Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem93.1 |
⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = - π ∧ ( 𝑝 ‘ 𝑚 ) = π ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) |
2 |
|
fourierdlem93.2 |
⊢ 𝐻 = ( 𝑖 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) |
3 |
|
fourierdlem93.3 |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
4 |
|
fourierdlem93.4 |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ) |
5 |
|
fourierdlem93.5 |
⊢ ( 𝜑 → 𝑋 ∈ ℝ ) |
6 |
|
fourierdlem93.6 |
⊢ ( 𝜑 → 𝐹 : ( - π [,] π ) ⟶ ℂ ) |
7 |
|
fourierdlem93.7 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
8 |
|
fourierdlem93.8 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
9 |
|
fourierdlem93.9 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
10 |
1
|
fourierdlem2 |
⊢ ( 𝑀 ∈ ℕ → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = - π ∧ ( 𝑄 ‘ 𝑀 ) = π ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
11 |
3 10
|
syl |
⊢ ( 𝜑 → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = - π ∧ ( 𝑄 ‘ 𝑀 ) = π ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
12 |
4 11
|
mpbid |
⊢ ( 𝜑 → ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = - π ∧ ( 𝑄 ‘ 𝑀 ) = π ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
13 |
12
|
simprd |
⊢ ( 𝜑 → ( ( ( 𝑄 ‘ 0 ) = - π ∧ ( 𝑄 ‘ 𝑀 ) = π ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
14 |
13
|
simplld |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) = - π ) |
15 |
14
|
eqcomd |
⊢ ( 𝜑 → - π = ( 𝑄 ‘ 0 ) ) |
16 |
13
|
simplrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) = π ) |
17 |
16
|
eqcomd |
⊢ ( 𝜑 → π = ( 𝑄 ‘ 𝑀 ) ) |
18 |
15 17
|
oveq12d |
⊢ ( 𝜑 → ( - π [,] π ) = ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
19 |
18
|
itgeq1d |
⊢ ( 𝜑 → ∫ ( - π [,] π ) ( 𝐹 ‘ 𝑡 ) d 𝑡 = ∫ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ( 𝐹 ‘ 𝑡 ) d 𝑡 ) |
20 |
|
0zd |
⊢ ( 𝜑 → 0 ∈ ℤ ) |
21 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
22 |
3 21
|
eleqtrdi |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 1 ) ) |
23 |
|
1e0p1 |
⊢ 1 = ( 0 + 1 ) |
24 |
23
|
a1i |
⊢ ( 𝜑 → 1 = ( 0 + 1 ) ) |
25 |
24
|
fveq2d |
⊢ ( 𝜑 → ( ℤ≥ ‘ 1 ) = ( ℤ≥ ‘ ( 0 + 1 ) ) ) |
26 |
22 25
|
eleqtrd |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ ( 0 + 1 ) ) ) |
27 |
1 3 4
|
fourierdlem15 |
⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( - π [,] π ) ) |
28 |
|
pire |
⊢ π ∈ ℝ |
29 |
28
|
renegcli |
⊢ - π ∈ ℝ |
30 |
|
iccssre |
⊢ ( ( - π ∈ ℝ ∧ π ∈ ℝ ) → ( - π [,] π ) ⊆ ℝ ) |
31 |
29 28 30
|
mp2an |
⊢ ( - π [,] π ) ⊆ ℝ |
32 |
31
|
a1i |
⊢ ( 𝜑 → ( - π [,] π ) ⊆ ℝ ) |
33 |
27 32
|
fssd |
⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
34 |
13
|
simprd |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
35 |
34
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
36 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → 𝐹 : ( - π [,] π ) ⟶ ℂ ) |
37 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → 𝑡 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
38 |
18
|
eqcomd |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) = ( - π [,] π ) ) |
39 |
38
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) = ( - π [,] π ) ) |
40 |
37 39
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → 𝑡 ∈ ( - π [,] π ) ) |
41 |
36 40
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → ( 𝐹 ‘ 𝑡 ) ∈ ℂ ) |
42 |
33
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
43 |
|
elfzofz |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
44 |
43
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
45 |
42 44
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
46 |
|
fzofzp1 |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
47 |
46
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
48 |
42 47
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
49 |
6
|
feqmptd |
⊢ ( 𝜑 → 𝐹 = ( 𝑡 ∈ ( - π [,] π ) ↦ ( 𝐹 ‘ 𝑡 ) ) ) |
50 |
49
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐹 = ( 𝑡 ∈ ( - π [,] π ) ↦ ( 𝐹 ‘ 𝑡 ) ) ) |
51 |
50
|
reseq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑡 ∈ ( - π [,] π ) ↦ ( 𝐹 ‘ 𝑡 ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
52 |
|
ioossicc |
⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
53 |
52
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
54 |
29
|
rexri |
⊢ - π ∈ ℝ* |
55 |
54
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → - π ∈ ℝ* ) |
56 |
28
|
rexri |
⊢ π ∈ ℝ* |
57 |
56
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → π ∈ ℝ* ) |
58 |
27
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( - π [,] π ) ) |
59 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑖 ∈ ( 0 ..^ 𝑀 ) ) |
60 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
61 |
55 57 58 59 60
|
fourierdlem1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑡 ∈ ( - π [,] π ) ) |
62 |
61
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∀ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑡 ∈ ( - π [,] π ) ) |
63 |
|
dfss3 |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( - π [,] π ) ↔ ∀ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑡 ∈ ( - π [,] π ) ) |
64 |
62 63
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( - π [,] π ) ) |
65 |
53 64
|
sstrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( - π [,] π ) ) |
66 |
65
|
resmptd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑡 ∈ ( - π [,] π ) ↦ ( 𝐹 ‘ 𝑡 ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑡 ) ) ) |
67 |
51 66
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑡 ) ) ) |
68 |
67
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑡 ) ) = ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
69 |
68 7
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑡 ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
70 |
67
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑡 ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
71 |
9 70
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑡 ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
72 |
67
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) = ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑡 ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
73 |
8 72
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑡 ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
74 |
45 48 69 71 73
|
iblcncfioo |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑡 ) ) ∈ 𝐿1 ) |
75 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐹 : ( - π [,] π ) ⟶ ℂ ) |
76 |
75 61
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ 𝑡 ) ∈ ℂ ) |
77 |
45 48 74 76
|
ibliooicc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑡 ) ) ∈ 𝐿1 ) |
78 |
20 26 33 35 41 77
|
itgspltprt |
⊢ ( 𝜑 → ∫ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ( 𝐹 ‘ 𝑡 ) d 𝑡 = Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑡 ) d 𝑡 ) |
79 |
|
fvres |
⊢ ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) = ( 𝐹 ‘ 𝑡 ) ) |
80 |
79
|
eqcomd |
⊢ ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐹 ‘ 𝑡 ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) ) |
81 |
80
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ 𝑡 ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) ) |
82 |
81
|
itgeq2dv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑡 ) d 𝑡 = ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) d 𝑡 ) |
83 |
|
eqid |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ if ( 𝑥 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , if ( 𝑥 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( ( 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 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑥 ) ) ) ) |
84 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐹 : ( - π [,] π ) ⟶ ℂ ) |
85 |
84 64
|
fssresd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ) |
86 |
53
|
resabs1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
87 |
86 7
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
88 |
86
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
89 |
45 48 35 85
|
limcicciooub |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
90 |
88 89
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
91 |
9 90
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
92 |
86
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
93 |
92
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) = ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
94 |
45 48 35 85
|
limciccioolb |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
95 |
93 94
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
96 |
8 95
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
97 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑋 ∈ ℝ ) |
98 |
83 45 48 35 85 87 91 96 97
|
fourierdlem82 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) d 𝑡 = ∫ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 ) |
99 |
45
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
100 |
48
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
101 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → 𝑋 ∈ ℝ ) |
102 |
99 101
|
resubcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ∈ ℝ ) |
103 |
100 101
|
resubcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ∈ ℝ ) |
104 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) |
105 |
|
eliccre |
⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ∈ ℝ ∧ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ∈ ℝ ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → 𝑡 ∈ ℝ ) |
106 |
102 103 104 105
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → 𝑡 ∈ ℝ ) |
107 |
101 106
|
readdcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → ( 𝑋 + 𝑡 ) ∈ ℝ ) |
108 |
|
elicc2 |
⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ∈ ℝ ∧ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ∈ ℝ ) → ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ↔ ( 𝑡 ∈ ℝ ∧ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ≤ 𝑡 ∧ 𝑡 ≤ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) ) |
109 |
102 103 108
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ↔ ( 𝑡 ∈ ℝ ∧ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ≤ 𝑡 ∧ 𝑡 ≤ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) ) |
110 |
104 109
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → ( 𝑡 ∈ ℝ ∧ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ≤ 𝑡 ∧ 𝑡 ≤ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) |
111 |
110
|
simp2d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ≤ 𝑡 ) |
112 |
99 101 106
|
lesubadd2d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ≤ 𝑡 ↔ ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑋 + 𝑡 ) ) ) |
113 |
111 112
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑋 + 𝑡 ) ) |
114 |
110
|
simp3d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → 𝑡 ≤ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) |
115 |
101 106 100
|
leaddsub2d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → ( ( 𝑋 + 𝑡 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ↔ 𝑡 ≤ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) |
116 |
114 115
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → ( 𝑋 + 𝑡 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
117 |
99 100 107 113 116
|
eliccd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → ( 𝑋 + 𝑡 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
118 |
|
fvres |
⊢ ( ( 𝑋 + 𝑡 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑋 + 𝑡 ) ) = ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) |
119 |
117 118
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑋 + 𝑡 ) ) = ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) |
120 |
119
|
itgeq2dv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 = ∫ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 ) |
121 |
82 98 120
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑡 ) d 𝑡 = ∫ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 ) |
122 |
121
|
sumeq2dv |
⊢ ( 𝜑 → Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑡 ) d 𝑡 = Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 ) |
123 |
|
oveq2 |
⊢ ( 𝑠 = 𝑡 → ( 𝑋 + 𝑠 ) = ( 𝑋 + 𝑡 ) ) |
124 |
123
|
fveq2d |
⊢ ( 𝑠 = 𝑡 → ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) = ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) |
125 |
124
|
cbvitgv |
⊢ ∫ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) d 𝑠 = ∫ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 |
126 |
125
|
a1i |
⊢ ( 𝜑 → ∫ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) d 𝑠 = ∫ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 ) |
127 |
2
|
a1i |
⊢ ( 𝜑 → 𝐻 = ( 𝑖 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) ) |
128 |
|
fveq2 |
⊢ ( 𝑖 = 0 → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 0 ) ) |
129 |
128
|
oveq1d |
⊢ ( 𝑖 = 0 → ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) = ( ( 𝑄 ‘ 0 ) − 𝑋 ) ) |
130 |
129
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 = 0 ) → ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) = ( ( 𝑄 ‘ 0 ) − 𝑋 ) ) |
131 |
3
|
nnzd |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
132 |
|
0le0 |
⊢ 0 ≤ 0 |
133 |
132
|
a1i |
⊢ ( 𝜑 → 0 ≤ 0 ) |
134 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
135 |
3
|
nnred |
⊢ ( 𝜑 → 𝑀 ∈ ℝ ) |
136 |
3
|
nngt0d |
⊢ ( 𝜑 → 0 < 𝑀 ) |
137 |
134 135 136
|
ltled |
⊢ ( 𝜑 → 0 ≤ 𝑀 ) |
138 |
20 131 20 133 137
|
elfzd |
⊢ ( 𝜑 → 0 ∈ ( 0 ... 𝑀 ) ) |
139 |
14 29
|
eqeltrdi |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) ∈ ℝ ) |
140 |
139 5
|
resubcld |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) − 𝑋 ) ∈ ℝ ) |
141 |
127 130 138 140
|
fvmptd |
⊢ ( 𝜑 → ( 𝐻 ‘ 0 ) = ( ( 𝑄 ‘ 0 ) − 𝑋 ) ) |
142 |
14
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) − 𝑋 ) = ( - π − 𝑋 ) ) |
143 |
141 142
|
eqtr2d |
⊢ ( 𝜑 → ( - π − 𝑋 ) = ( 𝐻 ‘ 0 ) ) |
144 |
|
fveq2 |
⊢ ( 𝑖 = 𝑀 → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 𝑀 ) ) |
145 |
144
|
oveq1d |
⊢ ( 𝑖 = 𝑀 → ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) = ( ( 𝑄 ‘ 𝑀 ) − 𝑋 ) ) |
146 |
145
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 = 𝑀 ) → ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) = ( ( 𝑄 ‘ 𝑀 ) − 𝑋 ) ) |
147 |
135
|
leidd |
⊢ ( 𝜑 → 𝑀 ≤ 𝑀 ) |
148 |
20 131 131 137 147
|
elfzd |
⊢ ( 𝜑 → 𝑀 ∈ ( 0 ... 𝑀 ) ) |
149 |
16 28
|
eqeltrdi |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) ∈ ℝ ) |
150 |
149 5
|
resubcld |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 𝑀 ) − 𝑋 ) ∈ ℝ ) |
151 |
127 146 148 150
|
fvmptd |
⊢ ( 𝜑 → ( 𝐻 ‘ 𝑀 ) = ( ( 𝑄 ‘ 𝑀 ) − 𝑋 ) ) |
152 |
16
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 𝑀 ) − 𝑋 ) = ( π − 𝑋 ) ) |
153 |
151 152
|
eqtr2d |
⊢ ( 𝜑 → ( π − 𝑋 ) = ( 𝐻 ‘ 𝑀 ) ) |
154 |
143 153
|
oveq12d |
⊢ ( 𝜑 → ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) = ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) |
155 |
154
|
itgeq1d |
⊢ ( 𝜑 → ∫ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 = ∫ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 ) |
156 |
33
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
157 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑀 ) ) → 𝑋 ∈ ℝ ) |
158 |
156 157
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ∈ ℝ ) |
159 |
158 2
|
fmptd |
⊢ ( 𝜑 → 𝐻 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
160 |
45 48 97 35
|
ltsub1dd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) |
161 |
44 158
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ∈ ℝ ) |
162 |
2
|
fvmpt2 |
⊢ ( ( 𝑖 ∈ ( 0 ... 𝑀 ) ∧ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ∈ ℝ ) → ( 𝐻 ‘ 𝑖 ) = ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) |
163 |
44 161 162
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐻 ‘ 𝑖 ) = ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) |
164 |
|
fveq2 |
⊢ ( 𝑖 = 𝑗 → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 𝑗 ) ) |
165 |
164
|
oveq1d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) = ( ( 𝑄 ‘ 𝑗 ) − 𝑋 ) ) |
166 |
165
|
cbvmptv |
⊢ ( 𝑖 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑗 ) − 𝑋 ) ) |
167 |
2 166
|
eqtri |
⊢ 𝐻 = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑗 ) − 𝑋 ) ) |
168 |
167
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐻 = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑗 ) − 𝑋 ) ) ) |
169 |
|
fveq2 |
⊢ ( 𝑗 = ( 𝑖 + 1 ) → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
170 |
169
|
oveq1d |
⊢ ( 𝑗 = ( 𝑖 + 1 ) → ( ( 𝑄 ‘ 𝑗 ) − 𝑋 ) = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) |
171 |
170
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 = ( 𝑖 + 1 ) ) → ( ( 𝑄 ‘ 𝑗 ) − 𝑋 ) = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) |
172 |
48 97
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ∈ ℝ ) |
173 |
168 171 47 172
|
fvmptd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐻 ‘ ( 𝑖 + 1 ) ) = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) |
174 |
160 163 173
|
3brtr4d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐻 ‘ 𝑖 ) < ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) |
175 |
|
frn |
⊢ ( 𝐹 : ( - π [,] π ) ⟶ ℂ → ran 𝐹 ⊆ ℂ ) |
176 |
6 175
|
syl |
⊢ ( 𝜑 → ran 𝐹 ⊆ ℂ ) |
177 |
176
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → ran 𝐹 ⊆ ℂ ) |
178 |
|
ffun |
⊢ ( 𝐹 : ( - π [,] π ) ⟶ ℂ → Fun 𝐹 ) |
179 |
6 178
|
syl |
⊢ ( 𝜑 → Fun 𝐹 ) |
180 |
179
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → Fun 𝐹 ) |
181 |
29
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → - π ∈ ℝ ) |
182 |
28
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → π ∈ ℝ ) |
183 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → 𝑋 ∈ ℝ ) |
184 |
141 140
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐻 ‘ 0 ) ∈ ℝ ) |
185 |
184
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → ( 𝐻 ‘ 0 ) ∈ ℝ ) |
186 |
151 150
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐻 ‘ 𝑀 ) ∈ ℝ ) |
187 |
186
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → ( 𝐻 ‘ 𝑀 ) ∈ ℝ ) |
188 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) |
189 |
|
eliccre |
⊢ ( ( ( 𝐻 ‘ 0 ) ∈ ℝ ∧ ( 𝐻 ‘ 𝑀 ) ∈ ℝ ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → 𝑡 ∈ ℝ ) |
190 |
185 187 188 189
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → 𝑡 ∈ ℝ ) |
191 |
183 190
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → ( 𝑋 + 𝑡 ) ∈ ℝ ) |
192 |
128
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 = 0 ) → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 0 ) ) |
193 |
192
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 = 0 ) → ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) = ( ( 𝑄 ‘ 0 ) − 𝑋 ) ) |
194 |
127 193 138 140
|
fvmptd |
⊢ ( 𝜑 → ( 𝐻 ‘ 0 ) = ( ( 𝑄 ‘ 0 ) − 𝑋 ) ) |
195 |
194
|
oveq2d |
⊢ ( 𝜑 → ( 𝑋 + ( 𝐻 ‘ 0 ) ) = ( 𝑋 + ( ( 𝑄 ‘ 0 ) − 𝑋 ) ) ) |
196 |
5
|
recnd |
⊢ ( 𝜑 → 𝑋 ∈ ℂ ) |
197 |
139
|
recnd |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) ∈ ℂ ) |
198 |
196 197
|
pncan3d |
⊢ ( 𝜑 → ( 𝑋 + ( ( 𝑄 ‘ 0 ) − 𝑋 ) ) = ( 𝑄 ‘ 0 ) ) |
199 |
195 198 14
|
3eqtrrd |
⊢ ( 𝜑 → - π = ( 𝑋 + ( 𝐻 ‘ 0 ) ) ) |
200 |
199
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → - π = ( 𝑋 + ( 𝐻 ‘ 0 ) ) ) |
201 |
|
elicc2 |
⊢ ( ( ( 𝐻 ‘ 0 ) ∈ ℝ ∧ ( 𝐻 ‘ 𝑀 ) ∈ ℝ ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ↔ ( 𝑡 ∈ ℝ ∧ ( 𝐻 ‘ 0 ) ≤ 𝑡 ∧ 𝑡 ≤ ( 𝐻 ‘ 𝑀 ) ) ) ) |
202 |
185 187 201
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ↔ ( 𝑡 ∈ ℝ ∧ ( 𝐻 ‘ 0 ) ≤ 𝑡 ∧ 𝑡 ≤ ( 𝐻 ‘ 𝑀 ) ) ) ) |
203 |
188 202
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → ( 𝑡 ∈ ℝ ∧ ( 𝐻 ‘ 0 ) ≤ 𝑡 ∧ 𝑡 ≤ ( 𝐻 ‘ 𝑀 ) ) ) |
204 |
203
|
simp2d |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → ( 𝐻 ‘ 0 ) ≤ 𝑡 ) |
205 |
185 190 183 204
|
leadd2dd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → ( 𝑋 + ( 𝐻 ‘ 0 ) ) ≤ ( 𝑋 + 𝑡 ) ) |
206 |
200 205
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → - π ≤ ( 𝑋 + 𝑡 ) ) |
207 |
203
|
simp3d |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → 𝑡 ≤ ( 𝐻 ‘ 𝑀 ) ) |
208 |
190 187 183 207
|
leadd2dd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → ( 𝑋 + 𝑡 ) ≤ ( 𝑋 + ( 𝐻 ‘ 𝑀 ) ) ) |
209 |
151
|
oveq2d |
⊢ ( 𝜑 → ( 𝑋 + ( 𝐻 ‘ 𝑀 ) ) = ( 𝑋 + ( ( 𝑄 ‘ 𝑀 ) − 𝑋 ) ) ) |
210 |
149
|
recnd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) ∈ ℂ ) |
211 |
196 210
|
pncan3d |
⊢ ( 𝜑 → ( 𝑋 + ( ( 𝑄 ‘ 𝑀 ) − 𝑋 ) ) = ( 𝑄 ‘ 𝑀 ) ) |
212 |
209 211 16
|
3eqtrrd |
⊢ ( 𝜑 → π = ( 𝑋 + ( 𝐻 ‘ 𝑀 ) ) ) |
213 |
212
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → π = ( 𝑋 + ( 𝐻 ‘ 𝑀 ) ) ) |
214 |
208 213
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → ( 𝑋 + 𝑡 ) ≤ π ) |
215 |
181 182 191 206 214
|
eliccd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → ( 𝑋 + 𝑡 ) ∈ ( - π [,] π ) ) |
216 |
|
fdm |
⊢ ( 𝐹 : ( - π [,] π ) ⟶ ℂ → dom 𝐹 = ( - π [,] π ) ) |
217 |
6 216
|
syl |
⊢ ( 𝜑 → dom 𝐹 = ( - π [,] π ) ) |
218 |
217
|
eqcomd |
⊢ ( 𝜑 → ( - π [,] π ) = dom 𝐹 ) |
219 |
218
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → ( - π [,] π ) = dom 𝐹 ) |
220 |
215 219
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → ( 𝑋 + 𝑡 ) ∈ dom 𝐹 ) |
221 |
|
fvelrn |
⊢ ( ( Fun 𝐹 ∧ ( 𝑋 + 𝑡 ) ∈ dom 𝐹 ) → ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ∈ ran 𝐹 ) |
222 |
180 220 221
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ∈ ran 𝐹 ) |
223 |
177 222
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ) → ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ∈ ℂ ) |
224 |
163 161
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐻 ‘ 𝑖 ) ∈ ℝ ) |
225 |
173 172
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
226 |
84 65
|
fssresd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ) |
227 |
45
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
228 |
227
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
229 |
48
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
230 |
229
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
231 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑋 ∈ ℝ ) |
232 |
|
elioore |
⊢ ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → 𝑡 ∈ ℝ ) |
233 |
232
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑡 ∈ ℝ ) |
234 |
231 233
|
readdcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑋 + 𝑡 ) ∈ ℝ ) |
235 |
163
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑋 + ( 𝐻 ‘ 𝑖 ) ) = ( 𝑋 + ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) ) |
236 |
196
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑋 ∈ ℂ ) |
237 |
45
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℂ ) |
238 |
236 237
|
pncan3d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑋 + ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) = ( 𝑄 ‘ 𝑖 ) ) |
239 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 𝑖 ) ) |
240 |
235 238 239
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) = ( 𝑋 + ( 𝐻 ‘ 𝑖 ) ) ) |
241 |
240
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) = ( 𝑋 + ( 𝐻 ‘ 𝑖 ) ) ) |
242 |
224
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐻 ‘ 𝑖 ) ∈ ℝ ) |
243 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
244 |
242
|
rexrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐻 ‘ 𝑖 ) ∈ ℝ* ) |
245 |
225
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
246 |
245
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
247 |
|
elioo2 |
⊢ ( ( ( 𝐻 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝑡 ∈ ℝ ∧ ( 𝐻 ‘ 𝑖 ) < 𝑡 ∧ 𝑡 < ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
248 |
244 246 247
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝑡 ∈ ℝ ∧ ( 𝐻 ‘ 𝑖 ) < 𝑡 ∧ 𝑡 < ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
249 |
243 248
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑡 ∈ ℝ ∧ ( 𝐻 ‘ 𝑖 ) < 𝑡 ∧ 𝑡 < ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
250 |
249
|
simp2d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐻 ‘ 𝑖 ) < 𝑡 ) |
251 |
242 233 231 250
|
ltadd2dd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑋 + ( 𝐻 ‘ 𝑖 ) ) < ( 𝑋 + 𝑡 ) ) |
252 |
241 251
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑋 + 𝑡 ) ) |
253 |
225
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
254 |
249
|
simp3d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑡 < ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) |
255 |
233 253 231 254
|
ltadd2dd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑋 + 𝑡 ) < ( 𝑋 + ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
256 |
173
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑋 + ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) = ( 𝑋 + ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) |
257 |
48
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℂ ) |
258 |
236 257
|
pncan3d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑋 + ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
259 |
256 258
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑋 + ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
260 |
259
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑋 + ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
261 |
255 260
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑋 + 𝑡 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
262 |
228 230 234 252 261
|
eliood |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑋 + 𝑡 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
263 |
|
eqid |
⊢ ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) = ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) |
264 |
262 263
|
fmptd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) : ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⟶ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
265 |
|
fcompt |
⊢ ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ∧ ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) : ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⟶ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∘ ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) = ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) ) ) |
266 |
226 264 265
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∘ ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) = ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) ) ) |
267 |
|
oveq2 |
⊢ ( 𝑡 = 𝑟 → ( 𝑋 + 𝑡 ) = ( 𝑋 + 𝑟 ) ) |
268 |
267
|
cbvmptv |
⊢ ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) = ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) |
269 |
268
|
fveq1i |
⊢ ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) = ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑠 ) |
270 |
269
|
fveq2i |
⊢ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑠 ) ) |
271 |
270
|
mpteq2i |
⊢ ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) ) = ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑠 ) ) ) |
272 |
271
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) ) = ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑠 ) ) ) ) |
273 |
|
fveq2 |
⊢ ( 𝑠 = 𝑡 → ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑠 ) = ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑡 ) ) |
274 |
273
|
fveq2d |
⊢ ( 𝑠 = 𝑡 → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑠 ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑡 ) ) ) |
275 |
274
|
cbvmptv |
⊢ ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑠 ) ) ) = ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑡 ) ) ) |
276 |
275
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑠 ) ) ) = ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑡 ) ) ) ) |
277 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) = ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ) |
278 |
|
oveq2 |
⊢ ( 𝑟 = 𝑡 → ( 𝑋 + 𝑟 ) = ( 𝑋 + 𝑡 ) ) |
279 |
278
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑟 = 𝑡 ) → ( 𝑋 + 𝑟 ) = ( 𝑋 + 𝑡 ) ) |
280 |
277 279 243 234
|
fvmptd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑡 ) = ( 𝑋 + 𝑡 ) ) |
281 |
280
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑡 ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑋 + 𝑡 ) ) ) |
282 |
|
fvres |
⊢ ( ( 𝑋 + 𝑡 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑋 + 𝑡 ) ) = ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) |
283 |
262 282
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝑋 + 𝑡 ) ) = ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) |
284 |
281 283
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑡 ) ) = ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) |
285 |
284
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑟 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑟 ) ) ‘ 𝑡 ) ) ) = ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) ) |
286 |
272 276 285
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) ) = ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) ) |
287 |
266 286
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∘ ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) ) |
288 |
|
eqid |
⊢ ( 𝑡 ∈ ℂ ↦ ( 𝑋 + 𝑡 ) ) = ( 𝑡 ∈ ℂ ↦ ( 𝑋 + 𝑡 ) ) |
289 |
|
ssid |
⊢ ℂ ⊆ ℂ |
290 |
289
|
a1i |
⊢ ( 𝑋 ∈ ℂ → ℂ ⊆ ℂ ) |
291 |
|
id |
⊢ ( 𝑋 ∈ ℂ → 𝑋 ∈ ℂ ) |
292 |
290 291 290
|
constcncfg |
⊢ ( 𝑋 ∈ ℂ → ( 𝑡 ∈ ℂ ↦ 𝑋 ) ∈ ( ℂ –cn→ ℂ ) ) |
293 |
|
cncfmptid |
⊢ ( ( ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( 𝑡 ∈ ℂ ↦ 𝑡 ) ∈ ( ℂ –cn→ ℂ ) ) |
294 |
289 289 293
|
mp2an |
⊢ ( 𝑡 ∈ ℂ ↦ 𝑡 ) ∈ ( ℂ –cn→ ℂ ) |
295 |
294
|
a1i |
⊢ ( 𝑋 ∈ ℂ → ( 𝑡 ∈ ℂ ↦ 𝑡 ) ∈ ( ℂ –cn→ ℂ ) ) |
296 |
292 295
|
addcncf |
⊢ ( 𝑋 ∈ ℂ → ( 𝑡 ∈ ℂ ↦ ( 𝑋 + 𝑡 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
297 |
236 296
|
syl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ℂ ↦ ( 𝑋 + 𝑡 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
298 |
|
ioosscn |
⊢ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ |
299 |
298
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ ) |
300 |
|
ioosscn |
⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ |
301 |
300
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ ) |
302 |
288 297 299 301 262
|
cncfmptssg |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) –cn→ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
303 |
302 7
|
cncfco |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∘ ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
304 |
287 303
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
305 |
227
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
306 |
229
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
307 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) |
308 |
|
vex |
⊢ 𝑟 ∈ V |
309 |
263
|
elrnmpt |
⊢ ( 𝑟 ∈ V → ( 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ↔ ∃ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) 𝑟 = ( 𝑋 + 𝑡 ) ) ) |
310 |
308 309
|
ax-mp |
⊢ ( 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ↔ ∃ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) 𝑟 = ( 𝑋 + 𝑡 ) ) |
311 |
307 310
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → ∃ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) 𝑟 = ( 𝑋 + 𝑡 ) ) |
312 |
|
nfv |
⊢ Ⅎ 𝑡 ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) |
313 |
|
nfmpt1 |
⊢ Ⅎ 𝑡 ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) |
314 |
313
|
nfrn |
⊢ Ⅎ 𝑡 ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) |
315 |
314
|
nfcri |
⊢ Ⅎ 𝑡 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) |
316 |
312 315
|
nfan |
⊢ Ⅎ 𝑡 ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) |
317 |
|
nfv |
⊢ Ⅎ 𝑡 𝑟 ∈ ℝ |
318 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑟 = ( 𝑋 + 𝑡 ) ) → 𝑟 = ( 𝑋 + 𝑡 ) ) |
319 |
5
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑟 = ( 𝑋 + 𝑡 ) ) → 𝑋 ∈ ℝ ) |
320 |
232
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑟 = ( 𝑋 + 𝑡 ) ) → 𝑡 ∈ ℝ ) |
321 |
319 320
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑟 = ( 𝑋 + 𝑡 ) ) → ( 𝑋 + 𝑡 ) ∈ ℝ ) |
322 |
318 321
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑟 = ( 𝑋 + 𝑡 ) ) → 𝑟 ∈ ℝ ) |
323 |
322
|
3exp |
⊢ ( 𝜑 → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑟 = ( 𝑋 + 𝑡 ) → 𝑟 ∈ ℝ ) ) ) |
324 |
323
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑟 = ( 𝑋 + 𝑡 ) → 𝑟 ∈ ℝ ) ) ) |
325 |
316 317 324
|
rexlimd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → ( ∃ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) 𝑟 = ( 𝑋 + 𝑡 ) → 𝑟 ∈ ℝ ) ) |
326 |
311 325
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → 𝑟 ∈ ℝ ) |
327 |
|
nfv |
⊢ Ⅎ 𝑡 ( 𝑄 ‘ 𝑖 ) < 𝑟 |
328 |
252
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑟 = ( 𝑋 + 𝑡 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑋 + 𝑡 ) ) |
329 |
|
simp3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑟 = ( 𝑋 + 𝑡 ) ) → 𝑟 = ( 𝑋 + 𝑡 ) ) |
330 |
328 329
|
breqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑟 = ( 𝑋 + 𝑡 ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑟 ) |
331 |
330
|
3exp |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑟 = ( 𝑋 + 𝑡 ) → ( 𝑄 ‘ 𝑖 ) < 𝑟 ) ) ) |
332 |
331
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑟 = ( 𝑋 + 𝑡 ) → ( 𝑄 ‘ 𝑖 ) < 𝑟 ) ) ) |
333 |
316 327 332
|
rexlimd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → ( ∃ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) 𝑟 = ( 𝑋 + 𝑡 ) → ( 𝑄 ‘ 𝑖 ) < 𝑟 ) ) |
334 |
311 333
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑟 ) |
335 |
|
nfv |
⊢ Ⅎ 𝑡 𝑟 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) |
336 |
261
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑟 = ( 𝑋 + 𝑡 ) ) → ( 𝑋 + 𝑡 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
337 |
329 336
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑟 = ( 𝑋 + 𝑡 ) ) → 𝑟 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
338 |
337
|
3exp |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑟 = ( 𝑋 + 𝑡 ) → 𝑟 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
339 |
338
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑟 = ( 𝑋 + 𝑡 ) → 𝑟 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
340 |
316 335 339
|
rexlimd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → ( ∃ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) 𝑟 = ( 𝑋 + 𝑡 ) → 𝑟 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
341 |
311 340
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → 𝑟 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
342 |
305 306 326 334 341
|
eliood |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
343 |
217
|
ineq2d |
⊢ ( 𝜑 → ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∩ dom 𝐹 ) = ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∩ ( - π [,] π ) ) ) |
344 |
343
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∩ dom 𝐹 ) = ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∩ ( - π [,] π ) ) ) |
345 |
|
dmres |
⊢ dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∩ dom 𝐹 ) |
346 |
345
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∩ dom 𝐹 ) ) |
347 |
|
dfss |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( - π [,] π ) ↔ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∩ ( - π [,] π ) ) ) |
348 |
65 347
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∩ ( - π [,] π ) ) ) |
349 |
344 346 348
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
350 |
349
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
351 |
342 350
|
eleqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → 𝑟 ∈ dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
352 |
326 341
|
ltned |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → 𝑟 ≠ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
353 |
352
|
neneqd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → ¬ 𝑟 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
354 |
|
velsn |
⊢ ( 𝑟 ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ↔ 𝑟 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
355 |
353 354
|
sylnibr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → ¬ 𝑟 ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) |
356 |
351 355
|
eldifd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → 𝑟 ∈ ( dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∖ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) |
357 |
356
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∀ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) 𝑟 ∈ ( dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∖ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) |
358 |
|
dfss3 |
⊢ ( ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ⊆ ( dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∖ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↔ ∀ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) 𝑟 ∈ ( dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∖ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) |
359 |
357 358
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ⊆ ( dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∖ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) |
360 |
|
eqid |
⊢ ( 𝑠 ∈ ℂ ↦ ( 𝑋 + 𝑠 ) ) = ( 𝑠 ∈ ℂ ↦ ( 𝑋 + 𝑠 ) ) |
361 |
196
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ℂ ) → 𝑋 ∈ ℂ ) |
362 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ℂ ) → 𝑠 ∈ ℂ ) |
363 |
361 362
|
addcomd |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ℂ ) → ( 𝑋 + 𝑠 ) = ( 𝑠 + 𝑋 ) ) |
364 |
363
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑠 ∈ ℂ ↦ ( 𝑋 + 𝑠 ) ) = ( 𝑠 ∈ ℂ ↦ ( 𝑠 + 𝑋 ) ) ) |
365 |
|
eqid |
⊢ ( 𝑠 ∈ ℂ ↦ ( 𝑠 + 𝑋 ) ) = ( 𝑠 ∈ ℂ ↦ ( 𝑠 + 𝑋 ) ) |
366 |
365
|
addccncf |
⊢ ( 𝑋 ∈ ℂ → ( 𝑠 ∈ ℂ ↦ ( 𝑠 + 𝑋 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
367 |
196 366
|
syl |
⊢ ( 𝜑 → ( 𝑠 ∈ ℂ ↦ ( 𝑠 + 𝑋 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
368 |
364 367
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑠 ∈ ℂ ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
369 |
368
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ℂ ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
370 |
224
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐻 ‘ 𝑖 ) ∈ ℝ* ) |
371 |
|
iocssre |
⊢ ( ( ( 𝐻 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) → ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ ) |
372 |
370 225 371
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ ) |
373 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
374 |
372 373
|
sstrdi |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ ) |
375 |
289
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ℂ ⊆ ℂ ) |
376 |
196
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑋 ∈ ℂ ) |
377 |
374
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑠 ∈ ℂ ) |
378 |
376 377
|
addcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑋 + 𝑠 ) ∈ ℂ ) |
379 |
360 369 374 375 378
|
cncfmptssg |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
380 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
381 |
|
eqid |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
382 |
380
|
cnfldtop |
⊢ ( TopOpen ‘ ℂfld ) ∈ Top |
383 |
|
unicntop |
⊢ ℂ = ∪ ( TopOpen ‘ ℂfld ) |
384 |
383
|
restid |
⊢ ( ( TopOpen ‘ ℂfld ) ∈ Top → ( ( TopOpen ‘ ℂfld ) ↾t ℂ ) = ( TopOpen ‘ ℂfld ) ) |
385 |
382 384
|
ax-mp |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ℂ ) = ( TopOpen ‘ ℂfld ) |
386 |
385
|
eqcomi |
⊢ ( TopOpen ‘ ℂfld ) = ( ( TopOpen ‘ ℂfld ) ↾t ℂ ) |
387 |
380 381 386
|
cncfcn |
⊢ ( ( ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) = ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
388 |
374 375 387
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) = ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
389 |
379 388
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
390 |
380
|
cnfldtopon |
⊢ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) |
391 |
390
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ) |
392 |
|
resttopon |
⊢ ( ( ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ∧ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ ) → ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( TopOn ‘ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
393 |
391 374 392
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( TopOn ‘ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
394 |
|
cncnp |
⊢ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( TopOn ‘ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ) → ( ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) Cn ( TopOpen ‘ ℂfld ) ) ↔ ( ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) : ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ∧ ∀ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑡 ) ) ) ) |
395 |
393 391 394
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) Cn ( TopOpen ‘ ℂfld ) ) ↔ ( ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) : ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ∧ ∀ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑡 ) ) ) ) |
396 |
389 395
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) : ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ∧ ∀ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑡 ) ) ) |
397 |
396
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∀ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑡 ) ) |
398 |
|
ubioc1 |
⊢ ( ( ( 𝐻 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ ( 𝐻 ‘ 𝑖 ) < ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
399 |
370 245 174 398
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
400 |
|
fveq2 |
⊢ ( 𝑡 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) → ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑡 ) = ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
401 |
400
|
eleq2d |
⊢ ( 𝑡 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) → ( ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑡 ) ↔ ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
402 |
401
|
rspccva |
⊢ ( ( ∀ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑡 ) ∧ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
403 |
397 399 402
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
404 |
|
ioounsn |
⊢ ( ( ( 𝐻 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ ( 𝐻 ‘ 𝑖 ) < ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) = ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
405 |
370 245 174 404
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) = ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
406 |
259
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑋 + ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
407 |
406
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑋 + ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
408 |
|
iftrue |
⊢ ( 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) → if ( 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) , ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
409 |
408
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) , ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
410 |
|
oveq2 |
⊢ ( 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) → ( 𝑋 + 𝑠 ) = ( 𝑋 + ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
411 |
410
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑋 + 𝑠 ) = ( 𝑋 + ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
412 |
407 409 411
|
3eqtr4d |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) , ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( 𝑋 + 𝑠 ) ) |
413 |
|
iffalse |
⊢ ( ¬ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) → if ( 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) , ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) |
414 |
413
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) , ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) |
415 |
|
eqidd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) = ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) |
416 |
|
oveq2 |
⊢ ( 𝑡 = 𝑠 → ( 𝑋 + 𝑡 ) = ( 𝑋 + 𝑠 ) ) |
417 |
416
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑡 = 𝑠 ) → ( 𝑋 + 𝑡 ) = ( 𝑋 + 𝑠 ) ) |
418 |
|
velsn |
⊢ ( 𝑠 ∈ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ↔ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) |
419 |
418
|
notbii |
⊢ ( ¬ 𝑠 ∈ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ↔ ¬ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) |
420 |
|
elun |
⊢ ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ↔ ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∨ 𝑠 ∈ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) |
421 |
420
|
biimpi |
⊢ ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) → ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∨ 𝑠 ∈ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) |
422 |
421
|
orcomd |
⊢ ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) → ( 𝑠 ∈ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ∨ 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
423 |
422
|
ord |
⊢ ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) → ( ¬ 𝑠 ∈ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } → 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
424 |
419 423
|
syl5bir |
⊢ ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) → ( ¬ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) → 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
425 |
424
|
imp |
⊢ ( ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ∧ ¬ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
426 |
425
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
427 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) → 𝑋 ∈ ℝ ) |
428 |
|
elioore |
⊢ ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → 𝑠 ∈ ℝ ) |
429 |
428
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑠 ∈ ℝ ) |
430 |
|
elsni |
⊢ ( 𝑠 ∈ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } → 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) |
431 |
430
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) → 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) |
432 |
225
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) → ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
433 |
431 432
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) → 𝑠 ∈ ℝ ) |
434 |
429 433
|
jaodan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∨ 𝑠 ∈ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) → 𝑠 ∈ ℝ ) |
435 |
420 434
|
sylan2b |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) → 𝑠 ∈ ℝ ) |
436 |
427 435
|
readdcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) → ( 𝑋 + 𝑠 ) ∈ ℝ ) |
437 |
436
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑋 + 𝑠 ) ∈ ℝ ) |
438 |
415 417 426 437
|
fvmptd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) = ( 𝑋 + 𝑠 ) ) |
439 |
414 438
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) , ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( 𝑋 + 𝑠 ) ) |
440 |
412 439
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) → if ( 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) , ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( 𝑋 + 𝑠 ) ) |
441 |
405 440
|
mpteq12dva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ↦ if ( 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) , ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) ) = ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ) |
442 |
405
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) = ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
443 |
442
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) = ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ) |
444 |
443
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) = ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) (,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
445 |
403 441 444
|
3eltr4d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ↦ if ( 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) , ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
446 |
|
eqid |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) = ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) |
447 |
|
eqid |
⊢ ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ↦ if ( 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) , ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) ) = ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ↦ if ( 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) , ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) ) |
448 |
264 301
|
fssd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) : ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ) |
449 |
225
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ℂ ) |
450 |
446 380 447 448 299 449
|
ellimc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) limℂ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ↦ if ( 𝑠 = ( 𝐻 ‘ ( 𝑖 + 1 ) ) , ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
451 |
445 450
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) limℂ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
452 |
359 451 9
|
limccog |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∘ ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) limℂ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
453 |
266 286
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∘ ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) = ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) ) |
454 |
453
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∘ ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) limℂ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) limℂ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
455 |
452 454
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) limℂ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
456 |
45
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
457 |
456 334
|
gtned |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → 𝑟 ≠ ( 𝑄 ‘ 𝑖 ) ) |
458 |
457
|
neneqd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → ¬ 𝑟 = ( 𝑄 ‘ 𝑖 ) ) |
459 |
|
velsn |
⊢ ( 𝑟 ∈ { ( 𝑄 ‘ 𝑖 ) } ↔ 𝑟 = ( 𝑄 ‘ 𝑖 ) ) |
460 |
458 459
|
sylnibr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → ¬ 𝑟 ∈ { ( 𝑄 ‘ 𝑖 ) } ) |
461 |
351 460
|
eldifd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) → 𝑟 ∈ ( dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∖ { ( 𝑄 ‘ 𝑖 ) } ) ) |
462 |
461
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∀ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) 𝑟 ∈ ( dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∖ { ( 𝑄 ‘ 𝑖 ) } ) ) |
463 |
|
dfss3 |
⊢ ( ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ⊆ ( dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∖ { ( 𝑄 ‘ 𝑖 ) } ) ↔ ∀ 𝑟 ∈ ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) 𝑟 ∈ ( dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∖ { ( 𝑄 ‘ 𝑖 ) } ) ) |
464 |
462 463
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ran ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ⊆ ( dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∖ { ( 𝑄 ‘ 𝑖 ) } ) ) |
465 |
|
icossre |
⊢ ( ( ( 𝐻 ‘ 𝑖 ) ∈ ℝ ∧ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) → ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ ) |
466 |
224 245 465
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ ) |
467 |
466 373
|
sstrdi |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ ) |
468 |
196
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑋 ∈ ℂ ) |
469 |
467
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑠 ∈ ℂ ) |
470 |
468 469
|
addcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑋 + 𝑠 ) ∈ ℂ ) |
471 |
360 369 467 375 470
|
cncfmptssg |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
472 |
|
eqid |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
473 |
380 472 386
|
cncfcn |
⊢ ( ( ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) = ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
474 |
467 375 473
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) = ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
475 |
471 474
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
476 |
|
resttopon |
⊢ ( ( ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ∧ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ ) → ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( TopOn ‘ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
477 |
391 467 476
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( TopOn ‘ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
478 |
|
cncnp |
⊢ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( TopOn ‘ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ) → ( ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) Cn ( TopOpen ‘ ℂfld ) ) ↔ ( ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) : ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ∧ ∀ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑡 ) ) ) ) |
479 |
477 391 478
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) Cn ( TopOpen ‘ ℂfld ) ) ↔ ( ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) : ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ∧ ∀ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑡 ) ) ) ) |
480 |
475 479
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) : ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ∧ ∀ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑡 ) ) ) |
481 |
480
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∀ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑡 ) ) |
482 |
|
lbico1 |
⊢ ( ( ( 𝐻 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ ( 𝐻 ‘ 𝑖 ) < ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐻 ‘ 𝑖 ) ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
483 |
370 245 174 482
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐻 ‘ 𝑖 ) ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
484 |
|
fveq2 |
⊢ ( 𝑡 = ( 𝐻 ‘ 𝑖 ) → ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑡 ) = ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ 𝑖 ) ) ) |
485 |
484
|
eleq2d |
⊢ ( 𝑡 = ( 𝐻 ‘ 𝑖 ) → ( ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑡 ) ↔ ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ 𝑖 ) ) ) ) |
486 |
485
|
rspccva |
⊢ ( ( ∀ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑡 ) ∧ ( 𝐻 ‘ 𝑖 ) ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ 𝑖 ) ) ) |
487 |
481 483 486
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ 𝑖 ) ) ) |
488 |
|
uncom |
⊢ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) = ( { ( 𝐻 ‘ 𝑖 ) } ∪ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
489 |
|
snunioo |
⊢ ( ( ( 𝐻 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝐻 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ ( 𝐻 ‘ 𝑖 ) < ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) → ( { ( 𝐻 ‘ 𝑖 ) } ∪ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
490 |
370 245 174 489
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( { ( 𝐻 ‘ 𝑖 ) } ∪ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
491 |
488 490
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) = ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
492 |
|
iftrue |
⊢ ( 𝑠 = ( 𝐻 ‘ 𝑖 ) → if ( 𝑠 = ( 𝐻 ‘ 𝑖 ) , ( 𝑄 ‘ 𝑖 ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( 𝑄 ‘ 𝑖 ) ) |
493 |
492
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 = ( 𝐻 ‘ 𝑖 ) ) → if ( 𝑠 = ( 𝐻 ‘ 𝑖 ) , ( 𝑄 ‘ 𝑖 ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( 𝑄 ‘ 𝑖 ) ) |
494 |
240
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 = ( 𝐻 ‘ 𝑖 ) ) → ( 𝑄 ‘ 𝑖 ) = ( 𝑋 + ( 𝐻 ‘ 𝑖 ) ) ) |
495 |
|
oveq2 |
⊢ ( 𝑠 = ( 𝐻 ‘ 𝑖 ) → ( 𝑋 + 𝑠 ) = ( 𝑋 + ( 𝐻 ‘ 𝑖 ) ) ) |
496 |
495
|
eqcomd |
⊢ ( 𝑠 = ( 𝐻 ‘ 𝑖 ) → ( 𝑋 + ( 𝐻 ‘ 𝑖 ) ) = ( 𝑋 + 𝑠 ) ) |
497 |
496
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 = ( 𝐻 ‘ 𝑖 ) ) → ( 𝑋 + ( 𝐻 ‘ 𝑖 ) ) = ( 𝑋 + 𝑠 ) ) |
498 |
493 494 497
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 = ( 𝐻 ‘ 𝑖 ) ) → if ( 𝑠 = ( 𝐻 ‘ 𝑖 ) , ( 𝑄 ‘ 𝑖 ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( 𝑋 + 𝑠 ) ) |
499 |
498
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) ∧ 𝑠 = ( 𝐻 ‘ 𝑖 ) ) → if ( 𝑠 = ( 𝐻 ‘ 𝑖 ) , ( 𝑄 ‘ 𝑖 ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( 𝑋 + 𝑠 ) ) |
500 |
|
iffalse |
⊢ ( ¬ 𝑠 = ( 𝐻 ‘ 𝑖 ) → if ( 𝑠 = ( 𝐻 ‘ 𝑖 ) , ( 𝑄 ‘ 𝑖 ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) |
501 |
500
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) ∧ ¬ 𝑠 = ( 𝐻 ‘ 𝑖 ) ) → if ( 𝑠 = ( 𝐻 ‘ 𝑖 ) , ( 𝑄 ‘ 𝑖 ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) |
502 |
|
eqidd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) ∧ ¬ 𝑠 = ( 𝐻 ‘ 𝑖 ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) = ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) |
503 |
416
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) ∧ ¬ 𝑠 = ( 𝐻 ‘ 𝑖 ) ) ∧ 𝑡 = 𝑠 ) → ( 𝑋 + 𝑡 ) = ( 𝑋 + 𝑠 ) ) |
504 |
|
velsn |
⊢ ( 𝑠 ∈ { ( 𝐻 ‘ 𝑖 ) } ↔ 𝑠 = ( 𝐻 ‘ 𝑖 ) ) |
505 |
504
|
notbii |
⊢ ( ¬ 𝑠 ∈ { ( 𝐻 ‘ 𝑖 ) } ↔ ¬ 𝑠 = ( 𝐻 ‘ 𝑖 ) ) |
506 |
|
elun |
⊢ ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ↔ ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∨ 𝑠 ∈ { ( 𝐻 ‘ 𝑖 ) } ) ) |
507 |
506
|
biimpi |
⊢ ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) → ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∨ 𝑠 ∈ { ( 𝐻 ‘ 𝑖 ) } ) ) |
508 |
507
|
orcomd |
⊢ ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) → ( 𝑠 ∈ { ( 𝐻 ‘ 𝑖 ) } ∨ 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
509 |
508
|
ord |
⊢ ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) → ( ¬ 𝑠 ∈ { ( 𝐻 ‘ 𝑖 ) } → 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
510 |
505 509
|
syl5bir |
⊢ ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) → ( ¬ 𝑠 = ( 𝐻 ‘ 𝑖 ) → 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
511 |
510
|
imp |
⊢ ( ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ∧ ¬ 𝑠 = ( 𝐻 ‘ 𝑖 ) ) → 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
512 |
511
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) ∧ ¬ 𝑠 = ( 𝐻 ‘ 𝑖 ) ) → 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
513 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) → 𝑋 ∈ ℝ ) |
514 |
|
elsni |
⊢ ( 𝑠 ∈ { ( 𝐻 ‘ 𝑖 ) } → 𝑠 = ( 𝐻 ‘ 𝑖 ) ) |
515 |
514
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ { ( 𝐻 ‘ 𝑖 ) } ) → 𝑠 = ( 𝐻 ‘ 𝑖 ) ) |
516 |
224
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ { ( 𝐻 ‘ 𝑖 ) } ) → ( 𝐻 ‘ 𝑖 ) ∈ ℝ ) |
517 |
515 516
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ { ( 𝐻 ‘ 𝑖 ) } ) → 𝑠 ∈ ℝ ) |
518 |
429 517
|
jaodan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∨ 𝑠 ∈ { ( 𝐻 ‘ 𝑖 ) } ) ) → 𝑠 ∈ ℝ ) |
519 |
506 518
|
sylan2b |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) → 𝑠 ∈ ℝ ) |
520 |
513 519
|
readdcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) → ( 𝑋 + 𝑠 ) ∈ ℝ ) |
521 |
520
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) ∧ ¬ 𝑠 = ( 𝐻 ‘ 𝑖 ) ) → ( 𝑋 + 𝑠 ) ∈ ℝ ) |
522 |
502 503 512 521
|
fvmptd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) ∧ ¬ 𝑠 = ( 𝐻 ‘ 𝑖 ) ) → ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) = ( 𝑋 + 𝑠 ) ) |
523 |
501 522
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) ∧ ¬ 𝑠 = ( 𝐻 ‘ 𝑖 ) ) → if ( 𝑠 = ( 𝐻 ‘ 𝑖 ) , ( 𝑄 ‘ 𝑖 ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( 𝑋 + 𝑠 ) ) |
524 |
499 523
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) → if ( 𝑠 = ( 𝐻 ‘ 𝑖 ) , ( 𝑄 ‘ 𝑖 ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) = ( 𝑋 + 𝑠 ) ) |
525 |
491 524
|
mpteq12dva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ↦ if ( 𝑠 = ( 𝐻 ‘ 𝑖 ) , ( 𝑄 ‘ 𝑖 ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) ) = ( 𝑠 ∈ ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑠 ) ) ) |
526 |
491
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) = ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) ) |
527 |
526
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) = ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ) |
528 |
527
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ 𝑖 ) ) = ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝐻 ‘ 𝑖 ) [,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ 𝑖 ) ) ) |
529 |
487 525 528
|
3eltr4d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ↦ if ( 𝑠 = ( 𝐻 ‘ 𝑖 ) , ( 𝑄 ‘ 𝑖 ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ 𝑖 ) ) ) |
530 |
|
eqid |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) = ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) |
531 |
|
eqid |
⊢ ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ↦ if ( 𝑠 = ( 𝐻 ‘ 𝑖 ) , ( 𝑄 ‘ 𝑖 ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) ) = ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ↦ if ( 𝑠 = ( 𝐻 ‘ 𝑖 ) , ( 𝑄 ‘ 𝑖 ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) ) |
532 |
224
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐻 ‘ 𝑖 ) ∈ ℂ ) |
533 |
530 380 531 448 299 532
|
ellimc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) ∈ ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) limℂ ( 𝐻 ‘ 𝑖 ) ) ↔ ( 𝑠 ∈ ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ↦ if ( 𝑠 = ( 𝐻 ‘ 𝑖 ) , ( 𝑄 ‘ 𝑖 ) , ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ‘ 𝑠 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝐻 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐻 ‘ 𝑖 ) ) ) ) |
534 |
529 533
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) limℂ ( 𝐻 ‘ 𝑖 ) ) ) |
535 |
464 534 8
|
limccog |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∘ ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) limℂ ( 𝐻 ‘ 𝑖 ) ) ) |
536 |
453
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∘ ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑋 + 𝑡 ) ) ) limℂ ( 𝐻 ‘ 𝑖 ) ) = ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) limℂ ( 𝐻 ‘ 𝑖 ) ) ) |
537 |
535 536
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) limℂ ( 𝐻 ‘ 𝑖 ) ) ) |
538 |
224 225 304 455 537
|
iblcncfioo |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) (,) ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) ∈ 𝐿1 ) |
539 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐹 : ( - π [,] π ) ⟶ ℂ ) |
540 |
54
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → - π ∈ ℝ* ) |
541 |
56
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → π ∈ ℝ* ) |
542 |
27
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( - π [,] π ) ) |
543 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑖 ∈ ( 0 ..^ 𝑀 ) ) |
544 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) |
545 |
163 173
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) = ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) |
546 |
545
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) = ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) |
547 |
544 546
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) |
548 |
547 117
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑋 + 𝑡 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
549 |
540 541 542 543 548
|
fourierdlem1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑋 + 𝑡 ) ∈ ( - π [,] π ) ) |
550 |
539 549
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ∈ ℂ ) |
551 |
224 225 538 550
|
ibliooicc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) ) ∈ 𝐿1 ) |
552 |
20 26 159 174 223 551
|
itgspltprt |
⊢ ( 𝜑 → ∫ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 = Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 ) |
553 |
545
|
itgeq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∫ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 = ∫ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 ) |
554 |
553
|
sumeq2dv |
⊢ ( 𝜑 → Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝐻 ‘ 𝑖 ) [,] ( 𝐻 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 = Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 ) |
555 |
552 554
|
eqtrd |
⊢ ( 𝜑 → ∫ ( ( 𝐻 ‘ 0 ) [,] ( 𝐻 ‘ 𝑀 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 = Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 ) |
556 |
126 155 555
|
3eqtrd |
⊢ ( 𝜑 → ∫ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) d 𝑠 = Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) [,] ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑡 ) ) d 𝑡 ) |
557 |
122 556
|
eqtr4d |
⊢ ( 𝜑 → Σ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∫ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ( 𝐹 ‘ 𝑡 ) d 𝑡 = ∫ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) d 𝑠 ) |
558 |
19 78 557
|
3eqtrd |
⊢ ( 𝜑 → ∫ ( - π [,] π ) ( 𝐹 ‘ 𝑡 ) d 𝑡 = ∫ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) d 𝑠 ) |