Step |
Hyp |
Ref |
Expression |
1 |
|
dvradcnv.g |
⊢ 𝐺 = ( 𝑥 ∈ ℂ ↦ ( 𝑛 ∈ ℕ0 ↦ ( ( 𝐴 ‘ 𝑛 ) · ( 𝑥 ↑ 𝑛 ) ) ) ) |
2 |
|
dvradcnv.r |
⊢ 𝑅 = sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝐺 ‘ 𝑟 ) ) ∈ dom ⇝ } , ℝ* , < ) |
3 |
|
dvradcnv.h |
⊢ 𝐻 = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 1 ) · ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) · ( 𝑋 ↑ 𝑛 ) ) ) |
4 |
|
dvradcnv.a |
⊢ ( 𝜑 → 𝐴 : ℕ0 ⟶ ℂ ) |
5 |
|
dvradcnv.x |
⊢ ( 𝜑 → 𝑋 ∈ ℂ ) |
6 |
|
dvradcnv.l |
⊢ ( 𝜑 → ( abs ‘ 𝑋 ) < 𝑅 ) |
7 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
8 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
9 |
8
|
a1i |
⊢ ( 𝜑 → 1 ∈ ℕ0 ) |
10 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
11 |
|
nn0cn |
⊢ ( 𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ ) |
12 |
11
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℂ ) |
13 |
|
nn0ex |
⊢ ℕ0 ∈ V |
14 |
13
|
mptex |
⊢ ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ∈ V |
15 |
14
|
shftval4 |
⊢ ( ( 1 ∈ ℂ ∧ 𝑘 ∈ ℂ ) → ( ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ‘ 𝑘 ) = ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ‘ ( 1 + 𝑘 ) ) ) |
16 |
10 12 15
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ‘ 𝑘 ) = ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ‘ ( 1 + 𝑘 ) ) ) |
17 |
|
addcom |
⊢ ( ( 1 ∈ ℂ ∧ 𝑘 ∈ ℂ ) → ( 1 + 𝑘 ) = ( 𝑘 + 1 ) ) |
18 |
10 12 17
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 1 + 𝑘 ) = ( 𝑘 + 1 ) ) |
19 |
18
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ‘ ( 1 + 𝑘 ) ) = ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) |
20 |
|
peano2nn0 |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑘 + 1 ) ∈ ℕ0 ) |
21 |
20
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 + 1 ) ∈ ℕ0 ) |
22 |
|
id |
⊢ ( 𝑖 = ( 𝑘 + 1 ) → 𝑖 = ( 𝑘 + 1 ) ) |
23 |
|
2fveq3 |
⊢ ( 𝑖 = ( 𝑘 + 1 ) → ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) = ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ ( 𝑘 + 1 ) ) ) ) |
24 |
22 23
|
oveq12d |
⊢ ( 𝑖 = ( 𝑘 + 1 ) → ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) = ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ ( 𝑘 + 1 ) ) ) ) ) |
25 |
|
eqid |
⊢ ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) = ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) |
26 |
|
ovex |
⊢ ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ ( 𝑘 + 1 ) ) ) ) ∈ V |
27 |
24 25 26
|
fvmpt |
⊢ ( ( 𝑘 + 1 ) ∈ ℕ0 → ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ ( 𝑘 + 1 ) ) ) ) ) |
28 |
21 27
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ ( 𝑘 + 1 ) ) ) ) ) |
29 |
1
|
pserval2 |
⊢ ( ( 𝑋 ∈ ℂ ∧ ( 𝑘 + 1 ) ∈ ℕ0 ) → ( ( 𝐺 ‘ 𝑋 ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) |
30 |
5 20 29
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐺 ‘ 𝑋 ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) |
31 |
30
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ ( 𝑘 + 1 ) ) ) = ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) |
32 |
31
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ ( 𝑘 + 1 ) ) ) ) = ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) |
33 |
28 32
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) |
34 |
16 19 33
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ‘ 𝑘 ) = ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) |
35 |
21
|
nn0red |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 + 1 ) ∈ ℝ ) |
36 |
|
ffvelrn |
⊢ ( ( 𝐴 : ℕ0 ⟶ ℂ ∧ ( 𝑘 + 1 ) ∈ ℕ0 ) → ( 𝐴 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
37 |
4 20 36
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
38 |
|
expcl |
⊢ ( ( 𝑋 ∈ ℂ ∧ ( 𝑘 + 1 ) ∈ ℕ0 ) → ( 𝑋 ↑ ( 𝑘 + 1 ) ) ∈ ℂ ) |
39 |
5 20 38
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑋 ↑ ( 𝑘 + 1 ) ) ∈ ℂ ) |
40 |
37 39
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ∈ ℂ ) |
41 |
40
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ∈ ℝ ) |
42 |
35 41
|
remulcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ∈ ℝ ) |
43 |
34 42
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ‘ 𝑘 ) ∈ ℝ ) |
44 |
|
oveq1 |
⊢ ( 𝑛 = 𝑘 → ( 𝑛 + 1 ) = ( 𝑘 + 1 ) ) |
45 |
44
|
fveq2d |
⊢ ( 𝑛 = 𝑘 → ( 𝐴 ‘ ( 𝑛 + 1 ) ) = ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) |
46 |
44 45
|
oveq12d |
⊢ ( 𝑛 = 𝑘 → ( ( 𝑛 + 1 ) · ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) = ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) ) |
47 |
|
oveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝑋 ↑ 𝑛 ) = ( 𝑋 ↑ 𝑘 ) ) |
48 |
46 47
|
oveq12d |
⊢ ( 𝑛 = 𝑘 → ( ( ( 𝑛 + 1 ) · ( 𝐴 ‘ ( 𝑛 + 1 ) ) ) · ( 𝑋 ↑ 𝑛 ) ) = ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) |
49 |
|
ovex |
⊢ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ∈ V |
50 |
48 3 49
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝐻 ‘ 𝑘 ) = ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) |
51 |
50
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐻 ‘ 𝑘 ) = ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) |
52 |
21
|
nn0cnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 + 1 ) ∈ ℂ ) |
53 |
52 37
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) ∈ ℂ ) |
54 |
|
expcl |
⊢ ( ( 𝑋 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝑋 ↑ 𝑘 ) ∈ ℂ ) |
55 |
5 54
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑋 ↑ 𝑘 ) ∈ ℂ ) |
56 |
53 55
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ∈ ℂ ) |
57 |
51 56
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐻 ‘ 𝑘 ) ∈ ℂ ) |
58 |
|
id |
⊢ ( 𝑖 = 𝑘 → 𝑖 = 𝑘 ) |
59 |
|
2fveq3 |
⊢ ( 𝑖 = 𝑘 → ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) = ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑘 ) ) ) |
60 |
58 59
|
oveq12d |
⊢ ( 𝑖 = 𝑘 → ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) = ( 𝑘 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑘 ) ) ) ) |
61 |
60
|
cbvmptv |
⊢ ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) = ( 𝑘 ∈ ℕ0 ↦ ( 𝑘 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑘 ) ) ) ) |
62 |
1 4 2 5 6 61
|
radcnvlt1 |
⊢ ( 𝜑 → ( seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ∈ dom ⇝ ∧ seq 0 ( + , ( abs ∘ ( 𝐺 ‘ 𝑋 ) ) ) ∈ dom ⇝ ) ) |
63 |
62
|
simpld |
⊢ ( 𝜑 → seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ∈ dom ⇝ ) |
64 |
|
climdm |
⊢ ( seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ∈ dom ⇝ ↔ seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ⇝ ( ⇝ ‘ seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ) ) |
65 |
63 64
|
sylib |
⊢ ( 𝜑 → seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ⇝ ( ⇝ ‘ seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ) ) |
66 |
|
0z |
⊢ 0 ∈ ℤ |
67 |
|
neg1z |
⊢ - 1 ∈ ℤ |
68 |
14
|
isershft |
⊢ ( ( 0 ∈ ℤ ∧ - 1 ∈ ℤ ) → ( seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ⇝ ( ⇝ ‘ seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ) ↔ seq ( 0 + - 1 ) ( + , ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ) ⇝ ( ⇝ ‘ seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ) ) ) |
69 |
66 67 68
|
mp2an |
⊢ ( seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ⇝ ( ⇝ ‘ seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ) ↔ seq ( 0 + - 1 ) ( + , ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ) ⇝ ( ⇝ ‘ seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ) ) |
70 |
65 69
|
sylib |
⊢ ( 𝜑 → seq ( 0 + - 1 ) ( + , ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ) ⇝ ( ⇝ ‘ seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ) ) |
71 |
|
seqex |
⊢ seq ( 0 + - 1 ) ( + , ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ) ∈ V |
72 |
|
fvex |
⊢ ( ⇝ ‘ seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ) ∈ V |
73 |
71 72
|
breldm |
⊢ ( seq ( 0 + - 1 ) ( + , ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ) ⇝ ( ⇝ ‘ seq 0 ( + , ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ) ) → seq ( 0 + - 1 ) ( + , ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ) ∈ dom ⇝ ) |
74 |
70 73
|
syl |
⊢ ( 𝜑 → seq ( 0 + - 1 ) ( + , ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ) ∈ dom ⇝ ) |
75 |
|
eqid |
⊢ ( ℤ≥ ‘ ( 0 + - 1 ) ) = ( ℤ≥ ‘ ( 0 + - 1 ) ) |
76 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
77 |
76
|
addid2i |
⊢ ( 0 + - 1 ) = - 1 |
78 |
|
0le1 |
⊢ 0 ≤ 1 |
79 |
|
1re |
⊢ 1 ∈ ℝ |
80 |
|
le0neg2 |
⊢ ( 1 ∈ ℝ → ( 0 ≤ 1 ↔ - 1 ≤ 0 ) ) |
81 |
79 80
|
ax-mp |
⊢ ( 0 ≤ 1 ↔ - 1 ≤ 0 ) |
82 |
78 81
|
mpbi |
⊢ - 1 ≤ 0 |
83 |
77 82
|
eqbrtri |
⊢ ( 0 + - 1 ) ≤ 0 |
84 |
77 67
|
eqeltri |
⊢ ( 0 + - 1 ) ∈ ℤ |
85 |
84
|
eluz1i |
⊢ ( 0 ∈ ( ℤ≥ ‘ ( 0 + - 1 ) ) ↔ ( 0 ∈ ℤ ∧ ( 0 + - 1 ) ≤ 0 ) ) |
86 |
66 83 85
|
mpbir2an |
⊢ 0 ∈ ( ℤ≥ ‘ ( 0 + - 1 ) ) |
87 |
86
|
a1i |
⊢ ( 𝜑 → 0 ∈ ( ℤ≥ ‘ ( 0 + - 1 ) ) ) |
88 |
|
eluzelcn |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ ( 0 + - 1 ) ) → 𝑘 ∈ ℂ ) |
89 |
88
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 0 + - 1 ) ) ) → 𝑘 ∈ ℂ ) |
90 |
10 89 15
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 0 + - 1 ) ) ) → ( ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ‘ 𝑘 ) = ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ‘ ( 1 + 𝑘 ) ) ) |
91 |
|
nn0re |
⊢ ( 𝑖 ∈ ℕ0 → 𝑖 ∈ ℝ ) |
92 |
91
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ0 ) → 𝑖 ∈ ℝ ) |
93 |
1 4 5
|
psergf |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑋 ) : ℕ0 ⟶ ℂ ) |
94 |
93
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ0 ) → ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ∈ ℂ ) |
95 |
94
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ0 ) → ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ∈ ℝ ) |
96 |
92 95
|
remulcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ0 ) → ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ∈ ℝ ) |
97 |
96
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ0 ) → ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ∈ ℂ ) |
98 |
97
|
fmpttd |
⊢ ( 𝜑 → ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) : ℕ0 ⟶ ℂ ) |
99 |
10 88 17
|
sylancr |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ ( 0 + - 1 ) ) → ( 1 + 𝑘 ) = ( 𝑘 + 1 ) ) |
100 |
|
eluzp1p1 |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ ( 0 + - 1 ) ) → ( 𝑘 + 1 ) ∈ ( ℤ≥ ‘ ( ( 0 + - 1 ) + 1 ) ) ) |
101 |
77
|
oveq1i |
⊢ ( ( 0 + - 1 ) + 1 ) = ( - 1 + 1 ) |
102 |
|
1pneg1e0 |
⊢ ( 1 + - 1 ) = 0 |
103 |
10 76 102
|
addcomli |
⊢ ( - 1 + 1 ) = 0 |
104 |
101 103
|
eqtri |
⊢ ( ( 0 + - 1 ) + 1 ) = 0 |
105 |
104
|
fveq2i |
⊢ ( ℤ≥ ‘ ( ( 0 + - 1 ) + 1 ) ) = ( ℤ≥ ‘ 0 ) |
106 |
7 105
|
eqtr4i |
⊢ ℕ0 = ( ℤ≥ ‘ ( ( 0 + - 1 ) + 1 ) ) |
107 |
100 106
|
eleqtrrdi |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ ( 0 + - 1 ) ) → ( 𝑘 + 1 ) ∈ ℕ0 ) |
108 |
99 107
|
eqeltrd |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ ( 0 + - 1 ) ) → ( 1 + 𝑘 ) ∈ ℕ0 ) |
109 |
|
ffvelrn |
⊢ ( ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) : ℕ0 ⟶ ℂ ∧ ( 1 + 𝑘 ) ∈ ℕ0 ) → ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ‘ ( 1 + 𝑘 ) ) ∈ ℂ ) |
110 |
98 108 109
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 0 + - 1 ) ) ) → ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) ‘ ( 1 + 𝑘 ) ) ∈ ℂ ) |
111 |
90 110
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 0 + - 1 ) ) ) → ( ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ‘ 𝑘 ) ∈ ℂ ) |
112 |
75 87 111
|
iserex |
⊢ ( 𝜑 → ( seq ( 0 + - 1 ) ( + , ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ) ∈ dom ⇝ ↔ seq 0 ( + , ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ) ∈ dom ⇝ ) ) |
113 |
74 112
|
mpbid |
⊢ ( 𝜑 → seq 0 ( + , ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ) ∈ dom ⇝ ) |
114 |
|
1red |
⊢ ( ( 𝜑 ∧ 𝑋 = 0 ) → 1 ∈ ℝ ) |
115 |
|
neqne |
⊢ ( ¬ 𝑋 = 0 → 𝑋 ≠ 0 ) |
116 |
|
absrpcl |
⊢ ( ( 𝑋 ∈ ℂ ∧ 𝑋 ≠ 0 ) → ( abs ‘ 𝑋 ) ∈ ℝ+ ) |
117 |
5 115 116
|
syl2an |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = 0 ) → ( abs ‘ 𝑋 ) ∈ ℝ+ ) |
118 |
117
|
rprecred |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = 0 ) → ( 1 / ( abs ‘ 𝑋 ) ) ∈ ℝ ) |
119 |
114 118
|
ifclda |
⊢ ( 𝜑 → if ( 𝑋 = 0 , 1 , ( 1 / ( abs ‘ 𝑋 ) ) ) ∈ ℝ ) |
120 |
|
oveq1 |
⊢ ( 1 = if ( 𝑋 = 0 , 1 , ( 1 / ( abs ‘ 𝑋 ) ) ) → ( 1 · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) = ( if ( 𝑋 = 0 , 1 , ( 1 / ( abs ‘ 𝑋 ) ) ) · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) ) |
121 |
120
|
breq2d |
⊢ ( 1 = if ( 𝑋 = 0 , 1 , ( 1 / ( abs ‘ 𝑋 ) ) ) → ( ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ≤ ( 1 · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) ↔ ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ≤ ( if ( 𝑋 = 0 , 1 , ( 1 / ( abs ‘ 𝑋 ) ) ) · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) ) ) |
122 |
|
oveq1 |
⊢ ( ( 1 / ( abs ‘ 𝑋 ) ) = if ( 𝑋 = 0 , 1 , ( 1 / ( abs ‘ 𝑋 ) ) ) → ( ( 1 / ( abs ‘ 𝑋 ) ) · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) = ( if ( 𝑋 = 0 , 1 , ( 1 / ( abs ‘ 𝑋 ) ) ) · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) ) |
123 |
122
|
breq2d |
⊢ ( ( 1 / ( abs ‘ 𝑋 ) ) = if ( 𝑋 = 0 , 1 , ( 1 / ( abs ‘ 𝑋 ) ) ) → ( ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ≤ ( ( 1 / ( abs ‘ 𝑋 ) ) · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) ↔ ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ≤ ( if ( 𝑋 = 0 , 1 , ( 1 / ( abs ‘ 𝑋 ) ) ) · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) ) ) |
124 |
|
elnnuz |
⊢ ( 𝑘 ∈ ℕ ↔ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) |
125 |
|
nnnn0 |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℕ0 ) |
126 |
124 125
|
sylbir |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 1 ) → 𝑘 ∈ ℕ0 ) |
127 |
21
|
nn0ge0d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 0 ≤ ( 𝑘 + 1 ) ) |
128 |
40
|
absge0d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 0 ≤ ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) |
129 |
35 41 127 128
|
mulge0d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 0 ≤ ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) |
130 |
126 129
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) → 0 ≤ ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) |
131 |
130
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) ∧ 𝑋 = 0 ) → 0 ≤ ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) |
132 |
|
oveq1 |
⊢ ( 𝑋 = 0 → ( 𝑋 ↑ 𝑘 ) = ( 0 ↑ 𝑘 ) ) |
133 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) → 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) |
134 |
133 124
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) → 𝑘 ∈ ℕ ) |
135 |
134
|
0expd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) → ( 0 ↑ 𝑘 ) = 0 ) |
136 |
132 135
|
sylan9eqr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) ∧ 𝑋 = 0 ) → ( 𝑋 ↑ 𝑘 ) = 0 ) |
137 |
136
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) ∧ 𝑋 = 0 ) → ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) = ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · 0 ) ) |
138 |
53
|
mul01d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · 0 ) = 0 ) |
139 |
126 138
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) → ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · 0 ) = 0 ) |
140 |
139
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) ∧ 𝑋 = 0 ) → ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · 0 ) = 0 ) |
141 |
137 140
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) ∧ 𝑋 = 0 ) → ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) = 0 ) |
142 |
141
|
abs00bd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) ∧ 𝑋 = 0 ) → ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) = 0 ) |
143 |
42
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ∈ ℂ ) |
144 |
143
|
mulid2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 1 · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) = ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) |
145 |
126 144
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) → ( 1 · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) = ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) |
146 |
145
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) ∧ 𝑋 = 0 ) → ( 1 · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) = ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) |
147 |
131 142 146
|
3brtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) ∧ 𝑋 = 0 ) → ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ≤ ( 1 · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) ) |
148 |
|
df-ne |
⊢ ( 𝑋 ≠ 0 ↔ ¬ 𝑋 = 0 ) |
149 |
56
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ∈ ℝ ) |
150 |
52 37 55
|
mulassd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) = ( ( 𝑘 + 1 ) · ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ) |
151 |
150
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) = ( abs ‘ ( ( 𝑘 + 1 ) · ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ) ) |
152 |
37 55
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) ∈ ℂ ) |
153 |
52 152
|
absmuld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( abs ‘ ( ( 𝑘 + 1 ) · ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ) = ( ( abs ‘ ( 𝑘 + 1 ) ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ) ) |
154 |
35 127
|
absidd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( abs ‘ ( 𝑘 + 1 ) ) = ( 𝑘 + 1 ) ) |
155 |
154
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( abs ‘ ( 𝑘 + 1 ) ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ) = ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ) ) |
156 |
151 153 155
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) = ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ) ) |
157 |
149 156
|
eqled |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ≤ ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ) ) |
158 |
157
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ≤ ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ) ) |
159 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑋 ∈ ℂ ) |
160 |
116
|
rpreccld |
⊢ ( ( 𝑋 ∈ ℂ ∧ 𝑋 ≠ 0 ) → ( 1 / ( abs ‘ 𝑋 ) ) ∈ ℝ+ ) |
161 |
159 160
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( 1 / ( abs ‘ 𝑋 ) ) ∈ ℝ+ ) |
162 |
161
|
rpcnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( 1 / ( abs ‘ 𝑋 ) ) ∈ ℂ ) |
163 |
52
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( 𝑘 + 1 ) ∈ ℂ ) |
164 |
41
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ∈ ℝ ) |
165 |
164
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ∈ ℂ ) |
166 |
162 163 165
|
mul12d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( ( 1 / ( abs ‘ 𝑋 ) ) · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) = ( ( 𝑘 + 1 ) · ( ( 1 / ( abs ‘ 𝑋 ) ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) ) |
167 |
40
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ∈ ℂ ) |
168 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ ℂ ) |
169 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → 𝑋 ≠ 0 ) |
170 |
167 168 169
|
absdivd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( abs ‘ ( ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) / 𝑋 ) ) = ( ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) / ( abs ‘ 𝑋 ) ) ) |
171 |
37
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( 𝐴 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
172 |
39
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( 𝑋 ↑ ( 𝑘 + 1 ) ) ∈ ℂ ) |
173 |
171 172 168 169
|
divassd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) / 𝑋 ) = ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( ( 𝑋 ↑ ( 𝑘 + 1 ) ) / 𝑋 ) ) ) |
174 |
12
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → 𝑘 ∈ ℂ ) |
175 |
|
pncan |
⊢ ( ( 𝑘 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑘 + 1 ) − 1 ) = 𝑘 ) |
176 |
174 10 175
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( ( 𝑘 + 1 ) − 1 ) = 𝑘 ) |
177 |
176
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( 𝑋 ↑ ( ( 𝑘 + 1 ) − 1 ) ) = ( 𝑋 ↑ 𝑘 ) ) |
178 |
21
|
nn0zd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 + 1 ) ∈ ℤ ) |
179 |
178
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( 𝑘 + 1 ) ∈ ℤ ) |
180 |
168 169 179
|
expm1d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( 𝑋 ↑ ( ( 𝑘 + 1 ) − 1 ) ) = ( ( 𝑋 ↑ ( 𝑘 + 1 ) ) / 𝑋 ) ) |
181 |
177 180
|
eqtr3d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( 𝑋 ↑ 𝑘 ) = ( ( 𝑋 ↑ ( 𝑘 + 1 ) ) / 𝑋 ) ) |
182 |
181
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) = ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( ( 𝑋 ↑ ( 𝑘 + 1 ) ) / 𝑋 ) ) ) |
183 |
173 182
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) / 𝑋 ) = ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) ) |
184 |
183
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( abs ‘ ( ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) / 𝑋 ) ) = ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ) |
185 |
5
|
abscld |
⊢ ( 𝜑 → ( abs ‘ 𝑋 ) ∈ ℝ ) |
186 |
185
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( abs ‘ 𝑋 ) ∈ ℝ ) |
187 |
186
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( abs ‘ 𝑋 ) ∈ ℂ ) |
188 |
159 116
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( abs ‘ 𝑋 ) ∈ ℝ+ ) |
189 |
188
|
rpne0d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( abs ‘ 𝑋 ) ≠ 0 ) |
190 |
165 187 189
|
divrec2d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) / ( abs ‘ 𝑋 ) ) = ( ( 1 / ( abs ‘ 𝑋 ) ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) |
191 |
170 184 190
|
3eqtr3rd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( ( 1 / ( abs ‘ 𝑋 ) ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) = ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ) |
192 |
191
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( ( 𝑘 + 1 ) · ( ( 1 / ( abs ‘ 𝑋 ) ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) = ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ) ) |
193 |
166 192
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( ( 1 / ( abs ‘ 𝑋 ) ) · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) = ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ) ) |
194 |
158 193
|
breqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑋 ≠ 0 ) → ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ≤ ( ( 1 / ( abs ‘ 𝑋 ) ) · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) ) |
195 |
126 194
|
sylanl2 |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) ∧ 𝑋 ≠ 0 ) → ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ≤ ( ( 1 / ( abs ‘ 𝑋 ) ) · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) ) |
196 |
148 195
|
sylan2br |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) ∧ ¬ 𝑋 = 0 ) → ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ≤ ( ( 1 / ( abs ‘ 𝑋 ) ) · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) ) |
197 |
121 123 147 196
|
ifbothda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) → ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ≤ ( if ( 𝑋 = 0 , 1 , ( 1 / ( abs ‘ 𝑋 ) ) ) · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) ) |
198 |
51
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( abs ‘ ( 𝐻 ‘ 𝑘 ) ) = ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ) |
199 |
126 198
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) → ( abs ‘ ( 𝐻 ‘ 𝑘 ) ) = ( abs ‘ ( ( ( 𝑘 + 1 ) · ( 𝐴 ‘ ( 𝑘 + 1 ) ) ) · ( 𝑋 ↑ 𝑘 ) ) ) ) |
200 |
34
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( if ( 𝑋 = 0 , 1 , ( 1 / ( abs ‘ 𝑋 ) ) ) · ( ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ‘ 𝑘 ) ) = ( if ( 𝑋 = 0 , 1 , ( 1 / ( abs ‘ 𝑋 ) ) ) · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) ) |
201 |
126 200
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) → ( if ( 𝑋 = 0 , 1 , ( 1 / ( abs ‘ 𝑋 ) ) ) · ( ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ‘ 𝑘 ) ) = ( if ( 𝑋 = 0 , 1 , ( 1 / ( abs ‘ 𝑋 ) ) ) · ( ( 𝑘 + 1 ) · ( abs ‘ ( ( 𝐴 ‘ ( 𝑘 + 1 ) ) · ( 𝑋 ↑ ( 𝑘 + 1 ) ) ) ) ) ) ) |
202 |
197 199 201
|
3brtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 1 ) ) → ( abs ‘ ( 𝐻 ‘ 𝑘 ) ) ≤ ( if ( 𝑋 = 0 , 1 , ( 1 / ( abs ‘ 𝑋 ) ) ) · ( ( ( 𝑖 ∈ ℕ0 ↦ ( 𝑖 · ( abs ‘ ( ( 𝐺 ‘ 𝑋 ) ‘ 𝑖 ) ) ) ) shift - 1 ) ‘ 𝑘 ) ) ) |
203 |
7 9 43 57 113 119 202
|
cvgcmpce |
⊢ ( 𝜑 → seq 0 ( + , 𝐻 ) ∈ dom ⇝ ) |