Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem73.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
fourierdlem73.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
fourierdlem73.f |
⊢ ( 𝜑 → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ ) |
4 |
|
fourierdlem73.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
5 |
|
fourierdlem73.qf |
⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( 𝐴 [,] 𝐵 ) ) |
6 |
|
fourierdlem73.q0 |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) = 𝐴 ) |
7 |
|
fourierdlem73.qm |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) = 𝐵 ) |
8 |
|
fourierdlem73.qilt |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
9 |
|
fourierdlem73.fcn |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
10 |
|
fourierdlem73.l |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
11 |
|
fourierdlem73.r |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
12 |
|
fourierdlem73.g |
⊢ 𝐺 = ( ℝ D 𝐹 ) |
13 |
|
fourierdlem73.gcn |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
14 |
|
fourierdlem73.gbd |
⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) |
15 |
|
fourierdlem73.s |
⊢ 𝑆 = ( 𝑟 ∈ ℝ+ ↦ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
16 |
|
fourierdlem73.d |
⊢ 𝐷 = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ) |
17 |
|
cncff |
⊢ ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) → ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ) |
18 |
13 17
|
syl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ) |
19 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
20 |
19
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ℝ ⊆ ℂ ) |
21 |
1 2
|
iccssred |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
22 |
5 21
|
fssd |
⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
23 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
24 |
|
elfzofz |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
25 |
24
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
26 |
23 25
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
27 |
|
fzofzp1 |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
28 |
27
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
29 |
23 28
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
30 |
26 29
|
iccssred |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ ) |
31 |
|
limccl |
⊢ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ⊆ ℂ |
32 |
31 11
|
sseldi |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ℂ ) |
33 |
32
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑅 ∈ ℂ ) |
34 |
|
limccl |
⊢ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ |
35 |
34 10
|
sseldi |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ℂ ) |
36 |
35
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐿 ∈ ℂ ) |
37 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ ) |
38 |
1
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐴 ∈ ℝ ) |
39 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐵 ∈ ℝ ) |
40 |
26
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
41 |
29
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
42 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
43 |
|
eliccre |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℝ ) |
44 |
40 41 42 43
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℝ ) |
45 |
1
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
46 |
45
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐴 ∈ ℝ* ) |
47 |
2
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
48 |
47
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐵 ∈ ℝ* ) |
49 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( 𝐴 [,] 𝐵 ) ) |
50 |
49 25
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ( 𝐴 [,] 𝐵 ) ) |
51 |
|
iccgelb |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝑄 ‘ 𝑖 ) ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ ( 𝑄 ‘ 𝑖 ) ) |
52 |
46 48 50 51
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐴 ≤ ( 𝑄 ‘ 𝑖 ) ) |
53 |
52
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐴 ≤ ( 𝑄 ‘ 𝑖 ) ) |
54 |
40
|
rexrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
55 |
41
|
rexrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
56 |
|
iccgelb |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ≤ 𝑥 ) |
57 |
54 55 42 56
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ≤ 𝑥 ) |
58 |
38 40 44 53 57
|
letrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐴 ≤ 𝑥 ) |
59 |
|
iccleub |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
60 |
54 55 42 59
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
61 |
45
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐴 ∈ ℝ* ) |
62 |
47
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐵 ∈ ℝ* ) |
63 |
49 28
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( 𝐴 [,] 𝐵 ) ) |
64 |
63
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( 𝐴 [,] 𝐵 ) ) |
65 |
|
iccleub |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ≤ 𝐵 ) |
66 |
61 62 64 65
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ≤ 𝐵 ) |
67 |
44 41 39 60 66
|
letrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ≤ 𝐵 ) |
68 |
38 39 44 58 67
|
eliccd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) |
69 |
37 68
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ ) |
70 |
36 69
|
ifcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ∈ ℂ ) |
71 |
33 70
|
ifcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ∈ ℂ ) |
72 |
71 16
|
fmptd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐷 : ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ) |
73 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
74 |
73
|
tgioo2 |
⊢ ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
75 |
|
iccntr |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
76 |
26 29 75
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
77 |
20 30 72 74 73 76
|
dvresntr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ℝ D 𝐷 ) = ( ℝ D ( 𝐷 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
78 |
|
ioossicc |
⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
79 |
78
|
sseli |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
80 |
79
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
81 |
|
fvres |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
82 |
80 81
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
83 |
80 71
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ∈ ℂ ) |
84 |
16
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ∈ ℂ ) → ( 𝐷 ‘ 𝑥 ) = if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ) |
85 |
80 83 84
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐷 ‘ 𝑥 ) = if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ) |
86 |
26
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
87 |
80 54
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
88 |
80 55
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
89 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
90 |
|
ioogtlb |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑥 ) |
91 |
87 88 89 90
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑥 ) |
92 |
86 91
|
gtned |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ≠ ( 𝑄 ‘ 𝑖 ) ) |
93 |
92
|
neneqd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) |
94 |
93
|
iffalsed |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) |
95 |
|
elioore |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ℝ ) |
96 |
95
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℝ ) |
97 |
|
iooltub |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
98 |
87 88 89 97
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
99 |
96 98
|
ltned |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ≠ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
100 |
99
|
neneqd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
101 |
100
|
iffalsed |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
102 |
85 94 101
|
3eqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐷 ‘ 𝑥 ) ) |
103 |
82 102
|
eqtr2d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐷 ‘ 𝑥 ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) |
104 |
103
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐷 ‘ 𝑥 ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) |
105 |
|
ffn |
⊢ ( 𝐷 : ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ → 𝐷 Fn ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
106 |
72 105
|
syl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐷 Fn ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
107 |
|
ffn |
⊢ ( 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ → 𝐹 Fn ( 𝐴 [,] 𝐵 ) ) |
108 |
3 107
|
syl |
⊢ ( 𝜑 → 𝐹 Fn ( 𝐴 [,] 𝐵 ) ) |
109 |
108
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) ) |
110 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ..^ 𝑀 ) ) |
111 |
46 48 49 110
|
fourierdlem8 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
112 |
|
fnssres |
⊢ ( ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) ∧ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) Fn ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
113 |
109 111 112
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) Fn ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
114 |
78
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
115 |
|
fvreseq |
⊢ ( ( ( 𝐷 Fn ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) Fn ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐷 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐷 ‘ 𝑥 ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) |
116 |
106 113 114 115
|
syl21anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐷 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐷 ‘ 𝑥 ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) |
117 |
104 116
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐷 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
118 |
114
|
resabs1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
119 |
117 118
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐷 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
120 |
119
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ℝ D ( 𝐷 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) = ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
121 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ ) |
122 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
123 |
114 30
|
sstrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ ) |
124 |
73 74
|
dvres |
⊢ ( ( ( ℝ ⊆ ℂ ∧ 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ ) ∧ ( ( 𝐴 [,] 𝐵 ) ⊆ ℝ ∧ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ ) ) → ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
125 |
20 121 122 123 124
|
syl22anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) = ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
126 |
12
|
eqcomi |
⊢ ( ℝ D 𝐹 ) = 𝐺 |
127 |
126
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ℝ D 𝐹 ) = 𝐺 ) |
128 |
|
iooretop |
⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∈ ( topGen ‘ ran (,) ) |
129 |
|
retop |
⊢ ( topGen ‘ ran (,) ) ∈ Top |
130 |
|
uniretop |
⊢ ℝ = ∪ ( topGen ‘ ran (,) ) |
131 |
130
|
isopn3 |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ Top ∧ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ ) → ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∈ ( topGen ‘ ran (,) ) ↔ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
132 |
129 123 131
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∈ ( topGen ‘ ran (,) ) ↔ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
133 |
128 132
|
mpbii |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
134 |
127 133
|
reseq12d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ℝ D 𝐹 ) ↾ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) = ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
135 |
125 134
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ℝ D ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) = ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
136 |
77 120 135
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ℝ D 𝐷 ) = ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
137 |
136
|
feq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ℝ D 𝐷 ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ↔ ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ) ) |
138 |
18 137
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ℝ D 𝐷 ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ) |
139 |
138
|
feqmptd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ℝ D 𝐷 ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ) |
140 |
139 136
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) = ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
141 |
|
ioombl |
⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∈ dom vol |
142 |
141
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∈ dom vol ) |
143 |
26 29 8
|
ltled |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
144 |
|
volioo |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ∧ ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( vol ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑄 ‘ 𝑖 ) ) ) |
145 |
26 29 143 144
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( vol ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑄 ‘ 𝑖 ) ) ) |
146 |
29 26
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑄 ‘ 𝑖 ) ) ∈ ℝ ) |
147 |
145 146
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( vol ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ℝ ) |
148 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) |
149 |
|
nfv |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ℝ ) |
150 |
|
nfra1 |
⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 |
151 |
149 150
|
nfan |
⊢ Ⅎ 𝑥 ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) |
152 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) → 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
153 |
|
fdm |
⊢ ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ → dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
154 |
18 153
|
syl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
155 |
154
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) → dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
156 |
152 155
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
157 |
|
fvres |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
158 |
156 157
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) → ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
159 |
158
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) → ( abs ‘ ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ) |
160 |
159
|
ad4ant14 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) → ( abs ‘ ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ) |
161 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) → ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) |
162 |
|
ssdmres |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ dom 𝐺 ↔ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
163 |
154 162
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ dom 𝐺 ) |
164 |
163
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ dom 𝐺 ) |
165 |
156 164
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) → 𝑥 ∈ dom 𝐺 ) |
166 |
165
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) → 𝑥 ∈ dom 𝐺 ) |
167 |
|
rsp |
⊢ ( ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 → ( 𝑥 ∈ dom 𝐺 → ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) ) |
168 |
161 166 167
|
sylc |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) |
169 |
168
|
adantllr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) |
170 |
160 169
|
eqbrtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) → ( abs ‘ ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ≤ 𝑦 ) |
171 |
170
|
ex |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) → ( 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ≤ 𝑦 ) ) |
172 |
151 171
|
ralrimi |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) → ∀ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ( abs ‘ ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ≤ 𝑦 ) |
173 |
172
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ℝ ) → ( ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 → ∀ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ( abs ‘ ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ≤ 𝑦 ) ) |
174 |
173
|
reximdva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ( abs ‘ ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ≤ 𝑦 ) ) |
175 |
148 174
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ dom ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ( abs ‘ ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ≤ 𝑦 ) |
176 |
142 147 13 175
|
cnbdibl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ 𝐿1 ) |
177 |
140 176
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ∈ 𝐿1 ) |
178 |
177
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ∈ 𝐿1 ) |
179 |
141
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∈ dom vol ) |
180 |
147
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( vol ‘ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ℝ ) |
181 |
140 13
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
182 |
181
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
183 |
|
coscn |
⊢ cos ∈ ( ℂ –cn→ ℂ ) |
184 |
183
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → cos ∈ ( ℂ –cn→ ℂ ) ) |
185 |
|
ioosscn |
⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ |
186 |
185
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ ) |
187 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → 𝑟 ∈ ℝ ) |
188 |
187
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → 𝑟 ∈ ℂ ) |
189 |
|
ssid |
⊢ ℂ ⊆ ℂ |
190 |
189
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ℂ ⊆ ℂ ) |
191 |
186 188 190
|
constcncfg |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ 𝑟 ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
192 |
185
|
a1i |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ ) |
193 |
189
|
a1i |
⊢ ( 𝜑 → ℂ ⊆ ℂ ) |
194 |
192 193
|
idcncfg |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ 𝑥 ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
195 |
194
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ 𝑥 ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
196 |
191 195
|
mulcncf |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑟 · 𝑥 ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
197 |
184 196
|
cncfmpt1f |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
198 |
197
|
negcncfg |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
199 |
182 198
|
mulcncf |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
200 |
|
nfv |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) |
201 |
200 150
|
nfan |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) |
202 |
136
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ℝ D 𝐷 ) ‘ 𝑥 ) = ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) |
203 |
202 157
|
sylan9eq |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ℝ D 𝐷 ) ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
204 |
203
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) = ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ) |
205 |
204
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) = ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ) |
206 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) |
207 |
164
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ dom 𝐺 ) |
208 |
206 207 167
|
sylc |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) |
209 |
205 208
|
eqbrtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) |
210 |
209
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) ) |
211 |
201 210
|
ralrimi |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) → ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) |
212 |
211
|
ex |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 → ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) ) |
213 |
212
|
reximdv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ dom 𝐺 ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) ) |
214 |
148 213
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) |
215 |
214
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) |
216 |
|
eqidd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ) |
217 |
|
fveq2 |
⊢ ( 𝑥 = 𝑧 → ( ( ℝ D 𝐷 ) ‘ 𝑥 ) = ( ( ℝ D 𝐷 ) ‘ 𝑧 ) ) |
218 |
|
eleq1w |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
219 |
218
|
anbi2d |
⊢ ( 𝑥 = 𝑧 → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
220 |
|
fveq2 |
⊢ ( 𝑥 = 𝑧 → ( 𝐺 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑧 ) ) |
221 |
217 220
|
eqeq12d |
⊢ ( 𝑥 = 𝑧 → ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ↔ ( ( ℝ D 𝐷 ) ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) |
222 |
219 221
|
imbi12d |
⊢ ( 𝑥 = 𝑧 → ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ℝ D 𝐷 ) ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) ↔ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ℝ D 𝐷 ) ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) ) |
223 |
222 203
|
chvarvv |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ℝ D 𝐷 ) ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) |
224 |
217 223
|
sylan9eqr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 = 𝑧 ) → ( ( ℝ D 𝐷 ) ‘ 𝑥 ) = ( 𝐺 ‘ 𝑧 ) ) |
225 |
|
oveq2 |
⊢ ( 𝑥 = 𝑧 → ( 𝑟 · 𝑥 ) = ( 𝑟 · 𝑧 ) ) |
226 |
225
|
fveq2d |
⊢ ( 𝑥 = 𝑧 → ( cos ‘ ( 𝑟 · 𝑥 ) ) = ( cos ‘ ( 𝑟 · 𝑧 ) ) ) |
227 |
226
|
negeqd |
⊢ ( 𝑥 = 𝑧 → - ( cos ‘ ( 𝑟 · 𝑥 ) ) = - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) |
228 |
227
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 = 𝑧 ) → - ( cos ‘ ( 𝑟 · 𝑥 ) ) = - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) |
229 |
224 228
|
oveq12d |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 = 𝑧 ) → ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) = ( ( 𝐺 ‘ 𝑧 ) · - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ) |
230 |
229
|
adantllr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 = 𝑧 ) → ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) = ( ( 𝐺 ‘ 𝑧 ) · - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ) |
231 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
232 |
|
fvres |
⊢ ( 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) |
233 |
232
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) |
234 |
18
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑧 ) ∈ ℂ ) |
235 |
233 234
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐺 ‘ 𝑧 ) ∈ ℂ ) |
236 |
235
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐺 ‘ 𝑧 ) ∈ ℂ ) |
237 |
|
simpl |
⊢ ( ( 𝑟 ∈ ℝ ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑟 ∈ ℝ ) |
238 |
|
elioore |
⊢ ( 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑧 ∈ ℝ ) |
239 |
238
|
adantl |
⊢ ( ( 𝑟 ∈ ℝ ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑧 ∈ ℝ ) |
240 |
237 239
|
remulcld |
⊢ ( ( 𝑟 ∈ ℝ ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑟 · 𝑧 ) ∈ ℝ ) |
241 |
240
|
recnd |
⊢ ( ( 𝑟 ∈ ℝ ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑟 · 𝑧 ) ∈ ℂ ) |
242 |
241
|
coscld |
⊢ ( ( 𝑟 ∈ ℝ ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( cos ‘ ( 𝑟 · 𝑧 ) ) ∈ ℂ ) |
243 |
242
|
negcld |
⊢ ( ( 𝑟 ∈ ℝ ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → - ( cos ‘ ( 𝑟 · 𝑧 ) ) ∈ ℂ ) |
244 |
243
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → - ( cos ‘ ( 𝑟 · 𝑧 ) ) ∈ ℂ ) |
245 |
236 244
|
mulcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐺 ‘ 𝑧 ) · - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ∈ ℂ ) |
246 |
216 230 231 245
|
fvmptd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ‘ 𝑧 ) = ( ( 𝐺 ‘ 𝑧 ) · - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ) |
247 |
246
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ‘ 𝑧 ) ) = ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) · - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ) ) |
248 |
247
|
ad4ant14 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ‘ 𝑧 ) ) = ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) · - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ) ) |
249 |
245
|
abscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) · - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ) ∈ ℝ ) |
250 |
249
|
ad4ant14 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) · - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ) ∈ ℝ ) |
251 |
236
|
abscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ∈ ℝ ) |
252 |
251
|
ad4ant14 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ∈ ℝ ) |
253 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑦 ∈ ℝ ) |
254 |
244
|
abscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ∈ ℝ ) |
255 |
|
1red |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 1 ∈ ℝ ) |
256 |
236
|
absge0d |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 0 ≤ ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ) |
257 |
242
|
absnegd |
⊢ ( ( 𝑟 ∈ ℝ ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) = ( abs ‘ ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ) |
258 |
|
abscosbd |
⊢ ( ( 𝑟 · 𝑧 ) ∈ ℝ → ( abs ‘ ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ≤ 1 ) |
259 |
240 258
|
syl |
⊢ ( ( 𝑟 ∈ ℝ ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ≤ 1 ) |
260 |
257 259
|
eqbrtrd |
⊢ ( ( 𝑟 ∈ ℝ ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ≤ 1 ) |
261 |
260
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ≤ 1 ) |
262 |
254 255 251 256 261
|
lemul2ad |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) · ( abs ‘ - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ) ≤ ( ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) · 1 ) ) |
263 |
236 244
|
absmuld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) · - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ) = ( ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) · ( abs ‘ - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ) ) |
264 |
251
|
recnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ∈ ℂ ) |
265 |
264
|
mulid1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) · 1 ) = ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ) |
266 |
265
|
eqcomd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) = ( ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) · 1 ) ) |
267 |
262 263 266
|
3brtr4d |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) · - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ) ≤ ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ) |
268 |
267
|
ad4ant14 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) · - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ) ≤ ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ) |
269 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) → ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) |
270 |
|
nfra1 |
⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 |
271 |
200 270
|
nfan |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) |
272 |
204
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) = ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ) |
273 |
272
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) → ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) = ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ) |
274 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) → ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) |
275 |
273 274
|
eqbrtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) → ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) |
276 |
275
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 → ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) ) |
277 |
276
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 → ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) ) |
278 |
271 277
|
ralimdaa |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) → ( ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 → ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) ) |
279 |
269 278
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) → ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ) |
280 |
220
|
fveq2d |
⊢ ( 𝑥 = 𝑧 → ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) = ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ) |
281 |
280
|
breq1d |
⊢ ( 𝑥 = 𝑧 → ( ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ↔ ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑦 ) ) |
282 |
281
|
cbvralvw |
⊢ ( ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( 𝐺 ‘ 𝑥 ) ) ≤ 𝑦 ↔ ∀ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑦 ) |
283 |
279 282
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) → ∀ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑦 ) |
284 |
283
|
ad4ant14 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) → ∀ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑦 ) |
285 |
284
|
r19.21bi |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( 𝐺 ‘ 𝑧 ) ) ≤ 𝑦 ) |
286 |
250 252 253 268 285
|
letrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( 𝐺 ‘ 𝑧 ) · - ( cos ‘ ( 𝑟 · 𝑧 ) ) ) ) ≤ 𝑦 ) |
287 |
248 286
|
eqbrtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) ∧ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( abs ‘ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ‘ 𝑧 ) ) ≤ 𝑦 ) |
288 |
287
|
ralrimiva |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) → ∀ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ‘ 𝑧 ) ) ≤ 𝑦 ) |
289 |
138
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ∈ ℂ ) |
290 |
289
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ∈ ℂ ) |
291 |
|
simpl |
⊢ ( ( 𝑟 ∈ ℝ ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑟 ∈ ℝ ) |
292 |
95
|
adantl |
⊢ ( ( 𝑟 ∈ ℝ ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℝ ) |
293 |
291 292
|
remulcld |
⊢ ( ( 𝑟 ∈ ℝ ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑟 · 𝑥 ) ∈ ℝ ) |
294 |
293
|
recnd |
⊢ ( ( 𝑟 ∈ ℝ ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑟 · 𝑥 ) ∈ ℂ ) |
295 |
294
|
coscld |
⊢ ( ( 𝑟 ∈ ℝ ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( cos ‘ ( 𝑟 · 𝑥 ) ) ∈ ℂ ) |
296 |
295
|
negcld |
⊢ ( ( 𝑟 ∈ ℝ ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → - ( cos ‘ ( 𝑟 · 𝑥 ) ) ∈ ℂ ) |
297 |
296
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → - ( cos ‘ ( 𝑟 · 𝑥 ) ) ∈ ℂ ) |
298 |
290 297
|
mulcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ∈ ℂ ) |
299 |
298
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ∈ ℂ ) |
300 |
|
dmmptg |
⊢ ( ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ∈ ℂ → dom ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
301 |
299 300
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → dom ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
302 |
301
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) → dom ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
303 |
302
|
raleqdv |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) → ( ∀ 𝑧 ∈ dom ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ( abs ‘ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ‘ 𝑧 ) ) ≤ 𝑦 ↔ ∀ 𝑧 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ‘ 𝑧 ) ) ≤ 𝑦 ) ) |
304 |
288 303
|
mpbird |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑦 ∈ ℝ ) ∧ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) → ∀ 𝑧 ∈ dom ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ( abs ‘ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ‘ 𝑧 ) ) ≤ 𝑦 ) |
305 |
304
|
ex |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) ∧ 𝑦 ∈ ℝ ) → ( ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 → ∀ 𝑧 ∈ dom ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ( abs ‘ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ‘ 𝑧 ) ) ≤ 𝑦 ) ) |
306 |
305
|
reximdva |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ dom ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ( abs ‘ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ‘ 𝑧 ) ) ≤ 𝑦 ) ) |
307 |
215 306
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ∃ 𝑦 ∈ ℝ ∀ 𝑧 ∈ dom ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ( abs ‘ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ‘ 𝑧 ) ) ≤ 𝑦 ) |
308 |
179 180 199 307
|
cnbdibl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ∈ 𝐿1 ) |
309 |
308
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑟 ∈ ℝ ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ) ∈ 𝐿1 ) |
310 |
289
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ∈ ℂ ) |
311 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑟 ∈ ℂ ) → 𝑟 ∈ ℂ ) |
312 |
185
|
sseli |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ℂ ) |
313 |
312
|
ad2antlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑟 ∈ ℂ ) → 𝑥 ∈ ℂ ) |
314 |
311 313
|
mulcld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑟 ∈ ℂ ) → ( 𝑟 · 𝑥 ) ∈ ℂ ) |
315 |
314
|
coscld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑟 ∈ ℂ ) → ( cos ‘ ( 𝑟 · 𝑥 ) ) ∈ ℂ ) |
316 |
293
|
ancoms |
⊢ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑟 ∈ ℝ ) → ( 𝑟 · 𝑥 ) ∈ ℝ ) |
317 |
|
abscosbd |
⊢ ( ( 𝑟 · 𝑥 ) ∈ ℝ → ( abs ‘ ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ≤ 1 ) |
318 |
316 317
|
syl |
⊢ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑟 ∈ ℝ ) → ( abs ‘ ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ≤ 1 ) |
319 |
318
|
adantll |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑟 ∈ ℝ ) → ( abs ‘ ( cos ‘ ( 𝑟 · 𝑥 ) ) ) ≤ 1 ) |
320 |
16
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐷 = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ) ) |
321 |
26
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
322 |
8
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
323 |
|
eqcom |
⊢ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) = 𝑥 ↔ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
324 |
323
|
biimpri |
⊢ ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = 𝑥 ) |
325 |
324
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = 𝑥 ) |
326 |
322 325
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑥 ) |
327 |
321 326
|
gtned |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ≠ ( 𝑄 ‘ 𝑖 ) ) |
328 |
327
|
neneqd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) |
329 |
328
|
iffalsed |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) |
330 |
|
iftrue |
⊢ ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) = 𝐿 ) |
331 |
330
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) = 𝐿 ) |
332 |
329 331
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = 𝐿 ) |
333 |
29
|
leidd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
334 |
26 29 29 143 333
|
eliccd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
335 |
320 332 334 10
|
fvmptd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = 𝐿 ) |
336 |
335 35
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∈ ℂ ) |
337 |
336
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) → ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∈ ℂ ) |
338 |
|
eqid |
⊢ ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
339 |
|
iftrue |
⊢ ( 𝑥 = ( 𝑄 ‘ 𝑖 ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = 𝑅 ) |
340 |
339
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = 𝑅 ) |
341 |
26
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
342 |
29
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
343 |
|
lbicc2 |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
344 |
341 342 143 343
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
345 |
320 340 344 11
|
fvmptd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) = 𝑅 ) |
346 |
345 32
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) ∈ ℂ ) |
347 |
346
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) → ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) ∈ ℂ ) |
348 |
|
eqid |
⊢ ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) ) = ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) ) |
349 |
|
eqid |
⊢ ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) d 𝑥 = ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) d 𝑥 |
350 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → 𝑒 ∈ ℝ+ ) |
351 |
4
|
nnrpd |
⊢ ( 𝜑 → 𝑀 ∈ ℝ+ ) |
352 |
351
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → 𝑀 ∈ ℝ+ ) |
353 |
350 352
|
rpdivcld |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → ( 𝑒 / 𝑀 ) ∈ ℝ+ ) |
354 |
353
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) → ( 𝑒 / 𝑀 ) ∈ ℝ+ ) |
355 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑟 ∈ ℂ ) → 𝑟 ∈ ℂ ) |
356 |
29
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℂ ) |
357 |
356
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑟 ∈ ℂ ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℂ ) |
358 |
355 357
|
mulcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑟 ∈ ℂ ) → ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∈ ℂ ) |
359 |
358
|
coscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑟 ∈ ℂ ) → ( cos ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ℂ ) |
360 |
29
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
361 |
187 360
|
remulcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∈ ℝ ) |
362 |
|
abscosbd |
⊢ ( ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∈ ℝ → ( abs ‘ ( cos ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ≤ 1 ) |
363 |
361 362
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( abs ‘ ( cos ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ≤ 1 ) |
364 |
363
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑟 ∈ ℝ ) → ( abs ‘ ( cos ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ≤ 1 ) |
365 |
26
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℂ ) |
366 |
365
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑟 ∈ ℂ ) → ( 𝑄 ‘ 𝑖 ) ∈ ℂ ) |
367 |
355 366
|
mulcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑟 ∈ ℂ ) → ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ∈ ℂ ) |
368 |
367
|
coscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑟 ∈ ℂ ) → ( cos ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) ∈ ℂ ) |
369 |
26
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
370 |
187 369
|
remulcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ∈ ℝ ) |
371 |
|
abscosbd |
⊢ ( ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ∈ ℝ → ( abs ‘ ( cos ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) ) ≤ 1 ) |
372 |
370 371
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ ) → ( abs ‘ ( cos ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) ) ≤ 1 ) |
373 |
372
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑟 ∈ ℝ ) → ( abs ‘ ( cos ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) ) ≤ 1 ) |
374 |
|
fveq2 |
⊢ ( 𝑧 = 𝑥 → ( ( ℝ D 𝐷 ) ‘ 𝑧 ) = ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) |
375 |
374
|
fveq2d |
⊢ ( 𝑧 = 𝑥 → ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑧 ) ) = ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ) |
376 |
375
|
cbvitgv |
⊢ ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑧 ) ) d 𝑧 = ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) d 𝑥 |
377 |
376
|
oveq2i |
⊢ ( ( ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) + ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) ) ) + ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑧 ) ) d 𝑧 ) = ( ( ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) + ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) ) ) + ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) d 𝑥 ) |
378 |
377
|
oveq1i |
⊢ ( ( ( ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) + ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) ) ) + ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑧 ) ) d 𝑧 ) / ( 𝑒 / 𝑀 ) ) = ( ( ( ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) + ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) ) ) + ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) d 𝑥 ) / ( 𝑒 / 𝑀 ) ) |
379 |
378
|
oveq1i |
⊢ ( ( ( ( ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) + ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) ) ) + ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑧 ) ) d 𝑧 ) / ( 𝑒 / 𝑀 ) ) + 1 ) = ( ( ( ( ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) + ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) ) ) + ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) d 𝑥 ) / ( 𝑒 / 𝑀 ) ) + 1 ) |
380 |
379
|
fveq2i |
⊢ ( ⌊ ‘ ( ( ( ( ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) + ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) ) ) + ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑧 ) ) d 𝑧 ) / ( 𝑒 / 𝑀 ) ) + 1 ) ) = ( ⌊ ‘ ( ( ( ( ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) + ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) ) ) + ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) d 𝑥 ) / ( 𝑒 / 𝑀 ) ) + 1 ) ) |
381 |
380
|
oveq1i |
⊢ ( ( ⌊ ‘ ( ( ( ( ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) + ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) ) ) + ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑧 ) ) d 𝑧 ) / ( 𝑒 / 𝑀 ) ) + 1 ) ) + 1 ) = ( ( ⌊ ‘ ( ( ( ( ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) + ( abs ‘ ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) ) ) + ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) d 𝑥 ) / ( 𝑒 / 𝑀 ) ) + 1 ) ) + 1 ) |
382 |
178 309 310 315 319 337 338 347 348 349 354 359 364 368 373 381
|
fourierdlem47 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) → ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ( ( ( ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) / 𝑟 ) ) − ( ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) / 𝑟 ) ) ) − ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( ( cos ‘ ( 𝑟 · 𝑥 ) ) / 𝑟 ) ) d 𝑥 ) ) < ( 𝑒 / 𝑀 ) ) |
383 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → 𝜑 ) |
384 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → 𝑖 ∈ ( 0 ..^ 𝑀 ) ) |
385 |
|
elioore |
⊢ ( 𝑟 ∈ ( 𝑚 (,) +∞ ) → 𝑟 ∈ ℝ ) |
386 |
385
|
adantl |
⊢ ( ( 𝑚 ∈ ℕ ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → 𝑟 ∈ ℝ ) |
387 |
|
0red |
⊢ ( ( 𝑚 ∈ ℕ ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → 0 ∈ ℝ ) |
388 |
|
nnre |
⊢ ( 𝑚 ∈ ℕ → 𝑚 ∈ ℝ ) |
389 |
388
|
adantr |
⊢ ( ( 𝑚 ∈ ℕ ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → 𝑚 ∈ ℝ ) |
390 |
|
nngt0 |
⊢ ( 𝑚 ∈ ℕ → 0 < 𝑚 ) |
391 |
390
|
adantr |
⊢ ( ( 𝑚 ∈ ℕ ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → 0 < 𝑚 ) |
392 |
389
|
rexrd |
⊢ ( ( 𝑚 ∈ ℕ ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → 𝑚 ∈ ℝ* ) |
393 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
394 |
393
|
a1i |
⊢ ( ( 𝑚 ∈ ℕ ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → +∞ ∈ ℝ* ) |
395 |
|
simpr |
⊢ ( ( 𝑚 ∈ ℕ ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → 𝑟 ∈ ( 𝑚 (,) +∞ ) ) |
396 |
|
ioogtlb |
⊢ ( ( 𝑚 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → 𝑚 < 𝑟 ) |
397 |
392 394 395 396
|
syl3anc |
⊢ ( ( 𝑚 ∈ ℕ ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → 𝑚 < 𝑟 ) |
398 |
387 389 386 391 397
|
lttrd |
⊢ ( ( 𝑚 ∈ ℕ ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → 0 < 𝑟 ) |
399 |
386 398
|
elrpd |
⊢ ( ( 𝑚 ∈ ℕ ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → 𝑟 ∈ ℝ+ ) |
400 |
399
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → 𝑟 ∈ ℝ+ ) |
401 |
26
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
402 |
29
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
403 |
72
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐷 ‘ 𝑥 ) ∈ ℂ ) |
404 |
403
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐷 ‘ 𝑥 ) ∈ ℂ ) |
405 |
|
rpcn |
⊢ ( 𝑟 ∈ ℝ+ → 𝑟 ∈ ℂ ) |
406 |
405
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑟 ∈ ℂ ) |
407 |
44
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℂ ) |
408 |
407
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℂ ) |
409 |
406 408
|
mulcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑟 · 𝑥 ) ∈ ℂ ) |
410 |
409
|
sincld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( sin ‘ ( 𝑟 · 𝑥 ) ) ∈ ℂ ) |
411 |
404 410
|
mulcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ∈ ℂ ) |
412 |
401 402 411
|
itgioo |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) → ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
413 |
143
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) → ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
414 |
72
|
feqmptd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐷 = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐷 ‘ 𝑥 ) ) ) |
415 |
|
iftrue |
⊢ ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = 𝐿 ) |
416 |
330 415
|
eqtr4d |
⊢ ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) = if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) |
417 |
416
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) = if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) |
418 |
|
iffalse |
⊢ ( ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) |
419 |
418
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) |
420 |
54
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
421 |
55
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
422 |
44
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ℝ ) |
423 |
26
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
424 |
44
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → 𝑥 ∈ ℝ ) |
425 |
57
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → ( 𝑄 ‘ 𝑖 ) ≤ 𝑥 ) |
426 |
|
neqne |
⊢ ( ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) → 𝑥 ≠ ( 𝑄 ‘ 𝑖 ) ) |
427 |
426
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → 𝑥 ≠ ( 𝑄 ‘ 𝑖 ) ) |
428 |
423 424 425 427
|
leneltd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑥 ) |
429 |
428
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑥 ) |
430 |
44
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ℝ ) |
431 |
29
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
432 |
60
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
433 |
323
|
biimpi |
⊢ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) = 𝑥 → 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
434 |
433
|
necon3bi |
⊢ ( ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ≠ 𝑥 ) |
435 |
434
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ≠ 𝑥 ) |
436 |
430 431 432 435
|
leneltd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
437 |
436
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
438 |
420 421 422 429 437
|
eliood |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
439 |
|
fvres |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
440 |
438 439
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
441 |
|
iffalse |
⊢ ( ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
442 |
441
|
eqcomd |
⊢ ( ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → ( 𝐹 ‘ 𝑥 ) = if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) |
443 |
442
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐹 ‘ 𝑥 ) = if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) |
444 |
419 440 443
|
3eqtrrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) = if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) |
445 |
417 444
|
pm2.61dan |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ¬ 𝑥 = ( 𝑄 ‘ 𝑖 ) ) → if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) = if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) |
446 |
445
|
ifeq2da |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) |
447 |
446
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) ) |
448 |
320 414 447
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐷 ‘ 𝑥 ) ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) ) |
449 |
|
eqid |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) |
450 |
200 449 26 29 9 10 11
|
cncfiooicc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
451 |
448 450
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐷 ‘ 𝑥 ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
452 |
414 451
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐷 ∈ ( ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
453 |
452
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) → 𝐷 ∈ ( ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
454 |
|
eqid |
⊢ ( ℝ D 𝐷 ) = ( ℝ D 𝐷 ) |
455 |
136 13
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ℝ D 𝐷 ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
456 |
455
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) → ( ℝ D 𝐷 ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
457 |
214
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( abs ‘ ( ( ℝ D 𝐷 ) ‘ 𝑥 ) ) ≤ 𝑦 ) |
458 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) → 𝑟 ∈ ℝ+ ) |
459 |
401 402 413 453 454 456 457 458
|
fourierdlem39 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) → ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = ( ( ( ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) / 𝑟 ) ) − ( ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) / 𝑟 ) ) ) − ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( ( cos ‘ ( 𝑟 · 𝑥 ) ) / 𝑟 ) ) d 𝑥 ) ) |
460 |
412 459
|
eqtr3d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ℝ+ ) → ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = ( ( ( ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) / 𝑟 ) ) − ( ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) / 𝑟 ) ) ) − ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( ( cos ‘ ( 𝑟 · 𝑥 ) ) / 𝑟 ) ) d 𝑥 ) ) |
461 |
383 384 400 460
|
syl21anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = ( ( ( ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) / 𝑟 ) ) − ( ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) / 𝑟 ) ) ) − ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( ( cos ‘ ( 𝑟 · 𝑥 ) ) / 𝑟 ) ) d 𝑥 ) ) |
462 |
461
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) = ( abs ‘ ( ( ( ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) / 𝑟 ) ) − ( ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) / 𝑟 ) ) ) − ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( ( cos ‘ ( 𝑟 · 𝑥 ) ) / 𝑟 ) ) d 𝑥 ) ) ) |
463 |
462
|
breq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → ( ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ↔ ( abs ‘ ( ( ( ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) / 𝑟 ) ) − ( ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) / 𝑟 ) ) ) − ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( ( cos ‘ ( 𝑟 · 𝑥 ) ) / 𝑟 ) ) d 𝑥 ) ) < ( 𝑒 / 𝑀 ) ) ) |
464 |
463
|
ralbidva |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑚 ∈ ℕ ) → ( ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ↔ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ( ( ( ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) / 𝑟 ) ) − ( ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) / 𝑟 ) ) ) − ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( ( cos ‘ ( 𝑟 · 𝑥 ) ) / 𝑟 ) ) d 𝑥 ) ) < ( 𝑒 / 𝑀 ) ) ) |
465 |
464
|
rexbidva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ↔ ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ( ( ( ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) / 𝑟 ) ) − ( ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) / 𝑟 ) ) ) − ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( ( cos ‘ ( 𝑟 · 𝑥 ) ) / 𝑟 ) ) d 𝑥 ) ) < ( 𝑒 / 𝑀 ) ) ) |
466 |
465
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) → ( ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ↔ ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ( ( ( ( 𝐷 ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) / 𝑟 ) ) − ( ( 𝐷 ‘ ( 𝑄 ‘ 𝑖 ) ) · - ( ( cos ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) / 𝑟 ) ) ) − ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( ( ℝ D 𝐷 ) ‘ 𝑥 ) · - ( ( cos ‘ ( 𝑟 · 𝑥 ) ) / 𝑟 ) ) d 𝑥 ) ) < ( 𝑒 / 𝑀 ) ) ) |
467 |
382 466
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑒 ∈ ℝ+ ) → ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
468 |
467
|
an32s |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
469 |
102
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) = ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ) |
470 |
469
|
itgeq2dv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
471 |
470
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
472 |
471
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
473 |
26
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
474 |
29
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
475 |
403
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐷 ‘ 𝑥 ) ∈ ℂ ) |
476 |
385
|
recnd |
⊢ ( 𝑟 ∈ ( 𝑚 (,) +∞ ) → 𝑟 ∈ ℂ ) |
477 |
476
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑟 ∈ ℂ ) |
478 |
407
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℂ ) |
479 |
477 478
|
mulcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑟 · 𝑥 ) ∈ ℂ ) |
480 |
479
|
sincld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( sin ‘ ( 𝑟 · 𝑥 ) ) ∈ ℂ ) |
481 |
475 480
|
mulcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ∈ ℂ ) |
482 |
473 474 481
|
itgioo |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
483 |
69
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ ) |
484 |
483 480
|
mulcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ∈ ℂ ) |
485 |
473 474 484
|
itgioo |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → ∫ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
486 |
472 482 485
|
3eqtr3d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
487 |
486
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) = ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ) |
488 |
487
|
breq1d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑚 (,) +∞ ) ) → ( ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ↔ ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ) |
489 |
488
|
ralbidva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ↔ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ) |
490 |
489
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ↔ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ) |
491 |
490
|
rexbidv |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐷 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ↔ ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ) |
492 |
468 491
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
493 |
492
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
494 |
493
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑒 ∈ ℝ+ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
495 |
|
nfv |
⊢ Ⅎ 𝑖 ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) |
496 |
|
nfra1 |
⊢ Ⅎ 𝑖 ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) |
497 |
495 496
|
nfan |
⊢ Ⅎ 𝑖 ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
498 |
|
nfv |
⊢ Ⅎ 𝑟 ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) |
499 |
|
nfcv |
⊢ Ⅎ 𝑟 ( 0 ..^ 𝑀 ) |
500 |
|
nfcv |
⊢ Ⅎ 𝑟 ℕ |
501 |
|
nfra1 |
⊢ Ⅎ 𝑟 ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) |
502 |
500 501
|
nfrex |
⊢ Ⅎ 𝑟 ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) |
503 |
499 502
|
nfralw |
⊢ Ⅎ 𝑟 ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) |
504 |
498 503
|
nfan |
⊢ Ⅎ 𝑟 ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
505 |
|
nfmpt1 |
⊢ Ⅎ 𝑖 ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ inf ( { 𝑚 ∈ ℕ ∣ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) } , ℝ , < ) ) |
506 |
|
fzofi |
⊢ ( 0 ..^ 𝑀 ) ∈ Fin |
507 |
506
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) → ( 0 ..^ 𝑀 ) ∈ Fin ) |
508 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
509 |
|
eqid |
⊢ { 𝑚 ∈ ℕ ∣ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) } = { 𝑚 ∈ ℕ ∣ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) } |
510 |
|
eqid |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ inf ( { 𝑚 ∈ ℕ ∣ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) } , ℝ , < ) ) = ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ inf ( { 𝑚 ∈ ℕ ∣ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) } , ℝ , < ) ) |
511 |
|
eqid |
⊢ sup ( ran ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ inf ( { 𝑚 ∈ ℕ ∣ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) } , ℝ , < ) ) , ℝ , < ) = sup ( ran ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ inf ( { 𝑚 ∈ ℕ ∣ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) } , ℝ , < ) ) , ℝ , < ) |
512 |
497 504 505 507 508 509 510 511
|
fourierdlem31 |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) → ∃ 𝑛 ∈ ℕ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
513 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ ∃ 𝑛 ∈ ℕ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) → ∃ 𝑛 ∈ ℕ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
514 |
|
nfv |
⊢ Ⅎ 𝑛 ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) |
515 |
|
nfre1 |
⊢ Ⅎ 𝑛 ∃ 𝑛 ∈ ℕ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) |
516 |
514 515
|
nfan |
⊢ Ⅎ 𝑛 ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ ∃ 𝑛 ∈ ℕ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
517 |
|
nfv |
⊢ Ⅎ 𝑟 𝑛 ∈ ℕ |
518 |
|
nfra1 |
⊢ Ⅎ 𝑟 ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) |
519 |
498 517 518
|
nf3an |
⊢ Ⅎ 𝑟 ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
520 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → 𝜑 ) |
521 |
|
elioore |
⊢ ( 𝑟 ∈ ( 𝑛 (,) +∞ ) → 𝑟 ∈ ℝ ) |
522 |
521
|
adantl |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → 𝑟 ∈ ℝ ) |
523 |
|
0red |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → 0 ∈ ℝ ) |
524 |
|
nnre |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℝ ) |
525 |
524
|
adantr |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → 𝑛 ∈ ℝ ) |
526 |
|
nngt0 |
⊢ ( 𝑛 ∈ ℕ → 0 < 𝑛 ) |
527 |
526
|
adantr |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → 0 < 𝑛 ) |
528 |
525
|
rexrd |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → 𝑛 ∈ ℝ* ) |
529 |
393
|
a1i |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → +∞ ∈ ℝ* ) |
530 |
|
simpr |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → 𝑟 ∈ ( 𝑛 (,) +∞ ) ) |
531 |
|
ioogtlb |
⊢ ( ( 𝑛 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → 𝑛 < 𝑟 ) |
532 |
528 529 530 531
|
syl3anc |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → 𝑛 < 𝑟 ) |
533 |
523 525 522 527 532
|
lttrd |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → 0 < 𝑟 ) |
534 |
522 533
|
elrpd |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → 𝑟 ∈ ℝ+ ) |
535 |
534
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → 𝑟 ∈ ℝ+ ) |
536 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → 𝐴 ∈ ℝ ) |
537 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → 𝐵 ∈ ℝ ) |
538 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ ) |
539 |
538
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ ) |
540 |
405
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑟 ∈ ℂ ) |
541 |
21
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑥 ∈ ℝ ) |
542 |
541
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑥 ∈ ℂ ) |
543 |
542
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑥 ∈ ℂ ) |
544 |
540 543
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑟 · 𝑥 ) ∈ ℂ ) |
545 |
544
|
sincld |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( sin ‘ ( 𝑟 · 𝑥 ) ) ∈ ℂ ) |
546 |
539 545
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ∈ ℂ ) |
547 |
536 537 546
|
itgioo |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = ∫ ( 𝐴 [,] 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
548 |
6
|
eqcomd |
⊢ ( 𝜑 → 𝐴 = ( 𝑄 ‘ 0 ) ) |
549 |
7
|
eqcomd |
⊢ ( 𝜑 → 𝐵 = ( 𝑄 ‘ 𝑀 ) ) |
550 |
548 549
|
oveq12d |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) = ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
551 |
550
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → ( 𝐴 [,] 𝐵 ) = ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
552 |
551
|
itgeq1d |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → ∫ ( 𝐴 [,] 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = ∫ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
553 |
|
0zd |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → 0 ∈ ℤ ) |
554 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
555 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
556 |
555
|
fveq2i |
⊢ ( ℤ≥ ‘ ( 0 + 1 ) ) = ( ℤ≥ ‘ 1 ) |
557 |
554 556
|
eqtr4i |
⊢ ℕ = ( ℤ≥ ‘ ( 0 + 1 ) ) |
558 |
4 557
|
eleqtrdi |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ ( 0 + 1 ) ) ) |
559 |
558
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → 𝑀 ∈ ( ℤ≥ ‘ ( 0 + 1 ) ) ) |
560 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
561 |
8
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
562 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
563 |
550
|
eqcomd |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) = ( 𝐴 [,] 𝐵 ) ) |
564 |
563
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) = ( 𝐴 [,] 𝐵 ) ) |
565 |
562 564
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) |
566 |
565
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) |
567 |
566 546
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ∈ ℂ ) |
568 |
26
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
569 |
29
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
570 |
114 111
|
sstrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
571 |
121 570
|
feqresmpt |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
572 |
571 9
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
573 |
572
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
574 |
|
sincn |
⊢ sin ∈ ( ℂ –cn→ ℂ ) |
575 |
574
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → sin ∈ ( ℂ –cn→ ℂ ) ) |
576 |
185
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ ) |
577 |
405
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → 𝑟 ∈ ℂ ) |
578 |
189
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → ℂ ⊆ ℂ ) |
579 |
576 577 578
|
constcncfg |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ 𝑟 ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
580 |
194
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ 𝑥 ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
581 |
579 580
|
mulcncf |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑟 · 𝑥 ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
582 |
581
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑟 · 𝑥 ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
583 |
575 582
|
cncfmpt1f |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
584 |
573 583
|
mulcncf |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
585 |
|
eqid |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) |
586 |
|
eqid |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( sin ‘ ( 𝑟 · 𝑥 ) ) ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( sin ‘ ( 𝑟 · 𝑥 ) ) ) |
587 |
|
eqid |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ) |
588 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ ) |
589 |
45
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐴 ∈ ℝ* ) |
590 |
47
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐵 ∈ ℝ* ) |
591 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( 𝐴 [,] 𝐵 ) ) |
592 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑖 ∈ ( 0 ..^ 𝑀 ) ) |
593 |
589 590 591 592 80
|
fourierdlem1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) |
594 |
588 593
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ ) |
595 |
594
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ ) |
596 |
577
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑟 ∈ ℂ ) |
597 |
312
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℂ ) |
598 |
596 597
|
mulcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑟 · 𝑥 ) ∈ ℂ ) |
599 |
598
|
sincld |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( sin ‘ ( 𝑟 · 𝑥 ) ) ∈ ℂ ) |
600 |
571
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
601 |
10 600
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
602 |
601
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
603 |
|
rpre |
⊢ ( 𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ ) |
604 |
603
|
adantr |
⊢ ( ( 𝑟 ∈ ℝ+ ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑟 ∈ ℝ ) |
605 |
95
|
adantl |
⊢ ( ( 𝑟 ∈ ℝ+ ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℝ ) |
606 |
604 605
|
remulcld |
⊢ ( ( 𝑟 ∈ ℝ+ ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑟 · 𝑥 ) ∈ ℝ ) |
607 |
606
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑟 · 𝑥 ) ∈ ℝ ) |
608 |
607
|
ad2ant2r |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝑟 · 𝑥 ) ≠ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) → ( 𝑟 · 𝑥 ) ∈ ℝ ) |
609 |
|
recn |
⊢ ( 𝑦 ∈ ℝ → 𝑦 ∈ ℂ ) |
610 |
609
|
sincld |
⊢ ( 𝑦 ∈ ℝ → ( sin ‘ 𝑦 ) ∈ ℂ ) |
611 |
610
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ℝ ) → ( sin ‘ 𝑦 ) ∈ ℂ ) |
612 |
|
eqid |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ 𝑟 ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ 𝑟 ) |
613 |
|
eqid |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ 𝑥 ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ 𝑥 ) |
614 |
|
eqid |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑟 · 𝑥 ) ) = ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑟 · 𝑥 ) ) |
615 |
185
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ ) |
616 |
577
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑟 ∈ ℂ ) |
617 |
569
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℂ ) |
618 |
612 615 616 617
|
constlimc |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑟 ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ 𝑟 ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
619 |
615 613 617
|
idlimc |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ 𝑥 ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
620 |
612 613 614 596 597 618 619
|
mullimc |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑟 · 𝑥 ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
621 |
|
eqid |
⊢ ( 𝑦 ∈ ℂ ↦ ( sin ‘ 𝑦 ) ) = ( 𝑦 ∈ ℂ ↦ ( sin ‘ 𝑦 ) ) |
622 |
|
sinf |
⊢ sin : ℂ ⟶ ℂ |
623 |
622
|
a1i |
⊢ ( ⊤ → sin : ℂ ⟶ ℂ ) |
624 |
623
|
feqmptd |
⊢ ( ⊤ → sin = ( 𝑦 ∈ ℂ ↦ ( sin ‘ 𝑦 ) ) ) |
625 |
624 574
|
eqeltrrdi |
⊢ ( ⊤ → ( 𝑦 ∈ ℂ ↦ ( sin ‘ 𝑦 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
626 |
19
|
a1i |
⊢ ( ⊤ → ℝ ⊆ ℂ ) |
627 |
|
resincl |
⊢ ( 𝑦 ∈ ℝ → ( sin ‘ 𝑦 ) ∈ ℝ ) |
628 |
627
|
adantl |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ℝ ) → ( sin ‘ 𝑦 ) ∈ ℝ ) |
629 |
621 625 626 626 628
|
cncfmptssg |
⊢ ( ⊤ → ( 𝑦 ∈ ℝ ↦ ( sin ‘ 𝑦 ) ) ∈ ( ℝ –cn→ ℝ ) ) |
630 |
629
|
mptru |
⊢ ( 𝑦 ∈ ℝ ↦ ( sin ‘ 𝑦 ) ) ∈ ( ℝ –cn→ ℝ ) |
631 |
630
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑦 ∈ ℝ ↦ ( sin ‘ 𝑦 ) ) ∈ ( ℝ –cn→ ℝ ) ) |
632 |
603
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑟 ∈ ℝ ) |
633 |
632 569
|
remulcld |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∈ ℝ ) |
634 |
|
fveq2 |
⊢ ( 𝑦 = ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( sin ‘ 𝑦 ) = ( sin ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
635 |
631 633 634
|
cnmptlimc |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( sin ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( 𝑦 ∈ ℝ ↦ ( sin ‘ 𝑦 ) ) limℂ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
636 |
|
fveq2 |
⊢ ( 𝑦 = ( 𝑟 · 𝑥 ) → ( sin ‘ 𝑦 ) = ( sin ‘ ( 𝑟 · 𝑥 ) ) ) |
637 |
|
fveq2 |
⊢ ( ( 𝑟 · 𝑥 ) = ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( sin ‘ ( 𝑟 · 𝑥 ) ) = ( sin ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
638 |
637
|
ad2antll |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝑟 · 𝑥 ) = ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) → ( sin ‘ ( 𝑟 · 𝑥 ) ) = ( sin ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
639 |
608 611 620 635 636 638
|
limcco |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( sin ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( sin ‘ ( 𝑟 · 𝑥 ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
640 |
585 586 587 595 599 602 639
|
mullimc |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐿 · ( sin ‘ ( 𝑟 · ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
641 |
571
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) = ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
642 |
11 641
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
643 |
642
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
644 |
607
|
ad2ant2r |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝑟 · 𝑥 ) ≠ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) ) → ( 𝑟 · 𝑥 ) ∈ ℝ ) |
645 |
568
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℂ ) |
646 |
612 615 616 645
|
constlimc |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑟 ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ 𝑟 ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
647 |
615 613 645
|
idlimc |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ 𝑥 ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
648 |
612 613 614 596 597 646 647
|
mullimc |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑟 · 𝑥 ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
649 |
632 568
|
remulcld |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ∈ ℝ ) |
650 |
|
fveq2 |
⊢ ( 𝑦 = ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) → ( sin ‘ 𝑦 ) = ( sin ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) ) |
651 |
631 649 650
|
cnmptlimc |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( sin ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) ∈ ( ( 𝑦 ∈ ℝ ↦ ( sin ‘ 𝑦 ) ) limℂ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) ) |
652 |
|
fveq2 |
⊢ ( ( 𝑟 · 𝑥 ) = ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) → ( sin ‘ ( 𝑟 · 𝑥 ) ) = ( sin ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) ) |
653 |
652
|
ad2antll |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝑟 · 𝑥 ) = ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) ) → ( sin ‘ ( 𝑟 · 𝑥 ) ) = ( sin ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) ) |
654 |
644 611 648 651 636 653
|
limcco |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( sin ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( sin ‘ ( 𝑟 · 𝑥 ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
655 |
585 586 587 595 599 643 654
|
mullimc |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑅 · ( sin ‘ ( 𝑟 · ( 𝑄 ‘ 𝑖 ) ) ) ) ∈ ( ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
656 |
568 569 584 640 655
|
iblcncfioo |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ) ∈ 𝐿1 ) |
657 |
|
simpll |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ) |
658 |
68
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) |
659 |
657 658 546
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ∈ ℂ ) |
660 |
568 569 656 659
|
ibliooicc |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ) ∈ 𝐿1 ) |
661 |
553 559 560 561 567 660
|
itgspltprt |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → ∫ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
662 |
547 552 661
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
663 |
520 535 662
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
664 |
506
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ( 0 ..^ 𝑀 ) ∈ Fin ) |
665 |
69
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ ) |
666 |
521
|
recnd |
⊢ ( 𝑟 ∈ ( 𝑛 (,) +∞ ) → 𝑟 ∈ ℂ ) |
667 |
666
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → 𝑟 ∈ ℂ ) |
668 |
667
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑟 ∈ ℂ ) |
669 |
407
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ ℂ ) |
670 |
668 669
|
mulcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑟 · 𝑥 ) ∈ ℂ ) |
671 |
670
|
sincld |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( sin ‘ ( 𝑟 · 𝑥 ) ) ∈ ℂ ) |
672 |
665 671
|
mulcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ∈ ℂ ) |
673 |
672
|
adantl3r |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ∈ ℂ ) |
674 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝜑 ) |
675 |
535
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑟 ∈ ℝ+ ) |
676 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ..^ 𝑀 ) ) |
677 |
674 675 676 660
|
syl21anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) ) ∈ 𝐿1 ) |
678 |
673 677
|
itgcl |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ∈ ℂ ) |
679 |
664 678
|
fsumcl |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ∈ ℂ ) |
680 |
663 679
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ∈ ℂ ) |
681 |
680
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ∈ ℂ ) |
682 |
681
|
3adantl3 |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ∈ ℂ ) |
683 |
682
|
abscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ∈ ℝ ) |
684 |
678
|
abscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ∈ ℝ ) |
685 |
664 684
|
fsumrecl |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ∈ ℝ ) |
686 |
685
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ∈ ℝ ) |
687 |
686
|
3adantl3 |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ∈ ℝ ) |
688 |
|
rpre |
⊢ ( 𝑒 ∈ ℝ+ → 𝑒 ∈ ℝ ) |
689 |
688
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → 𝑒 ∈ ℝ ) |
690 |
689
|
3ad2antl1 |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → 𝑒 ∈ ℝ ) |
691 |
663
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) = ( abs ‘ Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ) |
692 |
664 678
|
fsumabs |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ( abs ‘ Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ≤ Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ) |
693 |
691 692
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ≤ Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ) |
694 |
693
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ≤ Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ) |
695 |
694
|
3adantl3 |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ≤ Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ) |
696 |
506
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ( 0 ..^ 𝑀 ) ∈ Fin ) |
697 |
|
0zd |
⊢ ( 𝜑 → 0 ∈ ℤ ) |
698 |
4
|
nnzd |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
699 |
4
|
nngt0d |
⊢ ( 𝜑 → 0 < 𝑀 ) |
700 |
|
fzolb |
⊢ ( 0 ∈ ( 0 ..^ 𝑀 ) ↔ ( 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀 ) ) |
701 |
697 698 699 700
|
syl3anbrc |
⊢ ( 𝜑 → 0 ∈ ( 0 ..^ 𝑀 ) ) |
702 |
|
ne0i |
⊢ ( 0 ∈ ( 0 ..^ 𝑀 ) → ( 0 ..^ 𝑀 ) ≠ ∅ ) |
703 |
701 702
|
syl |
⊢ ( 𝜑 → ( 0 ..^ 𝑀 ) ≠ ∅ ) |
704 |
703
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ( 0 ..^ 𝑀 ) ≠ ∅ ) |
705 |
704
|
3ad2antl1 |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ( 0 ..^ 𝑀 ) ≠ ∅ ) |
706 |
|
simp1l |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) → 𝜑 ) |
707 |
706
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → 𝜑 ) |
708 |
|
simpll2 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → 𝑛 ∈ ℕ ) |
709 |
707 708
|
jca |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝜑 ∧ 𝑛 ∈ ℕ ) ) |
710 |
|
simplr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → 𝑟 ∈ ( 𝑛 (,) +∞ ) ) |
711 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → 𝑗 ∈ ( 0 ..^ 𝑀 ) ) |
712 |
|
eleq1w |
⊢ ( 𝑖 = 𝑗 → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↔ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) ) |
713 |
712
|
anbi2d |
⊢ ( 𝑖 = 𝑗 → ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ↔ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) ) ) |
714 |
|
fveq2 |
⊢ ( 𝑖 = 𝑗 → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 𝑗 ) ) |
715 |
|
oveq1 |
⊢ ( 𝑖 = 𝑗 → ( 𝑖 + 1 ) = ( 𝑗 + 1 ) ) |
716 |
715
|
fveq2d |
⊢ ( 𝑖 = 𝑗 → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
717 |
714 716
|
oveq12d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) |
718 |
717
|
itgeq1d |
⊢ ( 𝑖 = 𝑗 → ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
719 |
718
|
eleq1d |
⊢ ( 𝑖 = 𝑗 → ( ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ∈ ℂ ↔ ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ∈ ℂ ) ) |
720 |
713 719
|
imbi12d |
⊢ ( 𝑖 = 𝑗 → ( ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ∈ ℂ ) ↔ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ∈ ℂ ) ) ) |
721 |
720 678
|
chvarvv |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ∈ ℂ ) |
722 |
709 710 711 721
|
syl21anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ∈ ℂ ) |
723 |
722
|
abscld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ∈ ℝ ) |
724 |
353
|
rpred |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → ( 𝑒 / 𝑀 ) ∈ ℝ ) |
725 |
724
|
3ad2ant1 |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) → ( 𝑒 / 𝑀 ) ∈ ℝ ) |
726 |
725
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑒 / 𝑀 ) ∈ ℝ ) |
727 |
|
simpll3 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
728 |
|
rspa |
⊢ ( ( ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
729 |
728
|
adantr |
⊢ ( ( ( ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
730 |
718
|
fveq2d |
⊢ ( 𝑖 = 𝑗 → ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) = ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ) |
731 |
730
|
breq1d |
⊢ ( 𝑖 = 𝑗 → ( ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ↔ ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ) |
732 |
731
|
cbvralvw |
⊢ ( ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ↔ ∀ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
733 |
729 732
|
sylib |
⊢ ( ( ( ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ∀ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
734 |
|
rspa |
⊢ ( ( ∀ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
735 |
733 734
|
sylancom |
⊢ ( ( ( ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
736 |
727 710 711 735
|
syl21anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) |
737 |
696 705 723 726 736
|
fsumlt |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → Σ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < Σ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( 𝑒 / 𝑀 ) ) |
738 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ 𝑖 ) ) |
739 |
|
oveq1 |
⊢ ( 𝑗 = 𝑖 → ( 𝑗 + 1 ) = ( 𝑖 + 1 ) ) |
740 |
739
|
fveq2d |
⊢ ( 𝑗 = 𝑖 → ( 𝑄 ‘ ( 𝑗 + 1 ) ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
741 |
738 740
|
oveq12d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) = ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
742 |
741
|
itgeq1d |
⊢ ( 𝑗 = 𝑖 → ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 = ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
743 |
742
|
fveq2d |
⊢ ( 𝑗 = 𝑖 → ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) = ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ) |
744 |
743
|
cbvsumv |
⊢ Σ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) = Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) |
745 |
744
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → Σ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑗 ) [,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) = Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) ) |
746 |
353
|
rpcnd |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → ( 𝑒 / 𝑀 ) ∈ ℂ ) |
747 |
|
fsumconst |
⊢ ( ( ( 0 ..^ 𝑀 ) ∈ Fin ∧ ( 𝑒 / 𝑀 ) ∈ ℂ ) → Σ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( 𝑒 / 𝑀 ) = ( ( ♯ ‘ ( 0 ..^ 𝑀 ) ) · ( 𝑒 / 𝑀 ) ) ) |
748 |
506 746 747
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → Σ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( 𝑒 / 𝑀 ) = ( ( ♯ ‘ ( 0 ..^ 𝑀 ) ) · ( 𝑒 / 𝑀 ) ) ) |
749 |
4
|
nnnn0d |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
750 |
|
hashfzo0 |
⊢ ( 𝑀 ∈ ℕ0 → ( ♯ ‘ ( 0 ..^ 𝑀 ) ) = 𝑀 ) |
751 |
749 750
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ ( 0 ..^ 𝑀 ) ) = 𝑀 ) |
752 |
751
|
oveq1d |
⊢ ( 𝜑 → ( ( ♯ ‘ ( 0 ..^ 𝑀 ) ) · ( 𝑒 / 𝑀 ) ) = ( 𝑀 · ( 𝑒 / 𝑀 ) ) ) |
753 |
752
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → ( ( ♯ ‘ ( 0 ..^ 𝑀 ) ) · ( 𝑒 / 𝑀 ) ) = ( 𝑀 · ( 𝑒 / 𝑀 ) ) ) |
754 |
350
|
rpcnd |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → 𝑒 ∈ ℂ ) |
755 |
352
|
rpcnd |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → 𝑀 ∈ ℂ ) |
756 |
352
|
rpne0d |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → 𝑀 ≠ 0 ) |
757 |
754 755 756
|
divcan2d |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → ( 𝑀 · ( 𝑒 / 𝑀 ) ) = 𝑒 ) |
758 |
748 753 757
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → Σ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( 𝑒 / 𝑀 ) = 𝑒 ) |
759 |
758
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → Σ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( 𝑒 / 𝑀 ) = 𝑒 ) |
760 |
759
|
3ad2antl1 |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → Σ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( 𝑒 / 𝑀 ) = 𝑒 ) |
761 |
737 745 760
|
3brtr3d |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < 𝑒 ) |
762 |
683 687 690 695 761
|
lelttrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) ∧ 𝑟 ∈ ( 𝑛 (,) +∞ ) ) → ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < 𝑒 ) |
763 |
762
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) → ( 𝑟 ∈ ( 𝑛 (,) +∞ ) → ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < 𝑒 ) ) |
764 |
519 763
|
ralrimi |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ∧ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) → ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < 𝑒 ) |
765 |
764
|
3exp |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → ( 𝑛 ∈ ℕ → ( ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) → ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < 𝑒 ) ) ) |
766 |
765
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ ∃ 𝑛 ∈ ℕ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) → ( 𝑛 ∈ ℕ → ( ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) → ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < 𝑒 ) ) ) |
767 |
516 766
|
reximdai |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ ∃ 𝑛 ∈ ℕ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) → ( ∃ 𝑛 ∈ ℕ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) → ∃ 𝑛 ∈ ℕ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < 𝑒 ) ) |
768 |
513 767
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ ∃ 𝑛 ∈ ℕ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) → ∃ 𝑛 ∈ ℕ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < 𝑒 ) |
769 |
512 768
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) ) → ∃ 𝑛 ∈ ℕ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < 𝑒 ) |
770 |
769
|
ex |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ ℝ+ ) → ( ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) → ∃ 𝑛 ∈ ℕ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < 𝑒 ) ) |
771 |
770
|
ralimdva |
⊢ ( 𝜑 → ( ∀ 𝑒 ∈ ℝ+ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑚 ∈ ℕ ∀ 𝑟 ∈ ( 𝑚 (,) +∞ ) ( abs ‘ ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < ( 𝑒 / 𝑀 ) → ∀ 𝑒 ∈ ℝ+ ∃ 𝑛 ∈ ℕ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < 𝑒 ) ) |
772 |
494 771
|
mpd |
⊢ ( 𝜑 → ∀ 𝑒 ∈ ℝ+ ∃ 𝑛 ∈ ℕ ∀ 𝑟 ∈ ( 𝑛 (,) +∞ ) ( abs ‘ ∫ ( 𝐴 (,) 𝐵 ) ( ( 𝐹 ‘ 𝑥 ) · ( sin ‘ ( 𝑟 · 𝑥 ) ) ) d 𝑥 ) < 𝑒 ) |