Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem101.d |
⊢ 𝐷 = ( 𝑛 ∈ ℕ ↦ ( 𝑠 ∈ ℝ ↦ if ( ( 𝑠 mod ( 2 · π ) ) = 0 , ( ( ( 2 · 𝑛 ) + 1 ) / ( 2 · π ) ) , ( ( sin ‘ ( ( 𝑛 + ( 1 / 2 ) ) · 𝑠 ) ) / ( ( 2 · π ) · ( sin ‘ ( 𝑠 / 2 ) ) ) ) ) ) ) |
2 |
|
fourierdlem101.p |
⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = - π ∧ ( 𝑝 ‘ 𝑚 ) = π ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) |
3 |
|
fourierdlem101.g |
⊢ 𝐺 = ( 𝑡 ∈ ( - π [,] π ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) |
4 |
|
fourierdlem101.q |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ) |
5 |
|
fourierdlem101.6 |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
6 |
|
fourierdlem101.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
7 |
|
fourierdlem101.x |
⊢ ( 𝜑 → 𝑋 ∈ ℝ ) |
8 |
|
fourierdlem101.f |
⊢ ( 𝜑 → 𝐹 : ( - π [,] π ) ⟶ ℂ ) |
9 |
|
fourierdlem101.fcn |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
10 |
|
fourierdlem101.r |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
11 |
|
fourierdlem101.l |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
12 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( - π [,] π ) ) → 𝑡 ∈ ( - π [,] π ) ) |
13 |
8
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( - π [,] π ) ) → ( 𝐹 ‘ 𝑡 ) ∈ ℂ ) |
14 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( - π [,] π ) ) → 𝑁 ∈ ℕ ) |
15 |
|
pire |
⊢ π ∈ ℝ |
16 |
15
|
renegcli |
⊢ - π ∈ ℝ |
17 |
|
eliccre |
⊢ ( ( - π ∈ ℝ ∧ π ∈ ℝ ∧ 𝑡 ∈ ( - π [,] π ) ) → 𝑡 ∈ ℝ ) |
18 |
16 15 17
|
mp3an12 |
⊢ ( 𝑡 ∈ ( - π [,] π ) → 𝑡 ∈ ℝ ) |
19 |
18
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( - π [,] π ) ) → 𝑡 ∈ ℝ ) |
20 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( - π [,] π ) ) → 𝑋 ∈ ℝ ) |
21 |
19 20
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( - π [,] π ) ) → ( 𝑡 − 𝑋 ) ∈ ℝ ) |
22 |
1
|
dirkerre |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑡 − 𝑋 ) ∈ ℝ ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ∈ ℝ ) |
23 |
14 21 22
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( - π [,] π ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ∈ ℝ ) |
24 |
23
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( - π [,] π ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ∈ ℂ ) |
25 |
13 24
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( - π [,] π ) ) → ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ℂ ) |
26 |
3
|
fvmpt2 |
⊢ ( ( 𝑡 ∈ ( - π [,] π ) ∧ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ℂ ) → ( 𝐺 ‘ 𝑡 ) = ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) |
27 |
12 25 26
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( - π [,] π ) ) → ( 𝐺 ‘ 𝑡 ) = ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) |
28 |
27
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( - π [,] π ) ) → ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) = ( 𝐺 ‘ 𝑡 ) ) |
29 |
28
|
itgeq2dv |
⊢ ( 𝜑 → ∫ ( - π [,] π ) ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) d 𝑡 = ∫ ( - π [,] π ) ( 𝐺 ‘ 𝑡 ) d 𝑡 ) |
30 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ 𝑖 ) ) |
31 |
30
|
oveq1d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝑄 ‘ 𝑗 ) − 𝑋 ) = ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) |
32 |
31
|
cbvmptv |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑗 ) − 𝑋 ) ) = ( 𝑖 ∈ ( 0 ... 𝑀 ) ↦ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) |
33 |
25 3
|
fmptd |
⊢ ( 𝜑 → 𝐺 : ( - π [,] π ) ⟶ ℂ ) |
34 |
3
|
reseq1i |
⊢ ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑡 ∈ ( - π [,] π ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
35 |
|
ioossicc |
⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
36 |
16
|
a1i |
⊢ ( 𝜑 → - π ∈ ℝ ) |
37 |
36
|
rexrd |
⊢ ( 𝜑 → - π ∈ ℝ* ) |
38 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → - π ∈ ℝ* ) |
39 |
15
|
a1i |
⊢ ( 𝜑 → π ∈ ℝ ) |
40 |
39
|
rexrd |
⊢ ( 𝜑 → π ∈ ℝ* ) |
41 |
40
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → π ∈ ℝ* ) |
42 |
2 5 4
|
fourierdlem15 |
⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( - π [,] π ) ) |
43 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( - π [,] π ) ) |
44 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ..^ 𝑀 ) ) |
45 |
38 41 43 44
|
fourierdlem8 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) [,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( - π [,] π ) ) |
46 |
35 45
|
sstrid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( - π [,] π ) ) |
47 |
46
|
resmptd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑡 ∈ ( - π [,] π ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) ) |
48 |
34 47
|
eqtrid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) ) |
49 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐹 : ( - π [,] π ) ⟶ ℂ ) |
50 |
49 46
|
feqresmpt |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑡 ) ) ) |
51 |
50 9
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝐹 ‘ 𝑡 ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
52 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) = ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ) |
53 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑠 = ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ‘ 𝑟 ) ) → 𝑠 = ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ‘ 𝑟 ) ) |
54 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) = ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ) |
55 |
|
oveq1 |
⊢ ( 𝑡 = 𝑟 → ( 𝑡 − 𝑋 ) = ( 𝑟 − 𝑋 ) ) |
56 |
55
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑡 = 𝑟 ) → ( 𝑡 − 𝑋 ) = ( 𝑟 − 𝑋 ) ) |
57 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
58 |
|
elioore |
⊢ ( 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑟 ∈ ℝ ) |
59 |
58
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑟 ∈ ℝ ) |
60 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑋 ∈ ℝ ) |
61 |
59 60
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑟 − 𝑋 ) ∈ ℝ ) |
62 |
61
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑟 − 𝑋 ) ∈ ℝ ) |
63 |
54 56 57 62
|
fvmptd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ‘ 𝑟 ) = ( 𝑟 − 𝑋 ) ) |
64 |
63
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑠 = ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ‘ 𝑟 ) ) → ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ‘ 𝑟 ) = ( 𝑟 − 𝑋 ) ) |
65 |
53 64
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑠 = ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ‘ 𝑟 ) ) → 𝑠 = ( 𝑟 − 𝑋 ) ) |
66 |
65
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑠 = ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ‘ 𝑟 ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑟 − 𝑋 ) ) ) |
67 |
|
elioore |
⊢ ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑡 ∈ ℝ ) |
68 |
67
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑡 ∈ ℝ ) |
69 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑋 ∈ ℝ ) |
70 |
68 69
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑡 − 𝑋 ) ∈ ℝ ) |
71 |
70
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑡 − 𝑋 ) ∈ ℝ ) |
72 |
|
eqid |
⊢ ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) = ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) |
73 |
71 72
|
fmptd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℝ ) |
74 |
73
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ‘ 𝑟 ) ∈ ℝ ) |
75 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑁 ∈ ℕ ) |
76 |
1
|
dirkerre |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑟 − 𝑋 ) ∈ ℝ ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑟 − 𝑋 ) ) ∈ ℝ ) |
77 |
75 62 76
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑟 − 𝑋 ) ) ∈ ℝ ) |
78 |
52 66 74 77
|
fvmptd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ‘ ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ‘ 𝑟 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑟 − 𝑋 ) ) ) |
79 |
78
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑟 − 𝑋 ) ) = ( ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ‘ ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ‘ 𝑟 ) ) ) |
80 |
79
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑟 − 𝑋 ) ) ) = ( 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ‘ ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ‘ 𝑟 ) ) ) ) |
81 |
55
|
fveq2d |
⊢ ( 𝑡 = 𝑟 → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑟 − 𝑋 ) ) ) |
82 |
81
|
cbvmptv |
⊢ ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) = ( 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑟 − 𝑋 ) ) ) |
83 |
82
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) = ( 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑟 − 𝑋 ) ) ) ) |
84 |
1
|
dirkerre |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑠 ∈ ℝ ) → ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ∈ ℝ ) |
85 |
6 84
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ℝ ) → ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ∈ ℝ ) |
86 |
|
eqid |
⊢ ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) = ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) |
87 |
85 86
|
fmptd |
⊢ ( 𝜑 → ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) : ℝ ⟶ ℝ ) |
88 |
87
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) : ℝ ⟶ ℝ ) |
89 |
|
fcompt |
⊢ ( ( ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) : ℝ ⟶ ℝ ∧ ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℝ ) → ( ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ∘ ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ) = ( 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ‘ ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ‘ 𝑟 ) ) ) ) |
90 |
88 73 89
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ∘ ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ) = ( 𝑟 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ‘ ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ‘ 𝑟 ) ) ) ) |
91 |
80 83 90
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) = ( ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ∘ ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ) ) |
92 |
|
eqid |
⊢ ( 𝑡 ∈ ℂ ↦ ( 𝑡 − 𝑋 ) ) = ( 𝑡 ∈ ℂ ↦ ( 𝑡 − 𝑋 ) ) |
93 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ℂ ) → 𝑡 ∈ ℂ ) |
94 |
7
|
recnd |
⊢ ( 𝜑 → 𝑋 ∈ ℂ ) |
95 |
94
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ℂ ) → 𝑋 ∈ ℂ ) |
96 |
93 95
|
negsubd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ℂ ) → ( 𝑡 + - 𝑋 ) = ( 𝑡 − 𝑋 ) ) |
97 |
96
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ℂ ) → ( 𝑡 − 𝑋 ) = ( 𝑡 + - 𝑋 ) ) |
98 |
97
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑡 ∈ ℂ ↦ ( 𝑡 − 𝑋 ) ) = ( 𝑡 ∈ ℂ ↦ ( 𝑡 + - 𝑋 ) ) ) |
99 |
94
|
negcld |
⊢ ( 𝜑 → - 𝑋 ∈ ℂ ) |
100 |
|
eqid |
⊢ ( 𝑡 ∈ ℂ ↦ ( 𝑡 + - 𝑋 ) ) = ( 𝑡 ∈ ℂ ↦ ( 𝑡 + - 𝑋 ) ) |
101 |
100
|
addccncf |
⊢ ( - 𝑋 ∈ ℂ → ( 𝑡 ∈ ℂ ↦ ( 𝑡 + - 𝑋 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
102 |
99 101
|
syl |
⊢ ( 𝜑 → ( 𝑡 ∈ ℂ ↦ ( 𝑡 + - 𝑋 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
103 |
98 102
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑡 ∈ ℂ ↦ ( 𝑡 − 𝑋 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
104 |
103
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ℂ ↦ ( 𝑡 − 𝑋 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
105 |
|
ioossre |
⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ |
106 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
107 |
105 106
|
sstri |
⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ |
108 |
107
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ ) |
109 |
106
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ℝ ⊆ ℂ ) |
110 |
92 104 108 109 71
|
cncfmptssg |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℝ ) ) |
111 |
85
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ℝ ) → ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ∈ ℂ ) |
112 |
111 86
|
fmptd |
⊢ ( 𝜑 → ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) : ℝ ⟶ ℂ ) |
113 |
|
ssid |
⊢ ℂ ⊆ ℂ |
114 |
1
|
dirkerf |
⊢ ( 𝑁 ∈ ℕ → ( 𝐷 ‘ 𝑁 ) : ℝ ⟶ ℝ ) |
115 |
6 114
|
syl |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝑁 ) : ℝ ⟶ ℝ ) |
116 |
115
|
feqmptd |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝑁 ) = ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ) |
117 |
1
|
dirkercncf |
⊢ ( 𝑁 ∈ ℕ → ( 𝐷 ‘ 𝑁 ) ∈ ( ℝ –cn→ ℝ ) ) |
118 |
6 117
|
syl |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝑁 ) ∈ ( ℝ –cn→ ℝ ) ) |
119 |
116 118
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ∈ ( ℝ –cn→ ℝ ) ) |
120 |
|
cncffvrn |
⊢ ( ( ℂ ⊆ ℂ ∧ ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ∈ ( ℝ –cn→ ℝ ) ) → ( ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ∈ ( ℝ –cn→ ℂ ) ↔ ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) : ℝ ⟶ ℂ ) ) |
121 |
113 119 120
|
sylancr |
⊢ ( 𝜑 → ( ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ∈ ( ℝ –cn→ ℂ ) ↔ ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) : ℝ ⟶ ℂ ) ) |
122 |
112 121
|
mpbird |
⊢ ( 𝜑 → ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ∈ ( ℝ –cn→ ℂ ) ) |
123 |
122
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ∈ ( ℝ –cn→ ℂ ) ) |
124 |
110 123
|
cncfco |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑠 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ∘ ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
125 |
91 124
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
126 |
51 125
|
mulcncf |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
127 |
48 126
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
128 |
|
cncff |
⊢ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ) |
129 |
9 128
|
syl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ) |
130 |
115
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐷 ‘ 𝑁 ) : ℝ ⟶ ℝ ) |
131 |
|
elioore |
⊢ ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑠 ∈ ℝ ) |
132 |
131
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑠 ∈ ℝ ) |
133 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑋 ∈ ℝ ) |
134 |
132 133
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑠 − 𝑋 ) ∈ ℝ ) |
135 |
130 134
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ∈ ℝ ) |
136 |
135
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ∈ ℂ ) |
137 |
|
eqid |
⊢ ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) = ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) |
138 |
136 137
|
fmptd |
⊢ ( 𝜑 → ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ) |
139 |
138
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ) |
140 |
|
eqid |
⊢ ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) = ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) |
141 |
|
oveq1 |
⊢ ( 𝑡 = ( 𝑄 ‘ 𝑖 ) → ( 𝑡 − 𝑋 ) = ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) |
142 |
141
|
fveq2d |
⊢ ( 𝑡 = ( 𝑄 ‘ 𝑖 ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) ) |
143 |
142
|
eqcomd |
⊢ ( 𝑡 = ( 𝑄 ‘ 𝑖 ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
144 |
143
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) ∧ 𝑡 = ( 𝑄 ‘ 𝑖 ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
145 |
|
eqidd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ 𝑖 ) ) → ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) = ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ) |
146 |
|
oveq1 |
⊢ ( 𝑠 = 𝑡 → ( 𝑠 − 𝑋 ) = ( 𝑡 − 𝑋 ) ) |
147 |
146
|
fveq2d |
⊢ ( 𝑠 = 𝑡 → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
148 |
147
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ 𝑖 ) ) ∧ 𝑠 = 𝑡 ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
149 |
|
velsn |
⊢ ( 𝑡 ∈ { ( 𝑄 ‘ 𝑖 ) } ↔ 𝑡 = ( 𝑄 ‘ 𝑖 ) ) |
150 |
149
|
notbii |
⊢ ( ¬ 𝑡 ∈ { ( 𝑄 ‘ 𝑖 ) } ↔ ¬ 𝑡 = ( 𝑄 ‘ 𝑖 ) ) |
151 |
|
elunnel2 |
⊢ ( ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ∧ ¬ 𝑡 ∈ { ( 𝑄 ‘ 𝑖 ) } ) → 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
152 |
150 151
|
sylan2br |
⊢ ( ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ∧ ¬ 𝑡 = ( 𝑄 ‘ 𝑖 ) ) → 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
153 |
152
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ 𝑖 ) ) → 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
154 |
115
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → ( 𝐷 ‘ 𝑁 ) : ℝ ⟶ ℝ ) |
155 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 = ( 𝑄 ‘ 𝑖 ) ) → 𝑡 = ( 𝑄 ‘ 𝑖 ) ) |
156 |
18
|
ssriv |
⊢ ( - π [,] π ) ⊆ ℝ |
157 |
|
fzossfz |
⊢ ( 0 ..^ 𝑀 ) ⊆ ( 0 ... 𝑀 ) |
158 |
157 44
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
159 |
43 158
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ( - π [,] π ) ) |
160 |
156 159
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
161 |
160
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 = ( 𝑄 ‘ 𝑖 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
162 |
155 161
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 = ( 𝑄 ‘ 𝑖 ) ) → 𝑡 ∈ ℝ ) |
163 |
162
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) ∧ 𝑡 = ( 𝑄 ‘ 𝑖 ) ) → 𝑡 ∈ ℝ ) |
164 |
153 67
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ 𝑖 ) ) → 𝑡 ∈ ℝ ) |
165 |
163 164
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → 𝑡 ∈ ℝ ) |
166 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → 𝑋 ∈ ℝ ) |
167 |
165 166
|
resubcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → ( 𝑡 − 𝑋 ) ∈ ℝ ) |
168 |
154 167
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ∈ ℝ ) |
169 |
168
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ 𝑖 ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ∈ ℝ ) |
170 |
145 148 153 169
|
fvmptd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ 𝑖 ) ) → ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
171 |
144 170
|
ifeqda |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → if ( 𝑡 = ( 𝑄 ‘ 𝑖 ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
172 |
171
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ if ( 𝑡 = ( 𝑄 ‘ 𝑖 ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) = ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) |
173 |
115
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐷 ‘ 𝑁 ) : ℝ ⟶ ℝ ) |
174 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) |
175 |
|
elun |
⊢ ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↔ ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∨ 𝑠 ∈ { ( 𝑄 ‘ 𝑖 ) } ) ) |
176 |
174 175
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∨ 𝑠 ∈ { ( 𝑄 ‘ 𝑖 ) } ) ) |
177 |
176
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∨ 𝑠 ∈ { ( 𝑄 ‘ 𝑖 ) } ) ) |
178 |
|
elsni |
⊢ ( 𝑠 ∈ { ( 𝑄 ‘ 𝑖 ) } → 𝑠 = ( 𝑄 ‘ 𝑖 ) ) |
179 |
178
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ { ( 𝑄 ‘ 𝑖 ) } ) → 𝑠 = ( 𝑄 ‘ 𝑖 ) ) |
180 |
160
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ { ( 𝑄 ‘ 𝑖 ) } ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
181 |
179 180
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ { ( 𝑄 ‘ 𝑖 ) } ) → 𝑠 ∈ ℝ ) |
182 |
181
|
ex |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ { ( 𝑄 ‘ 𝑖 ) } → 𝑠 ∈ ℝ ) ) |
183 |
182
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → ( 𝑠 ∈ { ( 𝑄 ‘ 𝑖 ) } → 𝑠 ∈ ℝ ) ) |
184 |
|
pm3.44 |
⊢ ( ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑠 ∈ ℝ ) ∧ ( 𝑠 ∈ { ( 𝑄 ‘ 𝑖 ) } → 𝑠 ∈ ℝ ) ) → ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∨ 𝑠 ∈ { ( 𝑄 ‘ 𝑖 ) } ) → 𝑠 ∈ ℝ ) ) |
185 |
131 183 184
|
sylancr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∨ 𝑠 ∈ { ( 𝑄 ‘ 𝑖 ) } ) → 𝑠 ∈ ℝ ) ) |
186 |
177 185
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → 𝑠 ∈ ℝ ) |
187 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → 𝑋 ∈ ℝ ) |
188 |
186 187
|
resubcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → ( 𝑠 − 𝑋 ) ∈ ℝ ) |
189 |
|
eqid |
⊢ ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) = ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) |
190 |
188 189
|
fmptd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) : ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ⟶ ℝ ) |
191 |
|
fcompt |
⊢ ( ( ( 𝐷 ‘ 𝑁 ) : ℝ ⟶ ℝ ∧ ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) : ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ⟶ ℝ ) → ( ( 𝐷 ‘ 𝑁 ) ∘ ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) ) = ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) ‘ 𝑡 ) ) ) ) |
192 |
173 190 191
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐷 ‘ 𝑁 ) ∘ ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) ) = ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) ‘ 𝑡 ) ) ) ) |
193 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) = ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) ) |
194 |
146
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) ∧ 𝑠 = 𝑡 ) → ( 𝑠 − 𝑋 ) = ( 𝑡 − 𝑋 ) ) |
195 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) |
196 |
193 194 195 167
|
fvmptd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → ( ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) ‘ 𝑡 ) = ( 𝑡 − 𝑋 ) ) |
197 |
196
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) ‘ 𝑡 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
198 |
197
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) ‘ 𝑡 ) ) ) = ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) |
199 |
192 198
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) = ( ( 𝐷 ‘ 𝑁 ) ∘ ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) ) ) |
200 |
|
eqid |
⊢ ( 𝑠 ∈ ℂ ↦ ( 𝑠 − 𝑋 ) ) = ( 𝑠 ∈ ℂ ↦ ( 𝑠 − 𝑋 ) ) |
201 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ℂ ) → 𝑠 ∈ ℂ ) |
202 |
94
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ℂ ) → 𝑋 ∈ ℂ ) |
203 |
201 202
|
negsubd |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ℂ ) → ( 𝑠 + - 𝑋 ) = ( 𝑠 − 𝑋 ) ) |
204 |
203
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ℂ ) → ( 𝑠 − 𝑋 ) = ( 𝑠 + - 𝑋 ) ) |
205 |
204
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑠 ∈ ℂ ↦ ( 𝑠 − 𝑋 ) ) = ( 𝑠 ∈ ℂ ↦ ( 𝑠 + - 𝑋 ) ) ) |
206 |
|
eqid |
⊢ ( 𝑠 ∈ ℂ ↦ ( 𝑠 + - 𝑋 ) ) = ( 𝑠 ∈ ℂ ↦ ( 𝑠 + - 𝑋 ) ) |
207 |
206
|
addccncf |
⊢ ( - 𝑋 ∈ ℂ → ( 𝑠 ∈ ℂ ↦ ( 𝑠 + - 𝑋 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
208 |
99 207
|
syl |
⊢ ( 𝜑 → ( 𝑠 ∈ ℂ ↦ ( 𝑠 + - 𝑋 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
209 |
205 208
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑠 ∈ ℂ ↦ ( 𝑠 − 𝑋 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
210 |
209
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ℂ ↦ ( 𝑠 − 𝑋 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
211 |
160
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℂ ) |
212 |
211
|
snssd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → { ( 𝑄 ‘ 𝑖 ) } ⊆ ℂ ) |
213 |
108 212
|
unssd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ⊆ ℂ ) |
214 |
200 210 213 109 188
|
cncfmptssg |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) –cn→ ℝ ) ) |
215 |
116 122
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝑁 ) ∈ ( ℝ –cn→ ℂ ) ) |
216 |
215
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐷 ‘ 𝑁 ) ∈ ( ℝ –cn→ ℂ ) ) |
217 |
214 216
|
cncfco |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐷 ‘ 𝑁 ) ∘ ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) ) ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) –cn→ ℂ ) ) |
218 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
219 |
|
eqid |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) = ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) |
220 |
218
|
cnfldtop |
⊢ ( TopOpen ‘ ℂfld ) ∈ Top |
221 |
|
unicntop |
⊢ ℂ = ∪ ( TopOpen ‘ ℂfld ) |
222 |
221
|
restid |
⊢ ( ( TopOpen ‘ ℂfld ) ∈ Top → ( ( TopOpen ‘ ℂfld ) ↾t ℂ ) = ( TopOpen ‘ ℂfld ) ) |
223 |
220 222
|
ax-mp |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ℂ ) = ( TopOpen ‘ ℂfld ) |
224 |
223
|
eqcomi |
⊢ ( TopOpen ‘ ℂfld ) = ( ( TopOpen ‘ ℂfld ) ↾t ℂ ) |
225 |
218 219 224
|
cncfcn |
⊢ ( ( ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) –cn→ ℂ ) = ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
226 |
213 113 225
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) –cn→ ℂ ) = ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
227 |
217 226
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐷 ‘ 𝑁 ) ∘ ( 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( 𝑠 − 𝑋 ) ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
228 |
199 227
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
229 |
218
|
cnfldtopon |
⊢ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) |
230 |
|
resttopon |
⊢ ( ( ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ∧ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ⊆ ℂ ) → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) ∈ ( TopOn ‘ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) ) |
231 |
229 213 230
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) ∈ ( TopOn ‘ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) ) |
232 |
|
cncnp |
⊢ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) ∈ ( TopOn ‘ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) ∧ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ) → ( ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) Cn ( TopOpen ‘ ℂfld ) ) ↔ ( ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) : ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ⟶ ℂ ∧ ∀ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑠 ) ) ) ) |
233 |
231 229 232
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) Cn ( TopOpen ‘ ℂfld ) ) ↔ ( ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) : ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ⟶ ℂ ∧ ∀ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑠 ) ) ) ) |
234 |
228 233
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) : ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ⟶ ℂ ∧ ∀ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑠 ) ) ) |
235 |
234
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∀ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑠 ) ) |
236 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 𝑖 ) ) |
237 |
|
elsng |
⊢ ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ → ( ( 𝑄 ‘ 𝑖 ) ∈ { ( 𝑄 ‘ 𝑖 ) } ↔ ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 𝑖 ) ) ) |
238 |
160 237
|
syl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) ∈ { ( 𝑄 ‘ 𝑖 ) } ↔ ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 𝑖 ) ) ) |
239 |
236 238
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ { ( 𝑄 ‘ 𝑖 ) } ) |
240 |
239
|
olcd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∨ ( 𝑄 ‘ 𝑖 ) ∈ { ( 𝑄 ‘ 𝑖 ) } ) ) |
241 |
|
elun |
⊢ ( ( 𝑄 ‘ 𝑖 ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↔ ( ( 𝑄 ‘ 𝑖 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∨ ( 𝑄 ‘ 𝑖 ) ∈ { ( 𝑄 ‘ 𝑖 ) } ) ) |
242 |
240 241
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) |
243 |
|
fveq2 |
⊢ ( 𝑠 = ( 𝑄 ‘ 𝑖 ) → ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑠 ) = ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝑄 ‘ 𝑖 ) ) ) |
244 |
243
|
eleq2d |
⊢ ( 𝑠 = ( 𝑄 ‘ 𝑖 ) → ( ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑠 ) ↔ ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝑄 ‘ 𝑖 ) ) ) ) |
245 |
244
|
rspccva |
⊢ ( ( ∀ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑠 ) ∧ ( 𝑄 ‘ 𝑖 ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) → ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝑄 ‘ 𝑖 ) ) ) |
246 |
235 242 245
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝑄 ‘ 𝑖 ) ) ) |
247 |
172 246
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ if ( 𝑡 = ( 𝑄 ‘ 𝑖 ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝑄 ‘ 𝑖 ) ) ) |
248 |
|
eqid |
⊢ ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ if ( 𝑡 = ( 𝑄 ‘ 𝑖 ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) = ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ if ( 𝑡 = ( 𝑄 ‘ 𝑖 ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) |
249 |
219 218 248 139 108 211
|
ellimc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) ∈ ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ↔ ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ↦ if ( 𝑡 = ( 𝑄 ‘ 𝑖 ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ 𝑖 ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝑄 ‘ 𝑖 ) ) ) ) |
250 |
247 249
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) ∈ ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
251 |
129 139 140 10 250
|
mullimcf |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑅 · ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) ) ∈ ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
252 |
|
fvres |
⊢ ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) = ( 𝐹 ‘ 𝑡 ) ) |
253 |
252
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) = ( 𝐹 ‘ 𝑡 ) ) |
254 |
253
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) = ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) |
255 |
254
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) = ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) ) |
256 |
255
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) = ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
257 |
251 256
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑅 · ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) ) ∈ ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
258 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) = ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ) |
259 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑠 = 𝑡 ) → 𝑠 = 𝑡 ) |
260 |
259
|
oveq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑠 = 𝑡 ) → ( 𝑠 − 𝑋 ) = ( 𝑡 − 𝑋 ) ) |
261 |
260
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑠 = 𝑡 ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
262 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
263 |
115
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐷 ‘ 𝑁 ) : ℝ ⟶ ℝ ) |
264 |
263 71
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ∈ ℝ ) |
265 |
258 261 262 264
|
fvmptd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
266 |
265
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) = ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) |
267 |
266
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) = ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) ) |
268 |
267
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) = ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
269 |
257 268
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑅 · ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) ) ∈ ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
270 |
48
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) = ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
271 |
270
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) = ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
272 |
269 271
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑅 · ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ 𝑖 ) − 𝑋 ) ) ) ∈ ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
273 |
|
iftrue |
⊢ ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → if ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) |
274 |
|
oveq1 |
⊢ ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → ( 𝑡 − 𝑋 ) = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) |
275 |
274
|
eqcomd |
⊢ ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) = ( 𝑡 − 𝑋 ) ) |
276 |
275
|
fveq2d |
⊢ ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
277 |
273 276
|
eqtrd |
⊢ ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → if ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
278 |
277
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
279 |
|
iffalse |
⊢ ( ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → if ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) = ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) |
280 |
279
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) = ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) |
281 |
|
eqidd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) = ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ) |
282 |
147
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑠 = 𝑡 ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
283 |
|
elun |
⊢ ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↔ ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∨ 𝑡 ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) |
284 |
283
|
biimpi |
⊢ ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∨ 𝑡 ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) |
285 |
284
|
orcomd |
⊢ ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) → ( 𝑡 ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ∨ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
286 |
285
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑡 ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ∨ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
287 |
|
velsn |
⊢ ( 𝑡 ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ↔ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
288 |
287
|
notbii |
⊢ ( ¬ 𝑡 ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ↔ ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
289 |
288
|
biimpri |
⊢ ( ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → ¬ 𝑡 ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) |
290 |
289
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ¬ 𝑡 ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) |
291 |
|
pm2.53 |
⊢ ( ( 𝑡 ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ∨ 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ¬ 𝑡 ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } → 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
292 |
286 290 291
|
sylc |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
293 |
173
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐷 ‘ 𝑁 ) : ℝ ⟶ ℝ ) |
294 |
292 67
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑡 ∈ ℝ ) |
295 |
7
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝑋 ∈ ℝ ) |
296 |
294 295
|
resubcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑡 − 𝑋 ) ∈ ℝ ) |
297 |
293 296
|
ffvelrnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ∈ ℝ ) |
298 |
281 282 292 297
|
fvmptd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
299 |
280 298
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ¬ 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → if ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
300 |
278 299
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) → if ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
301 |
300
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ if ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) = ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) |
302 |
|
eqid |
⊢ ( 𝑡 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) = ( 𝑡 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) |
303 |
106
|
a1i |
⊢ ( 𝜑 → ℝ ⊆ ℂ ) |
304 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ℝ ) → 𝑡 ∈ ℝ ) |
305 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ℝ ) → 𝑋 ∈ ℝ ) |
306 |
304 305
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ℝ ) → ( 𝑡 − 𝑋 ) ∈ ℝ ) |
307 |
92 103 303 303 306
|
cncfmptssg |
⊢ ( 𝜑 → ( 𝑡 ∈ ℝ ↦ ( 𝑡 − 𝑋 ) ) ∈ ( ℝ –cn→ ℝ ) ) |
308 |
307 215
|
cncfcompt |
⊢ ( 𝜑 → ( 𝑡 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ℝ –cn→ ℂ ) ) |
309 |
308
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ℝ ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ℝ –cn→ ℂ ) ) |
310 |
105
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ ) |
311 |
|
fzofzp1 |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
312 |
311
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
313 |
43 312
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( - π [,] π ) ) |
314 |
156 313
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
315 |
314
|
snssd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ⊆ ℝ ) |
316 |
310 315
|
unssd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ⊆ ℝ ) |
317 |
113
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ℂ ⊆ ℂ ) |
318 |
173
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) → ( 𝐷 ‘ 𝑁 ) : ℝ ⟶ ℝ ) |
319 |
316
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) → 𝑡 ∈ ℝ ) |
320 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) → 𝑋 ∈ ℝ ) |
321 |
319 320
|
resubcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) → ( 𝑡 − 𝑋 ) ∈ ℝ ) |
322 |
318 321
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ∈ ℝ ) |
323 |
322
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ∈ ℂ ) |
324 |
302 309 316 317 323
|
cncfmptssg |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) –cn→ ℂ ) ) |
325 |
156 106
|
sstri |
⊢ ( - π [,] π ) ⊆ ℂ |
326 |
325 313
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℂ ) |
327 |
326
|
snssd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ⊆ ℂ ) |
328 |
108 327
|
unssd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ⊆ ℂ ) |
329 |
|
eqid |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) = ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) |
330 |
218 329 224
|
cncfcn |
⊢ ( ( ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) –cn→ ℂ ) = ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
331 |
328 113 330
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) –cn→ ℂ ) = ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
332 |
324 331
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
333 |
|
resttopon |
⊢ ( ( ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ∧ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ⊆ ℂ ) → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∈ ( TopOn ‘ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ) |
334 |
229 328 333
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∈ ( TopOn ‘ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ) |
335 |
|
cncnp |
⊢ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∈ ( TopOn ‘ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ) → ( ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) Cn ( TopOpen ‘ ℂfld ) ) ↔ ( ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) : ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ⟶ ℂ ∧ ∀ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑠 ) ) ) ) |
336 |
334 229 335
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) Cn ( TopOpen ‘ ℂfld ) ) ↔ ( ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) : ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ⟶ ℂ ∧ ∀ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑠 ) ) ) ) |
337 |
332 336
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) : ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ⟶ ℂ ∧ ∀ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑠 ) ) ) |
338 |
337
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∀ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑠 ) ) |
339 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
340 |
|
elsng |
⊢ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ↔ ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
341 |
314 340
|
syl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ↔ ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
342 |
339 341
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) |
343 |
342
|
olcd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∨ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) |
344 |
|
elun |
⊢ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↔ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∨ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) |
345 |
343 344
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) |
346 |
|
fveq2 |
⊢ ( 𝑠 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑠 ) = ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
347 |
346
|
eleq2d |
⊢ ( 𝑠 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) → ( ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑠 ) ↔ ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
348 |
347
|
rspccva |
⊢ ( ( ∀ 𝑠 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑠 ) ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) → ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
349 |
338 345 348
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
350 |
301 349
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ if ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
351 |
|
eqid |
⊢ ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ if ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) = ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ if ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) |
352 |
329 218 351 139 108 326
|
ellimc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ∈ ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝑡 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ↦ if ( 𝑡 = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) , ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) CnP ( TopOpen ‘ ℂfld ) ) ‘ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
353 |
350 352
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ∈ ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
354 |
129 139 140 11 353
|
mullimcf |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐿 · ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) ∈ ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
355 |
267 255 48
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) = ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
356 |
355
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑡 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑡 ) · ( ( 𝑠 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↦ ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑠 − 𝑋 ) ) ) ‘ 𝑡 ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
357 |
354 356
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐿 · ( ( 𝐷 ‘ 𝑁 ) ‘ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − 𝑋 ) ) ) ∈ ( ( 𝐺 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
358 |
2 32 5 4 7 33 127 272 357
|
fourierdlem93 |
⊢ ( 𝜑 → ∫ ( - π [,] π ) ( 𝐺 ‘ 𝑡 ) d 𝑡 = ∫ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ( 𝐺 ‘ ( 𝑋 + 𝑠 ) ) d 𝑠 ) |
359 |
3
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → 𝐺 = ( 𝑡 ∈ ( - π [,] π ) ↦ ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) ) |
360 |
|
fveq2 |
⊢ ( 𝑡 = ( 𝑋 + 𝑠 ) → ( 𝐹 ‘ 𝑡 ) = ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) ) |
361 |
360
|
oveq1d |
⊢ ( 𝑡 = ( 𝑋 + 𝑠 ) → ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) = ( ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) |
362 |
361
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) ∧ 𝑡 = ( 𝑋 + 𝑠 ) ) → ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) = ( ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) ) |
363 |
|
oveq1 |
⊢ ( 𝑡 = ( 𝑋 + 𝑠 ) → ( 𝑡 − 𝑋 ) = ( ( 𝑋 + 𝑠 ) − 𝑋 ) ) |
364 |
94
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → 𝑋 ∈ ℂ ) |
365 |
36 7
|
resubcld |
⊢ ( 𝜑 → ( - π − 𝑋 ) ∈ ℝ ) |
366 |
365
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( - π − 𝑋 ) ∈ ℝ ) |
367 |
39 7
|
resubcld |
⊢ ( 𝜑 → ( π − 𝑋 ) ∈ ℝ ) |
368 |
367
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( π − 𝑋 ) ∈ ℝ ) |
369 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) |
370 |
|
eliccre |
⊢ ( ( ( - π − 𝑋 ) ∈ ℝ ∧ ( π − 𝑋 ) ∈ ℝ ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → 𝑠 ∈ ℝ ) |
371 |
366 368 369 370
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → 𝑠 ∈ ℝ ) |
372 |
371
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → 𝑠 ∈ ℂ ) |
373 |
364 372
|
pncan2d |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( ( 𝑋 + 𝑠 ) − 𝑋 ) = 𝑠 ) |
374 |
363 373
|
sylan9eqr |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) ∧ 𝑡 = ( 𝑋 + 𝑠 ) ) → ( 𝑡 − 𝑋 ) = 𝑠 ) |
375 |
374
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) ∧ 𝑡 = ( 𝑋 + 𝑠 ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) = ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) |
376 |
375
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) ∧ 𝑡 = ( 𝑋 + 𝑠 ) ) → ( ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) = ( ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) · ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ) |
377 |
362 376
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) ∧ 𝑡 = ( 𝑋 + 𝑠 ) ) → ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) = ( ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) · ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ) |
378 |
16
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → - π ∈ ℝ ) |
379 |
15
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → π ∈ ℝ ) |
380 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → 𝑋 ∈ ℝ ) |
381 |
380 371
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( 𝑋 + 𝑠 ) ∈ ℝ ) |
382 |
36
|
recnd |
⊢ ( 𝜑 → - π ∈ ℂ ) |
383 |
94 382
|
pncan3d |
⊢ ( 𝜑 → ( 𝑋 + ( - π − 𝑋 ) ) = - π ) |
384 |
383
|
eqcomd |
⊢ ( 𝜑 → - π = ( 𝑋 + ( - π − 𝑋 ) ) ) |
385 |
384
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → - π = ( 𝑋 + ( - π − 𝑋 ) ) ) |
386 |
|
elicc2 |
⊢ ( ( ( - π − 𝑋 ) ∈ ℝ ∧ ( π − 𝑋 ) ∈ ℝ ) → ( 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ↔ ( 𝑠 ∈ ℝ ∧ ( - π − 𝑋 ) ≤ 𝑠 ∧ 𝑠 ≤ ( π − 𝑋 ) ) ) ) |
387 |
366 368 386
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ↔ ( 𝑠 ∈ ℝ ∧ ( - π − 𝑋 ) ≤ 𝑠 ∧ 𝑠 ≤ ( π − 𝑋 ) ) ) ) |
388 |
369 387
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( 𝑠 ∈ ℝ ∧ ( - π − 𝑋 ) ≤ 𝑠 ∧ 𝑠 ≤ ( π − 𝑋 ) ) ) |
389 |
388
|
simp2d |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( - π − 𝑋 ) ≤ 𝑠 ) |
390 |
366 371 380 389
|
leadd2dd |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( 𝑋 + ( - π − 𝑋 ) ) ≤ ( 𝑋 + 𝑠 ) ) |
391 |
385 390
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → - π ≤ ( 𝑋 + 𝑠 ) ) |
392 |
388
|
simp3d |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → 𝑠 ≤ ( π − 𝑋 ) ) |
393 |
371 368 380 392
|
leadd2dd |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( 𝑋 + 𝑠 ) ≤ ( 𝑋 + ( π − 𝑋 ) ) ) |
394 |
|
picn |
⊢ π ∈ ℂ |
395 |
394
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → π ∈ ℂ ) |
396 |
364 395
|
pncan3d |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( 𝑋 + ( π − 𝑋 ) ) = π ) |
397 |
393 396
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( 𝑋 + 𝑠 ) ≤ π ) |
398 |
378 379 381 391 397
|
eliccd |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( 𝑋 + 𝑠 ) ∈ ( - π [,] π ) ) |
399 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → 𝐹 : ( - π [,] π ) ⟶ ℂ ) |
400 |
399 398
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) ∈ ℂ ) |
401 |
371 111
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ∈ ℂ ) |
402 |
400 401
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) · ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ∈ ℂ ) |
403 |
359 377 398 402
|
fvmptd |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ) → ( 𝐺 ‘ ( 𝑋 + 𝑠 ) ) = ( ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) · ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) ) |
404 |
403
|
itgeq2dv |
⊢ ( 𝜑 → ∫ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ( 𝐺 ‘ ( 𝑋 + 𝑠 ) ) d 𝑠 = ∫ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ( ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) · ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) d 𝑠 ) |
405 |
29 358 404
|
3eqtrd |
⊢ ( 𝜑 → ∫ ( - π [,] π ) ( ( 𝐹 ‘ 𝑡 ) · ( ( 𝐷 ‘ 𝑁 ) ‘ ( 𝑡 − 𝑋 ) ) ) d 𝑡 = ∫ ( ( - π − 𝑋 ) [,] ( π − 𝑋 ) ) ( ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) · ( ( 𝐷 ‘ 𝑁 ) ‘ 𝑠 ) ) d 𝑠 ) |