑁 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑁 ) ) ) ) |
104 |
6 103
|
mpcom |
⊢ ( 𝜑 → ( seq 𝑀 ( · , 𝐺 ) ‘ 𝑁 ) = ( 1 / ( seq 𝑀 ( · , 𝐹 ) ‘ 𝑁 ) ) ) |