Step |
Hyp |
Ref |
Expression |
1 |
|
ftalem.1 |
⊢ 𝐴 = ( coeff ‘ 𝐹 ) |
2 |
|
ftalem.2 |
⊢ 𝑁 = ( deg ‘ 𝐹 ) |
3 |
|
ftalem.3 |
⊢ ( 𝜑 → 𝐹 ∈ ( Poly ‘ 𝑆 ) ) |
4 |
|
ftalem.4 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
5 |
|
ftalem4.5 |
⊢ ( 𝜑 → ( 𝐹 ‘ 0 ) ≠ 0 ) |
6 |
|
ftalem4.6 |
⊢ 𝐾 = inf ( { 𝑛 ∈ ℕ ∣ ( 𝐴 ‘ 𝑛 ) ≠ 0 } , ℝ , < ) |
7 |
|
ftalem4.7 |
⊢ 𝑇 = ( - ( ( 𝐹 ‘ 0 ) / ( 𝐴 ‘ 𝐾 ) ) ↑𝑐 ( 1 / 𝐾 ) ) |
8 |
|
ftalem4.8 |
⊢ 𝑈 = ( ( abs ‘ ( 𝐹 ‘ 0 ) ) / ( Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( abs ‘ ( ( 𝐴 ‘ 𝑘 ) · ( 𝑇 ↑ 𝑘 ) ) ) + 1 ) ) |
9 |
|
ftalem4.9 |
⊢ 𝑋 = if ( 1 ≤ 𝑈 , 1 , 𝑈 ) |
10 |
|
ssrab2 |
⊢ { 𝑛 ∈ ℕ ∣ ( 𝐴 ‘ 𝑛 ) ≠ 0 } ⊆ ℕ |
11 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
12 |
10 11
|
sseqtri |
⊢ { 𝑛 ∈ ℕ ∣ ( 𝐴 ‘ 𝑛 ) ≠ 0 } ⊆ ( ℤ≥ ‘ 1 ) |
13 |
|
fveq2 |
⊢ ( 𝑛 = 𝑁 → ( 𝐴 ‘ 𝑛 ) = ( 𝐴 ‘ 𝑁 ) ) |
14 |
13
|
neeq1d |
⊢ ( 𝑛 = 𝑁 → ( ( 𝐴 ‘ 𝑛 ) ≠ 0 ↔ ( 𝐴 ‘ 𝑁 ) ≠ 0 ) ) |
15 |
4
|
nnne0d |
⊢ ( 𝜑 → 𝑁 ≠ 0 ) |
16 |
2 1
|
dgreq0 |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ( 𝐹 = 0𝑝 ↔ ( 𝐴 ‘ 𝑁 ) = 0 ) ) |
17 |
3 16
|
syl |
⊢ ( 𝜑 → ( 𝐹 = 0𝑝 ↔ ( 𝐴 ‘ 𝑁 ) = 0 ) ) |
18 |
|
fveq2 |
⊢ ( 𝐹 = 0𝑝 → ( deg ‘ 𝐹 ) = ( deg ‘ 0𝑝 ) ) |
19 |
|
dgr0 |
⊢ ( deg ‘ 0𝑝 ) = 0 |
20 |
18 19
|
eqtrdi |
⊢ ( 𝐹 = 0𝑝 → ( deg ‘ 𝐹 ) = 0 ) |
21 |
2 20
|
syl5eq |
⊢ ( 𝐹 = 0𝑝 → 𝑁 = 0 ) |
22 |
17 21
|
syl6bir |
⊢ ( 𝜑 → ( ( 𝐴 ‘ 𝑁 ) = 0 → 𝑁 = 0 ) ) |
23 |
22
|
necon3d |
⊢ ( 𝜑 → ( 𝑁 ≠ 0 → ( 𝐴 ‘ 𝑁 ) ≠ 0 ) ) |
24 |
15 23
|
mpd |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝑁 ) ≠ 0 ) |
25 |
14 4 24
|
elrabd |
⊢ ( 𝜑 → 𝑁 ∈ { 𝑛 ∈ ℕ ∣ ( 𝐴 ‘ 𝑛 ) ≠ 0 } ) |
26 |
25
|
ne0d |
⊢ ( 𝜑 → { 𝑛 ∈ ℕ ∣ ( 𝐴 ‘ 𝑛 ) ≠ 0 } ≠ ∅ ) |
27 |
|
infssuzcl |
⊢ ( ( { 𝑛 ∈ ℕ ∣ ( 𝐴 ‘ 𝑛 ) ≠ 0 } ⊆ ( ℤ≥ ‘ 1 ) ∧ { 𝑛 ∈ ℕ ∣ ( 𝐴 ‘ 𝑛 ) ≠ 0 } ≠ ∅ ) → inf ( { 𝑛 ∈ ℕ ∣ ( 𝐴 ‘ 𝑛 ) ≠ 0 } , ℝ , < ) ∈ { 𝑛 ∈ ℕ ∣ ( 𝐴 ‘ 𝑛 ) ≠ 0 } ) |
28 |
12 26 27
|
sylancr |
⊢ ( 𝜑 → inf ( { 𝑛 ∈ ℕ ∣ ( 𝐴 ‘ 𝑛 ) ≠ 0 } , ℝ , < ) ∈ { 𝑛 ∈ ℕ ∣ ( 𝐴 ‘ 𝑛 ) ≠ 0 } ) |
29 |
6 28
|
eqeltrid |
⊢ ( 𝜑 → 𝐾 ∈ { 𝑛 ∈ ℕ ∣ ( 𝐴 ‘ 𝑛 ) ≠ 0 } ) |
30 |
|
fveq2 |
⊢ ( 𝑛 = 𝐾 → ( 𝐴 ‘ 𝑛 ) = ( 𝐴 ‘ 𝐾 ) ) |
31 |
30
|
neeq1d |
⊢ ( 𝑛 = 𝐾 → ( ( 𝐴 ‘ 𝑛 ) ≠ 0 ↔ ( 𝐴 ‘ 𝐾 ) ≠ 0 ) ) |
32 |
31
|
elrab |
⊢ ( 𝐾 ∈ { 𝑛 ∈ ℕ ∣ ( 𝐴 ‘ 𝑛 ) ≠ 0 } ↔ ( 𝐾 ∈ ℕ ∧ ( 𝐴 ‘ 𝐾 ) ≠ 0 ) ) |
33 |
29 32
|
sylib |
⊢ ( 𝜑 → ( 𝐾 ∈ ℕ ∧ ( 𝐴 ‘ 𝐾 ) ≠ 0 ) ) |
34 |
|
plyf |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → 𝐹 : ℂ ⟶ ℂ ) |
35 |
3 34
|
syl |
⊢ ( 𝜑 → 𝐹 : ℂ ⟶ ℂ ) |
36 |
|
0cn |
⊢ 0 ∈ ℂ |
37 |
|
ffvelrn |
⊢ ( ( 𝐹 : ℂ ⟶ ℂ ∧ 0 ∈ ℂ ) → ( 𝐹 ‘ 0 ) ∈ ℂ ) |
38 |
35 36 37
|
sylancl |
⊢ ( 𝜑 → ( 𝐹 ‘ 0 ) ∈ ℂ ) |
39 |
1
|
coef3 |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → 𝐴 : ℕ0 ⟶ ℂ ) |
40 |
3 39
|
syl |
⊢ ( 𝜑 → 𝐴 : ℕ0 ⟶ ℂ ) |
41 |
33
|
simpld |
⊢ ( 𝜑 → 𝐾 ∈ ℕ ) |
42 |
41
|
nnnn0d |
⊢ ( 𝜑 → 𝐾 ∈ ℕ0 ) |
43 |
40 42
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝐾 ) ∈ ℂ ) |
44 |
33
|
simprd |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝐾 ) ≠ 0 ) |
45 |
38 43 44
|
divcld |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 0 ) / ( 𝐴 ‘ 𝐾 ) ) ∈ ℂ ) |
46 |
45
|
negcld |
⊢ ( 𝜑 → - ( ( 𝐹 ‘ 0 ) / ( 𝐴 ‘ 𝐾 ) ) ∈ ℂ ) |
47 |
41
|
nnrecred |
⊢ ( 𝜑 → ( 1 / 𝐾 ) ∈ ℝ ) |
48 |
47
|
recnd |
⊢ ( 𝜑 → ( 1 / 𝐾 ) ∈ ℂ ) |
49 |
46 48
|
cxpcld |
⊢ ( 𝜑 → ( - ( ( 𝐹 ‘ 0 ) / ( 𝐴 ‘ 𝐾 ) ) ↑𝑐 ( 1 / 𝐾 ) ) ∈ ℂ ) |
50 |
7 49
|
eqeltrid |
⊢ ( 𝜑 → 𝑇 ∈ ℂ ) |
51 |
38 5
|
absrpcld |
⊢ ( 𝜑 → ( abs ‘ ( 𝐹 ‘ 0 ) ) ∈ ℝ+ ) |
52 |
|
fzfid |
⊢ ( 𝜑 → ( ( 𝐾 + 1 ) ... 𝑁 ) ∈ Fin ) |
53 |
|
peano2nn0 |
⊢ ( 𝐾 ∈ ℕ0 → ( 𝐾 + 1 ) ∈ ℕ0 ) |
54 |
42 53
|
syl |
⊢ ( 𝜑 → ( 𝐾 + 1 ) ∈ ℕ0 ) |
55 |
|
elfzuz |
⊢ ( 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) → 𝑘 ∈ ( ℤ≥ ‘ ( 𝐾 + 1 ) ) ) |
56 |
|
eluznn0 |
⊢ ( ( ( 𝐾 + 1 ) ∈ ℕ0 ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐾 + 1 ) ) ) → 𝑘 ∈ ℕ0 ) |
57 |
54 55 56
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → 𝑘 ∈ ℕ0 ) |
58 |
40
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℂ ) |
59 |
57 58
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → ( 𝐴 ‘ 𝑘 ) ∈ ℂ ) |
60 |
|
expcl |
⊢ ( ( 𝑇 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝑇 ↑ 𝑘 ) ∈ ℂ ) |
61 |
50 57 60
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → ( 𝑇 ↑ 𝑘 ) ∈ ℂ ) |
62 |
59 61
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝑇 ↑ 𝑘 ) ) ∈ ℂ ) |
63 |
62
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → ( abs ‘ ( ( 𝐴 ‘ 𝑘 ) · ( 𝑇 ↑ 𝑘 ) ) ) ∈ ℝ ) |
64 |
52 63
|
fsumrecl |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( abs ‘ ( ( 𝐴 ‘ 𝑘 ) · ( 𝑇 ↑ 𝑘 ) ) ) ∈ ℝ ) |
65 |
62
|
absge0d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → 0 ≤ ( abs ‘ ( ( 𝐴 ‘ 𝑘 ) · ( 𝑇 ↑ 𝑘 ) ) ) ) |
66 |
52 63 65
|
fsumge0 |
⊢ ( 𝜑 → 0 ≤ Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( abs ‘ ( ( 𝐴 ‘ 𝑘 ) · ( 𝑇 ↑ 𝑘 ) ) ) ) |
67 |
64 66
|
ge0p1rpd |
⊢ ( 𝜑 → ( Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( abs ‘ ( ( 𝐴 ‘ 𝑘 ) · ( 𝑇 ↑ 𝑘 ) ) ) + 1 ) ∈ ℝ+ ) |
68 |
51 67
|
rpdivcld |
⊢ ( 𝜑 → ( ( abs ‘ ( 𝐹 ‘ 0 ) ) / ( Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( abs ‘ ( ( 𝐴 ‘ 𝑘 ) · ( 𝑇 ↑ 𝑘 ) ) ) + 1 ) ) ∈ ℝ+ ) |
69 |
8 68
|
eqeltrid |
⊢ ( 𝜑 → 𝑈 ∈ ℝ+ ) |
70 |
|
1rp |
⊢ 1 ∈ ℝ+ |
71 |
|
ifcl |
⊢ ( ( 1 ∈ ℝ+ ∧ 𝑈 ∈ ℝ+ ) → if ( 1 ≤ 𝑈 , 1 , 𝑈 ) ∈ ℝ+ ) |
72 |
70 69 71
|
sylancr |
⊢ ( 𝜑 → if ( 1 ≤ 𝑈 , 1 , 𝑈 ) ∈ ℝ+ ) |
73 |
9 72
|
eqeltrid |
⊢ ( 𝜑 → 𝑋 ∈ ℝ+ ) |
74 |
50 69 73
|
3jca |
⊢ ( 𝜑 → ( 𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ+ ∧ 𝑋 ∈ ℝ+ ) ) |
75 |
33 74
|
jca |
⊢ ( 𝜑 → ( ( 𝐾 ∈ ℕ ∧ ( 𝐴 ‘ 𝐾 ) ≠ 0 ) ∧ ( 𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ+ ∧ 𝑋 ∈ ℝ+ ) ) ) |