Step |
Hyp |
Ref |
Expression |
1 |
|
fourierlemiblglemlem.p |
⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) |
2 |
|
fourierdlem100.t |
⊢ 𝑇 = ( 𝐵 − 𝐴 ) |
3 |
|
fourierdlem100.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
4 |
|
fourierdlem100.q |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ) |
5 |
|
fourierdlem100.f |
⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℂ ) |
6 |
|
fourierdlem100.per |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
7 |
|
fourierdlem100.fcn |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
8 |
|
fourierdlem100.r |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
9 |
|
fourierdlem100.l |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
10 |
|
fourierdlem100.c |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
11 |
|
fourierdlem100.d |
⊢ ( 𝜑 → 𝐷 ∈ ( 𝐶 (,) +∞ ) ) |
12 |
|
fourierdlem100.o |
⊢ 𝑂 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐶 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐷 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) |
13 |
|
fourierdlem100.n |
⊢ 𝑁 = ( ( ♯ ‘ 𝐻 ) − 1 ) |
14 |
|
fourierdlem100.h |
⊢ 𝐻 = ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) |
15 |
|
fourierdlem100.s |
⊢ 𝑆 = ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , 𝐻 ) ) |
16 |
|
fourierdlem100.e |
⊢ 𝐸 = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
17 |
|
fourierdlem100.j |
⊢ 𝐽 = ( 𝑦 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑦 = 𝐵 , 𝐴 , 𝑦 ) ) |
18 |
|
fourierdlem100.i |
⊢ 𝐼 = ( 𝑥 ∈ ℝ ↦ sup ( { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( 𝐽 ‘ ( 𝐸 ‘ 𝑥 ) ) } , ℝ , < ) ) |
19 |
|
elioore |
⊢ ( 𝐷 ∈ ( 𝐶 (,) +∞ ) → 𝐷 ∈ ℝ ) |
20 |
11 19
|
syl |
⊢ ( 𝜑 → 𝐷 ∈ ℝ ) |
21 |
10 20
|
iccssred |
⊢ ( 𝜑 → ( 𝐶 [,] 𝐷 ) ⊆ ℝ ) |
22 |
5 21
|
feqresmpt |
⊢ ( 𝜑 → ( 𝐹 ↾ ( 𝐶 [,] 𝐷 ) ) = ( 𝑥 ∈ ( 𝐶 [,] 𝐷 ) ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
23 |
|
fveq2 |
⊢ ( 𝑖 = 𝑗 → ( 𝑝 ‘ 𝑖 ) = ( 𝑝 ‘ 𝑗 ) ) |
24 |
|
oveq1 |
⊢ ( 𝑖 = 𝑗 → ( 𝑖 + 1 ) = ( 𝑗 + 1 ) ) |
25 |
24
|
fveq2d |
⊢ ( 𝑖 = 𝑗 → ( 𝑝 ‘ ( 𝑖 + 1 ) ) = ( 𝑝 ‘ ( 𝑗 + 1 ) ) ) |
26 |
23 25
|
breq12d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑝 ‘ 𝑗 ) < ( 𝑝 ‘ ( 𝑗 + 1 ) ) ) ) |
27 |
26
|
cbvralvw |
⊢ ( ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ↔ ∀ 𝑗 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑗 ) < ( 𝑝 ‘ ( 𝑗 + 1 ) ) ) |
28 |
27
|
anbi2i |
⊢ ( ( ( ( 𝑝 ‘ 0 ) = 𝐶 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐷 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ ( ( ( 𝑝 ‘ 0 ) = 𝐶 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐷 ) ∧ ∀ 𝑗 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑗 ) < ( 𝑝 ‘ ( 𝑗 + 1 ) ) ) ) |
29 |
28
|
a1i |
⊢ ( 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) → ( ( ( ( 𝑝 ‘ 0 ) = 𝐶 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐷 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) ↔ ( ( ( 𝑝 ‘ 0 ) = 𝐶 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐷 ) ∧ ∀ 𝑗 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑗 ) < ( 𝑝 ‘ ( 𝑗 + 1 ) ) ) ) ) |
30 |
29
|
rabbiia |
⊢ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐶 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐷 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } = { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐶 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐷 ) ∧ ∀ 𝑗 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑗 ) < ( 𝑝 ‘ ( 𝑗 + 1 ) ) ) } |
31 |
30
|
mpteq2i |
⊢ ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐶 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐷 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐶 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐷 ) ∧ ∀ 𝑗 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑗 ) < ( 𝑝 ‘ ( 𝑗 + 1 ) ) ) } ) |
32 |
12 31
|
eqtri |
⊢ 𝑂 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐶 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐷 ) ∧ ∀ 𝑗 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑗 ) < ( 𝑝 ‘ ( 𝑗 + 1 ) ) ) } ) |
33 |
|
elioo4g |
⊢ ( 𝐷 ∈ ( 𝐶 (,) +∞ ) ↔ ( ( 𝐶 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐷 ∈ ℝ ) ∧ ( 𝐶 < 𝐷 ∧ 𝐷 < +∞ ) ) ) |
34 |
11 33
|
sylib |
⊢ ( 𝜑 → ( ( 𝐶 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐷 ∈ ℝ ) ∧ ( 𝐶 < 𝐷 ∧ 𝐷 < +∞ ) ) ) |
35 |
34
|
simprd |
⊢ ( 𝜑 → ( 𝐶 < 𝐷 ∧ 𝐷 < +∞ ) ) |
36 |
35
|
simpld |
⊢ ( 𝜑 → 𝐶 < 𝐷 ) |
37 |
|
id |
⊢ ( 𝑦 = 𝑥 → 𝑦 = 𝑥 ) |
38 |
2
|
eqcomi |
⊢ ( 𝐵 − 𝐴 ) = 𝑇 |
39 |
38
|
oveq2i |
⊢ ( 𝑘 · ( 𝐵 − 𝐴 ) ) = ( 𝑘 · 𝑇 ) |
40 |
39
|
a1i |
⊢ ( 𝑦 = 𝑥 → ( 𝑘 · ( 𝐵 − 𝐴 ) ) = ( 𝑘 · 𝑇 ) ) |
41 |
37 40
|
oveq12d |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) = ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) |
42 |
41
|
eleq1d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 ↔ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ) ) |
43 |
42
|
rexbidv |
⊢ ( 𝑦 = 𝑥 → ( ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 ↔ ∃ 𝑘 ∈ ℤ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ) ) |
44 |
43
|
cbvrabv |
⊢ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } = { 𝑥 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } |
45 |
44
|
uneq2i |
⊢ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) = ( { 𝐶 , 𝐷 } ∪ { 𝑥 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) |
46 |
39
|
eqcomi |
⊢ ( 𝑘 · 𝑇 ) = ( 𝑘 · ( 𝐵 − 𝐴 ) ) |
47 |
46
|
oveq2i |
⊢ ( 𝑦 + ( 𝑘 · 𝑇 ) ) = ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) |
48 |
47
|
eleq1i |
⊢ ( ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ↔ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 ) |
49 |
48
|
rexbii |
⊢ ( ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 ) |
50 |
49
|
rgenw |
⊢ ∀ 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ( ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 ) |
51 |
|
rabbi |
⊢ ( ∀ 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ( ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 ) ↔ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } = { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) |
52 |
50 51
|
mpbi |
⊢ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } = { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } |
53 |
52
|
uneq2i |
⊢ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) = ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) |
54 |
14 53
|
eqtri |
⊢ 𝐻 = ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) |
55 |
54
|
fveq2i |
⊢ ( ♯ ‘ 𝐻 ) = ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) ) |
56 |
55
|
oveq1i |
⊢ ( ( ♯ ‘ 𝐻 ) − 1 ) = ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) ) − 1 ) |
57 |
13 56
|
eqtri |
⊢ 𝑁 = ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) ) − 1 ) |
58 |
|
isoeq5 |
⊢ ( 𝐻 = ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) → ( 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , 𝐻 ) ↔ 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) ) ) ) |
59 |
54 58
|
ax-mp |
⊢ ( 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , 𝐻 ) ↔ 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) ) ) |
60 |
59
|
iotabii |
⊢ ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , 𝐻 ) ) = ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) ) ) |
61 |
15 60
|
eqtri |
⊢ 𝑆 = ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) ) ) |
62 |
2 1 3 4 10 20 36 12 45 57 61
|
fourierdlem54 |
⊢ ( 𝜑 → ( ( 𝑁 ∈ ℕ ∧ 𝑆 ∈ ( 𝑂 ‘ 𝑁 ) ) ∧ 𝑆 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) ) ) ) |
63 |
62
|
simpld |
⊢ ( 𝜑 → ( 𝑁 ∈ ℕ ∧ 𝑆 ∈ ( 𝑂 ‘ 𝑁 ) ) ) |
64 |
63
|
simpld |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
65 |
63
|
simprd |
⊢ ( 𝜑 → 𝑆 ∈ ( 𝑂 ‘ 𝑁 ) ) |
66 |
5 21
|
fssresd |
⊢ ( 𝜑 → ( 𝐹 ↾ ( 𝐶 [,] 𝐷 ) ) : ( 𝐶 [,] 𝐷 ) ⟶ ℂ ) |
67 |
|
ioossicc |
⊢ ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ⊆ ( ( 𝑆 ‘ 𝑗 ) [,] ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) |
68 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → 𝐶 ∈ ℝ ) |
69 |
68
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → 𝐶 ∈ ℝ* ) |
70 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → 𝐷 ∈ ( 𝐶 (,) +∞ ) ) |
71 |
70 19
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → 𝐷 ∈ ℝ ) |
72 |
71
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → 𝐷 ∈ ℝ* ) |
73 |
12 64 65
|
fourierdlem15 |
⊢ ( 𝜑 → 𝑆 : ( 0 ... 𝑁 ) ⟶ ( 𝐶 [,] 𝐷 ) ) |
74 |
73
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → 𝑆 : ( 0 ... 𝑁 ) ⟶ ( 𝐶 [,] 𝐷 ) ) |
75 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → 𝑗 ∈ ( 0 ..^ 𝑁 ) ) |
76 |
69 72 74 75
|
fourierdlem8 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝑆 ‘ 𝑗 ) [,] ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ⊆ ( 𝐶 [,] 𝐷 ) ) |
77 |
67 76
|
sstrid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ⊆ ( 𝐶 [,] 𝐷 ) ) |
78 |
77
|
resabs1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝐹 ↾ ( 𝐶 [,] 𝐷 ) ) ↾ ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) = ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ) |
79 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → 𝑀 ∈ ℕ ) |
80 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ) |
81 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → 𝐹 : ℝ ⟶ ℂ ) |
82 |
6
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
83 |
7
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
84 |
|
eqid |
⊢ ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) = ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) |
85 |
|
eqid |
⊢ ( 𝐹 ↾ ( ( 𝐽 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ) = ( 𝐹 ↾ ( ( 𝐽 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ) |
86 |
|
eqid |
⊢ ( 𝑦 ∈ ( ( ( 𝐽 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) + ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ) (,) ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) + ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ) ) ↦ ( ( 𝐹 ↾ ( ( 𝐽 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ) ‘ ( 𝑦 − ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ) ) ) = ( 𝑦 ∈ ( ( ( 𝐽 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) + ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ) (,) ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) + ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ) ) ↦ ( ( 𝐹 ↾ ( ( 𝐽 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ) ‘ ( 𝑦 − ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ) ) ) |
87 |
1 2 79 80 81 82 83 68 70 12 14 13 15 16 17 75 84 85 86 18
|
fourierdlem90 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ∈ ( ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) –cn→ ℂ ) ) |
88 |
78 87
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝐹 ↾ ( 𝐶 [,] 𝐷 ) ) ↾ ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ∈ ( ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) –cn→ ℂ ) ) |
89 |
8
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
90 |
|
eqid |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ 𝑅 ) = ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ 𝑅 ) |
91 |
1 2 79 80 81 82 83 89 68 70 12 14 13 15 16 17 75 84 18 90
|
fourierdlem89 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → if ( ( 𝐽 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) = ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) ) , ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ 𝑅 ) ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) ) , ( 𝐹 ‘ ( 𝐽 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) limℂ ( 𝑆 ‘ 𝑗 ) ) ) |
92 |
78
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) = ( ( 𝐹 ↾ ( 𝐶 [,] 𝐷 ) ) ↾ ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ) |
93 |
92
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) limℂ ( 𝑆 ‘ 𝑗 ) ) = ( ( ( 𝐹 ↾ ( 𝐶 [,] 𝐷 ) ) ↾ ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) limℂ ( 𝑆 ‘ 𝑗 ) ) ) |
94 |
91 93
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → if ( ( 𝐽 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) = ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) ) , ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ 𝑅 ) ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) ) , ( 𝐹 ‘ ( 𝐽 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) ) ) ∈ ( ( ( 𝐹 ↾ ( 𝐶 [,] 𝐷 ) ) ↾ ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) limℂ ( 𝑆 ‘ 𝑗 ) ) ) |
95 |
9
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
96 |
|
eqid |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ 𝐿 ) = ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ 𝐿 ) |
97 |
1 2 79 80 81 82 83 95 68 70 12 14 13 15 16 17 75 84 18 96
|
fourierdlem91 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → if ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) + 1 ) ) , ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ 𝐿 ) ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) ) , ( 𝐹 ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) limℂ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) |
98 |
92
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) limℂ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) = ( ( ( 𝐹 ↾ ( 𝐶 [,] 𝐷 ) ) ↾ ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) limℂ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) |
99 |
97 98
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → if ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) + 1 ) ) , ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ 𝐿 ) ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) ) , ( 𝐹 ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ) ∈ ( ( ( 𝐹 ↾ ( 𝐶 [,] 𝐷 ) ) ↾ ( ( 𝑆 ‘ 𝑗 ) (,) ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) limℂ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) |
100 |
32 64 65 66 88 94 99
|
fourierdlem69 |
⊢ ( 𝜑 → ( 𝐹 ↾ ( 𝐶 [,] 𝐷 ) ) ∈ 𝐿1 ) |
101 |
22 100
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐶 [,] 𝐷 ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ 𝐿1 ) |