Step |
Hyp |
Ref |
Expression |
1 |
|
lmclim2.2 |
⊢ ( 𝜑 → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
2 |
|
lmclim2.3 |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ 𝑋 ) |
3 |
|
geomcau.4 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
4 |
|
geomcau.5 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) |
5 |
|
geomcau.6 |
⊢ ( 𝜑 → 𝐵 < 1 ) |
6 |
|
geomcau.7 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ) |
7 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
8 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
9 |
4
|
rpcnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
10 |
4
|
rpred |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
11 |
4
|
rpge0d |
⊢ ( 𝜑 → 0 ≤ 𝐵 ) |
12 |
10 11
|
absidd |
⊢ ( 𝜑 → ( abs ‘ 𝐵 ) = 𝐵 ) |
13 |
12 5
|
eqbrtrd |
⊢ ( 𝜑 → ( abs ‘ 𝐵 ) < 1 ) |
14 |
9 13
|
expcnv |
⊢ ( 𝜑 → ( 𝑚 ∈ ℕ0 ↦ ( 𝐵 ↑ 𝑚 ) ) ⇝ 0 ) |
15 |
|
1re |
⊢ 1 ∈ ℝ |
16 |
|
resubcl |
⊢ ( ( 1 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 1 − 𝐵 ) ∈ ℝ ) |
17 |
15 10 16
|
sylancr |
⊢ ( 𝜑 → ( 1 − 𝐵 ) ∈ ℝ ) |
18 |
|
posdif |
⊢ ( ( 𝐵 ∈ ℝ ∧ 1 ∈ ℝ ) → ( 𝐵 < 1 ↔ 0 < ( 1 − 𝐵 ) ) ) |
19 |
10 15 18
|
sylancl |
⊢ ( 𝜑 → ( 𝐵 < 1 ↔ 0 < ( 1 − 𝐵 ) ) ) |
20 |
5 19
|
mpbid |
⊢ ( 𝜑 → 0 < ( 1 − 𝐵 ) ) |
21 |
17 20
|
elrpd |
⊢ ( 𝜑 → ( 1 − 𝐵 ) ∈ ℝ+ ) |
22 |
3 21
|
rerpdivcld |
⊢ ( 𝜑 → ( 𝐴 / ( 1 − 𝐵 ) ) ∈ ℝ ) |
23 |
22
|
recnd |
⊢ ( 𝜑 → ( 𝐴 / ( 1 − 𝐵 ) ) ∈ ℂ ) |
24 |
|
nnex |
⊢ ℕ ∈ V |
25 |
24
|
mptex |
⊢ ( 𝑚 ∈ ℕ ↦ ( ( 𝐵 ↑ 𝑚 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ∈ V |
26 |
25
|
a1i |
⊢ ( 𝜑 → ( 𝑚 ∈ ℕ ↦ ( ( 𝐵 ↑ 𝑚 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ∈ V ) |
27 |
|
nnnn0 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 ) |
28 |
27
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℕ0 ) |
29 |
|
oveq2 |
⊢ ( 𝑚 = 𝑛 → ( 𝐵 ↑ 𝑚 ) = ( 𝐵 ↑ 𝑛 ) ) |
30 |
|
eqid |
⊢ ( 𝑚 ∈ ℕ0 ↦ ( 𝐵 ↑ 𝑚 ) ) = ( 𝑚 ∈ ℕ0 ↦ ( 𝐵 ↑ 𝑚 ) ) |
31 |
|
ovex |
⊢ ( 𝐵 ↑ 𝑛 ) ∈ V |
32 |
29 30 31
|
fvmpt |
⊢ ( 𝑛 ∈ ℕ0 → ( ( 𝑚 ∈ ℕ0 ↦ ( 𝐵 ↑ 𝑚 ) ) ‘ 𝑛 ) = ( 𝐵 ↑ 𝑛 ) ) |
33 |
28 32
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 𝑚 ∈ ℕ0 ↦ ( 𝐵 ↑ 𝑚 ) ) ‘ 𝑛 ) = ( 𝐵 ↑ 𝑛 ) ) |
34 |
|
nnz |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℤ ) |
35 |
|
rpexpcl |
⊢ ( ( 𝐵 ∈ ℝ+ ∧ 𝑛 ∈ ℤ ) → ( 𝐵 ↑ 𝑛 ) ∈ ℝ+ ) |
36 |
4 34 35
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐵 ↑ 𝑛 ) ∈ ℝ+ ) |
37 |
36
|
rpcnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐵 ↑ 𝑛 ) ∈ ℂ ) |
38 |
33 37
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 𝑚 ∈ ℕ0 ↦ ( 𝐵 ↑ 𝑚 ) ) ‘ 𝑛 ) ∈ ℂ ) |
39 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 / ( 1 − 𝐵 ) ) ∈ ℂ ) |
40 |
37 39
|
mulcomd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) = ( ( 𝐴 / ( 1 − 𝐵 ) ) · ( 𝐵 ↑ 𝑛 ) ) ) |
41 |
29
|
oveq1d |
⊢ ( 𝑚 = 𝑛 → ( ( 𝐵 ↑ 𝑚 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) = ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) |
42 |
|
eqid |
⊢ ( 𝑚 ∈ ℕ ↦ ( ( 𝐵 ↑ 𝑚 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) = ( 𝑚 ∈ ℕ ↦ ( ( 𝐵 ↑ 𝑚 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) |
43 |
|
ovex |
⊢ ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ∈ V |
44 |
41 42 43
|
fvmpt |
⊢ ( 𝑛 ∈ ℕ → ( ( 𝑚 ∈ ℕ ↦ ( ( 𝐵 ↑ 𝑚 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ‘ 𝑛 ) = ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) |
45 |
44
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 𝑚 ∈ ℕ ↦ ( ( 𝐵 ↑ 𝑚 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ‘ 𝑛 ) = ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) |
46 |
33
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 𝐴 / ( 1 − 𝐵 ) ) · ( ( 𝑚 ∈ ℕ0 ↦ ( 𝐵 ↑ 𝑚 ) ) ‘ 𝑛 ) ) = ( ( 𝐴 / ( 1 − 𝐵 ) ) · ( 𝐵 ↑ 𝑛 ) ) ) |
47 |
40 45 46
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 𝑚 ∈ ℕ ↦ ( ( 𝐵 ↑ 𝑚 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ‘ 𝑛 ) = ( ( 𝐴 / ( 1 − 𝐵 ) ) · ( ( 𝑚 ∈ ℕ0 ↦ ( 𝐵 ↑ 𝑚 ) ) ‘ 𝑛 ) ) ) |
48 |
7 8 14 23 26 38 47
|
climmulc2 |
⊢ ( 𝜑 → ( 𝑚 ∈ ℕ ↦ ( ( 𝐵 ↑ 𝑚 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ⇝ ( ( 𝐴 / ( 1 − 𝐵 ) ) · 0 ) ) |
49 |
23
|
mul01d |
⊢ ( 𝜑 → ( ( 𝐴 / ( 1 − 𝐵 ) ) · 0 ) = 0 ) |
50 |
48 49
|
breqtrd |
⊢ ( 𝜑 → ( 𝑚 ∈ ℕ ↦ ( ( 𝐵 ↑ 𝑚 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ⇝ 0 ) |
51 |
36
|
rpred |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐵 ↑ 𝑛 ) ∈ ℝ ) |
52 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 / ( 1 − 𝐵 ) ) ∈ ℝ ) |
53 |
51 52
|
remulcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ∈ ℝ ) |
54 |
53
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ∈ ℂ ) |
55 |
7 8 26 45 54
|
clim0c |
⊢ ( 𝜑 → ( ( 𝑚 ∈ ℕ ↦ ( ( 𝐵 ↑ 𝑚 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ⇝ 0 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℕ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 ) ) |
56 |
50 55
|
mpbid |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℕ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 ) |
57 |
|
nnz |
⊢ ( 𝑗 ∈ ℕ → 𝑗 ∈ ℤ ) |
58 |
57
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ℤ ) |
59 |
|
uzid |
⊢ ( 𝑗 ∈ ℤ → 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
60 |
|
oveq2 |
⊢ ( 𝑛 = 𝑗 → ( 𝐵 ↑ 𝑛 ) = ( 𝐵 ↑ 𝑗 ) ) |
61 |
60
|
fvoveq1d |
⊢ ( 𝑛 = 𝑗 → ( abs ‘ ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) = ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ) |
62 |
61
|
breq1d |
⊢ ( 𝑛 = 𝑗 → ( ( abs ‘ ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 ↔ ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 ) ) |
63 |
62
|
rspcv |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑗 ) → ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 → ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 ) ) |
64 |
58 59 63
|
3syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 → ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 ) ) |
65 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
66 |
|
simpl |
⊢ ( ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑗 ∈ ℕ ) |
67 |
|
ffvelrn |
⊢ ( ( 𝐹 : ℕ ⟶ 𝑋 ∧ 𝑗 ∈ ℕ ) → ( 𝐹 ‘ 𝑗 ) ∈ 𝑋 ) |
68 |
2 66 67
|
syl2an |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( 𝐹 ‘ 𝑗 ) ∈ 𝑋 ) |
69 |
|
eluznn |
⊢ ( ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑛 ∈ ℕ ) |
70 |
|
ffvelrn |
⊢ ( ( 𝐹 : ℕ ⟶ 𝑋 ∧ 𝑛 ∈ ℕ ) → ( 𝐹 ‘ 𝑛 ) ∈ 𝑋 ) |
71 |
2 69 70
|
syl2an |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( 𝐹 ‘ 𝑛 ) ∈ 𝑋 ) |
72 |
|
metcl |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑗 ) ∈ 𝑋 ∧ ( 𝐹 ‘ 𝑛 ) ∈ 𝑋 ) → ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ) |
73 |
65 68 71 72
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ) |
74 |
|
eqid |
⊢ ( ℤ≥ ‘ 𝑗 ) = ( ℤ≥ ‘ 𝑗 ) |
75 |
|
nnnn0 |
⊢ ( 𝑗 ∈ ℕ → 𝑗 ∈ ℕ0 ) |
76 |
75
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → 𝑗 ∈ ℕ0 ) |
77 |
76
|
nn0zd |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → 𝑗 ∈ ℤ ) |
78 |
|
oveq2 |
⊢ ( 𝑚 = 𝑘 → ( 𝐵 ↑ 𝑚 ) = ( 𝐵 ↑ 𝑘 ) ) |
79 |
78
|
oveq2d |
⊢ ( 𝑚 = 𝑘 → ( 𝐴 · ( 𝐵 ↑ 𝑚 ) ) = ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ) |
80 |
|
eqid |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐴 · ( 𝐵 ↑ 𝑚 ) ) ) = ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐴 · ( 𝐵 ↑ 𝑚 ) ) ) |
81 |
|
ovex |
⊢ ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ∈ V |
82 |
79 80 81
|
fvmpt |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) → ( ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐴 · ( 𝐵 ↑ 𝑚 ) ) ) ‘ 𝑘 ) = ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ) |
83 |
82
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐴 · ( 𝐵 ↑ 𝑚 ) ) ) ‘ 𝑘 ) = ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ) |
84 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝐴 ∈ ℝ ) |
85 |
10
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝐵 ∈ ℝ ) |
86 |
|
eluznn0 |
⊢ ( ( 𝑗 ∈ ℕ0 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ ℕ0 ) |
87 |
76 86
|
sylan |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ ℕ0 ) |
88 |
85 87
|
reexpcld |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐵 ↑ 𝑘 ) ∈ ℝ ) |
89 |
84 88
|
remulcld |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ∈ ℝ ) |
90 |
89
|
recnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ∈ ℂ ) |
91 |
3
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
92 |
91
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → 𝐴 ∈ ℂ ) |
93 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → 𝐵 ∈ ℂ ) |
94 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( abs ‘ 𝐵 ) < 1 ) |
95 |
|
eqid |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐵 ↑ 𝑚 ) ) = ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐵 ↑ 𝑚 ) ) |
96 |
|
ovex |
⊢ ( 𝐵 ↑ 𝑘 ) ∈ V |
97 |
78 95 96
|
fvmpt |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) → ( ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐵 ↑ 𝑚 ) ) ‘ 𝑘 ) = ( 𝐵 ↑ 𝑘 ) ) |
98 |
97
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐵 ↑ 𝑚 ) ) ‘ 𝑘 ) = ( 𝐵 ↑ 𝑘 ) ) |
99 |
93 94 76 98
|
geolim2 |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → seq 𝑗 ( + , ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐵 ↑ 𝑚 ) ) ) ⇝ ( ( 𝐵 ↑ 𝑗 ) / ( 1 − 𝐵 ) ) ) |
100 |
88
|
recnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐵 ↑ 𝑘 ) ∈ ℂ ) |
101 |
98 100
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐵 ↑ 𝑚 ) ) ‘ 𝑘 ) ∈ ℂ ) |
102 |
98
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐴 · ( ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐵 ↑ 𝑚 ) ) ‘ 𝑘 ) ) = ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ) |
103 |
83 102
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐴 · ( 𝐵 ↑ 𝑚 ) ) ) ‘ 𝑘 ) = ( 𝐴 · ( ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐵 ↑ 𝑚 ) ) ‘ 𝑘 ) ) ) |
104 |
74 77 92 99 101 103
|
isermulc2 |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → seq 𝑗 ( + , ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐴 · ( 𝐵 ↑ 𝑚 ) ) ) ) ⇝ ( 𝐴 · ( ( 𝐵 ↑ 𝑗 ) / ( 1 − 𝐵 ) ) ) ) |
105 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → 𝐵 ∈ ℝ+ ) |
106 |
105 77
|
rpexpcld |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( 𝐵 ↑ 𝑗 ) ∈ ℝ+ ) |
107 |
106
|
rpcnd |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( 𝐵 ↑ 𝑗 ) ∈ ℂ ) |
108 |
17
|
recnd |
⊢ ( 𝜑 → ( 1 − 𝐵 ) ∈ ℂ ) |
109 |
108
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( 1 − 𝐵 ) ∈ ℂ ) |
110 |
21
|
rpne0d |
⊢ ( 𝜑 → ( 1 − 𝐵 ) ≠ 0 ) |
111 |
110
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( 1 − 𝐵 ) ≠ 0 ) |
112 |
92 107 109 111
|
div12d |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( 𝐴 · ( ( 𝐵 ↑ 𝑗 ) / ( 1 − 𝐵 ) ) ) = ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) |
113 |
104 112
|
breqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → seq 𝑗 ( + , ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐴 · ( 𝐵 ↑ 𝑚 ) ) ) ) ⇝ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) |
114 |
74 77 83 90 113
|
isumclim |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → Σ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) = ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) |
115 |
|
seqex |
⊢ seq 𝑗 ( + , ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐴 · ( 𝐵 ↑ 𝑚 ) ) ) ) ∈ V |
116 |
|
ovex |
⊢ ( 𝐴 · ( ( 𝐵 ↑ 𝑗 ) / ( 1 − 𝐵 ) ) ) ∈ V |
117 |
115 116
|
breldm |
⊢ ( seq 𝑗 ( + , ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐴 · ( 𝐵 ↑ 𝑚 ) ) ) ) ⇝ ( 𝐴 · ( ( 𝐵 ↑ 𝑗 ) / ( 1 − 𝐵 ) ) ) → seq 𝑗 ( + , ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐴 · ( 𝐵 ↑ 𝑚 ) ) ) ) ∈ dom ⇝ ) |
118 |
104 117
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → seq 𝑗 ( + , ( 𝑚 ∈ ( ℤ≥ ‘ 𝑗 ) ↦ ( 𝐴 · ( 𝐵 ↑ 𝑚 ) ) ) ) ∈ dom ⇝ ) |
119 |
74 77 83 89 118
|
isumrecl |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → Σ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ∈ ℝ ) |
120 |
114 119
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ∈ ℝ ) |
121 |
120
|
recnd |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ∈ ℂ ) |
122 |
121
|
abscld |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ∈ ℝ ) |
123 |
|
fzfid |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( 𝑗 ... ( 𝑛 − 1 ) ) ∈ Fin ) |
124 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( 𝑗 ... ( 𝑛 − 1 ) ) ) → 𝜑 ) |
125 |
|
elfzuz |
⊢ ( 𝑘 ∈ ( 𝑗 ... ( 𝑛 − 1 ) ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
126 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → 𝑗 ∈ ℕ ) |
127 |
|
eluznn |
⊢ ( ( 𝑗 ∈ ℕ ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ ℕ ) |
128 |
126 127
|
sylan |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ ℕ ) |
129 |
125 128
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( 𝑗 ... ( 𝑛 − 1 ) ) ) → 𝑘 ∈ ℕ ) |
130 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
131 |
2
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) ∈ 𝑋 ) |
132 |
|
peano2nn |
⊢ ( 𝑘 ∈ ℕ → ( 𝑘 + 1 ) ∈ ℕ ) |
133 |
|
ffvelrn |
⊢ ( ( 𝐹 : ℕ ⟶ 𝑋 ∧ ( 𝑘 + 1 ) ∈ ℕ ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ 𝑋 ) |
134 |
2 132 133
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ 𝑋 ) |
135 |
|
metcl |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑋 ∧ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ 𝑋 ) → ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ∈ ℝ ) |
136 |
130 131 134 135
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ∈ ℝ ) |
137 |
124 129 136
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( 𝑗 ... ( 𝑛 − 1 ) ) ) → ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ∈ ℝ ) |
138 |
123 137
|
fsumrecl |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → Σ 𝑘 ∈ ( 𝑗 ... ( 𝑛 − 1 ) ) ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ∈ ℝ ) |
139 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
140 |
|
elfzuz |
⊢ ( 𝑘 ∈ ( 𝑗 ... 𝑛 ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
141 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝜑 ) |
142 |
141 128 131
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ 𝑋 ) |
143 |
140 142
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( 𝑗 ... 𝑛 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ 𝑋 ) |
144 |
65 139 143
|
mettrifi |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ≤ Σ 𝑘 ∈ ( 𝑗 ... ( 𝑛 − 1 ) ) ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
145 |
125 89
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( 𝑗 ... ( 𝑛 − 1 ) ) ) → ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ∈ ℝ ) |
146 |
123 145
|
fsumrecl |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → Σ 𝑘 ∈ ( 𝑗 ... ( 𝑛 − 1 ) ) ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ∈ ℝ ) |
147 |
124 129 6
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( 𝑗 ... ( 𝑛 − 1 ) ) ) → ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ) |
148 |
123 137 145 147
|
fsumle |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → Σ 𝑘 ∈ ( 𝑗 ... ( 𝑛 − 1 ) ) ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ Σ 𝑘 ∈ ( 𝑗 ... ( 𝑛 − 1 ) ) ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ) |
149 |
|
fzssuz |
⊢ ( 𝑗 ... ( 𝑛 − 1 ) ) ⊆ ( ℤ≥ ‘ 𝑗 ) |
150 |
149
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( 𝑗 ... ( 𝑛 − 1 ) ) ⊆ ( ℤ≥ ‘ 𝑗 ) ) |
151 |
|
0red |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 0 ∈ ℝ ) |
152 |
|
nnz |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℤ ) |
153 |
|
rpexpcl |
⊢ ( ( 𝐵 ∈ ℝ+ ∧ 𝑘 ∈ ℤ ) → ( 𝐵 ↑ 𝑘 ) ∈ ℝ+ ) |
154 |
4 152 153
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐵 ↑ 𝑘 ) ∈ ℝ+ ) |
155 |
136 154
|
rerpdivcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) / ( 𝐵 ↑ 𝑘 ) ) ∈ ℝ ) |
156 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝐴 ∈ ℝ ) |
157 |
|
metge0 |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑋 ∧ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ 𝑋 ) → 0 ≤ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
158 |
130 131 134 157
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 0 ≤ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
159 |
136 154 158
|
divge0d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 0 ≤ ( ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) / ( 𝐵 ↑ 𝑘 ) ) ) |
160 |
136 156 154
|
ledivmul2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) / ( 𝐵 ↑ 𝑘 ) ) ≤ 𝐴 ↔ ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ) ) |
161 |
6 160
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) / ( 𝐵 ↑ 𝑘 ) ) ≤ 𝐴 ) |
162 |
151 155 156 159 161
|
letrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 0 ≤ 𝐴 ) |
163 |
141 128 162
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 0 ≤ 𝐴 ) |
164 |
141 128 154
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐵 ↑ 𝑘 ) ∈ ℝ+ ) |
165 |
164
|
rpge0d |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 0 ≤ ( 𝐵 ↑ 𝑘 ) ) |
166 |
84 88 163 165
|
mulge0d |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 0 ≤ ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ) |
167 |
74 77 123 150 83 89 166 118
|
isumless |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → Σ 𝑘 ∈ ( 𝑗 ... ( 𝑛 − 1 ) ) ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ≤ Σ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ) |
168 |
138 146 119 148 167
|
letrd |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → Σ 𝑘 ∈ ( 𝑗 ... ( 𝑛 − 1 ) ) ( ( 𝐹 ‘ 𝑘 ) 𝐷 ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ≤ Σ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ) |
169 |
73 138 119 144 168
|
letrd |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ≤ Σ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐴 · ( 𝐵 ↑ 𝑘 ) ) ) |
170 |
169 114
|
breqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ≤ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) |
171 |
120
|
leabsd |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ≤ ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ) |
172 |
73 120 122 170 171
|
letrd |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ≤ ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ) |
173 |
172
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ≤ ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ) |
174 |
73
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ) |
175 |
122
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ∈ ℝ ) |
176 |
|
rpre |
⊢ ( 𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ ) |
177 |
176
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → 𝑥 ∈ ℝ ) |
178 |
|
lelttr |
⊢ ( ( ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( ( ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ≤ ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 ) → ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) ) |
179 |
174 175 177 178
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ≤ ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) ∧ ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 ) → ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) ) |
180 |
173 179
|
mpand |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑗 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) ) → ( ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 → ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) ) |
181 |
180
|
anassrs |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 → ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) ) |
182 |
181
|
ralrimdva |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ( abs ‘ ( ( 𝐵 ↑ 𝑗 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 → ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) ) |
183 |
64 182
|
syld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑗 ∈ ℕ ) → ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 → ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) ) |
184 |
183
|
reximdva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ∃ 𝑗 ∈ ℕ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 → ∃ 𝑗 ∈ ℕ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) ) |
185 |
184
|
ralimdva |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℕ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( 𝐵 ↑ 𝑛 ) · ( 𝐴 / ( 1 − 𝐵 ) ) ) ) < 𝑥 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℕ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) ) |
186 |
56 185
|
mpd |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℕ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) |
187 |
|
metxmet |
⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
188 |
1 187
|
syl |
⊢ ( 𝜑 → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
189 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑛 ) ) |
190 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐹 ‘ 𝑗 ) ) |
191 |
7 188 8 189 190 2
|
iscauf |
⊢ ( 𝜑 → ( 𝐹 ∈ ( Cau ‘ 𝐷 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℕ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) ) |
192 |
186 191
|
mpbird |
⊢ ( 𝜑 → 𝐹 ∈ ( Cau ‘ 𝐷 ) ) |