Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem98.f |
⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℝ ) |
2 |
|
fourierdlem98.p |
⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) |
3 |
|
fourierdlem98.t |
⊢ 𝑇 = ( 𝐵 − 𝐴 ) |
4 |
|
fourierdlem98.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
5 |
|
fourierdlem98.q |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ) |
6 |
|
fourierdlem98.fper |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
7 |
|
fourierdlem98.qcn |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
8 |
|
fourierdlem98.c |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
9 |
|
fourierdlem98.d |
⊢ ( 𝜑 → 𝐷 ∈ ( 𝐶 (,) +∞ ) ) |
10 |
|
fourierdlem98.j |
⊢ ( 𝜑 → 𝐽 ∈ ( 0 ..^ ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) ) |
11 |
|
fourierdlem98.v |
⊢ 𝑉 = ( ℩ 𝑔 𝑔 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ ℎ ∈ ℤ ( 𝑦 + ( ℎ · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) |
12 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
13 |
12
|
a1i |
⊢ ( 𝜑 → ℝ ⊆ ℂ ) |
14 |
1 13
|
fssd |
⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℂ ) |
15 |
|
eqid |
⊢ ( 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 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) |
16 |
|
oveq1 |
⊢ ( 𝑧 = 𝑦 → ( 𝑧 + ( 𝑙 · 𝑇 ) ) = ( 𝑦 + ( 𝑙 · 𝑇 ) ) ) |
17 |
16
|
eleq1d |
⊢ ( 𝑧 = 𝑦 → ( ( 𝑧 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 ↔ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 ) ) |
18 |
17
|
rexbidv |
⊢ ( 𝑧 = 𝑦 → ( ∃ 𝑙 ∈ ℤ ( 𝑧 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 ) ) |
19 |
18
|
cbvrabv |
⊢ { 𝑧 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑧 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } = { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } |
20 |
19
|
uneq2i |
⊢ ( { 𝐶 , 𝐷 } ∪ { 𝑧 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑧 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) = ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) |
21 |
20
|
eqcomi |
⊢ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) = ( { 𝐶 , 𝐷 } ∪ { 𝑧 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑧 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) |
22 |
|
oveq1 |
⊢ ( 𝑘 = 𝑙 → ( 𝑘 · 𝑇 ) = ( 𝑙 · 𝑇 ) ) |
23 |
22
|
oveq2d |
⊢ ( 𝑘 = 𝑙 → ( 𝑦 + ( 𝑘 · 𝑇 ) ) = ( 𝑦 + ( 𝑙 · 𝑇 ) ) ) |
24 |
23
|
eleq1d |
⊢ ( 𝑘 = 𝑙 → ( ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ↔ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 ) ) |
25 |
24
|
cbvrexvw |
⊢ ( ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 ) |
26 |
25
|
a1i |
⊢ ( 𝑦 ∈ ( 𝐶 [,] 𝐷 ) → ( ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 ) ) |
27 |
26
|
rabbiia |
⊢ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } = { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } |
28 |
27
|
uneq2i |
⊢ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) = ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) |
29 |
28
|
fveq2i |
⊢ ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) = ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) ) |
30 |
29
|
oveq1i |
⊢ ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) = ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) |
31 |
|
oveq1 |
⊢ ( 𝑙 = ℎ → ( 𝑙 · 𝑇 ) = ( ℎ · 𝑇 ) ) |
32 |
31
|
oveq2d |
⊢ ( 𝑙 = ℎ → ( 𝑦 + ( 𝑙 · 𝑇 ) ) = ( 𝑦 + ( ℎ · 𝑇 ) ) ) |
33 |
32
|
eleq1d |
⊢ ( 𝑙 = ℎ → ( ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 ↔ ( 𝑦 + ( ℎ · 𝑇 ) ) ∈ ran 𝑄 ) ) |
34 |
33
|
cbvrexvw |
⊢ ( ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ ℎ ∈ ℤ ( 𝑦 + ( ℎ · 𝑇 ) ) ∈ ran 𝑄 ) |
35 |
34
|
a1i |
⊢ ( 𝑦 ∈ ( 𝐶 [,] 𝐷 ) → ( ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ ℎ ∈ ℤ ( 𝑦 + ( ℎ · 𝑇 ) ) ∈ ran 𝑄 ) ) |
36 |
35
|
rabbiia |
⊢ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } = { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ ℎ ∈ ℤ ( 𝑦 + ( ℎ · 𝑇 ) ) ∈ ran 𝑄 } |
37 |
36
|
uneq2i |
⊢ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) = ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ ℎ ∈ ℤ ( 𝑦 + ( ℎ · 𝑇 ) ) ∈ ran 𝑄 } ) |
38 |
|
isoeq5 |
⊢ ( ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) = ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ ℎ ∈ ℤ ( 𝑦 + ( ℎ · 𝑇 ) ) ∈ ran 𝑄 } ) → ( 𝑔 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ↔ 𝑔 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ ℎ ∈ ℤ ( 𝑦 + ( ℎ · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) ) |
39 |
37 38
|
ax-mp |
⊢ ( 𝑔 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ↔ 𝑔 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ ℎ ∈ ℤ ( 𝑦 + ( ℎ · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) |
40 |
39
|
iotabii |
⊢ ( ℩ 𝑔 𝑔 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) = ( ℩ 𝑔 𝑔 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ ℎ ∈ ℤ ( 𝑦 + ( ℎ · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) |
41 |
|
isoeq1 |
⊢ ( 𝑓 = 𝑔 → ( 𝑓 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ↔ 𝑔 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) ) |
42 |
41
|
cbviotavw |
⊢ ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) = ( ℩ 𝑔 𝑔 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) |
43 |
40 42 11
|
3eqtr4ri |
⊢ 𝑉 = ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑙 ∈ ℤ ( 𝑦 + ( 𝑙 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) |
44 |
|
id |
⊢ ( 𝑣 = 𝑥 → 𝑣 = 𝑥 ) |
45 |
|
oveq2 |
⊢ ( 𝑣 = 𝑥 → ( 𝐵 − 𝑣 ) = ( 𝐵 − 𝑥 ) ) |
46 |
45
|
oveq1d |
⊢ ( 𝑣 = 𝑥 → ( ( 𝐵 − 𝑣 ) / 𝑇 ) = ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) |
47 |
46
|
fveq2d |
⊢ ( 𝑣 = 𝑥 → ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) = ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) ) |
48 |
47
|
oveq1d |
⊢ ( 𝑣 = 𝑥 → ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) |
49 |
44 48
|
oveq12d |
⊢ ( 𝑣 = 𝑥 → ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) = ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
50 |
49
|
cbvmptv |
⊢ ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
51 |
|
eqeq1 |
⊢ ( 𝑢 = 𝑧 → ( 𝑢 = 𝐵 ↔ 𝑧 = 𝐵 ) ) |
52 |
|
id |
⊢ ( 𝑢 = 𝑧 → 𝑢 = 𝑧 ) |
53 |
51 52
|
ifbieq2d |
⊢ ( 𝑢 = 𝑧 → if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) = if ( 𝑧 = 𝐵 , 𝐴 , 𝑧 ) ) |
54 |
53
|
cbvmptv |
⊢ ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) = ( 𝑧 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑧 = 𝐵 , 𝐴 , 𝑧 ) ) |
55 |
|
eqid |
⊢ ( ( 𝑉 ‘ ( 𝐽 + 1 ) ) − ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) ) = ( ( 𝑉 ‘ ( 𝐽 + 1 ) ) − ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) ) |
56 |
|
eqid |
⊢ ( 𝐹 ↾ ( ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ 𝐽 ) ) ) (,) ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) ) ) = ( 𝐹 ↾ ( ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ 𝐽 ) ) ) (,) ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) ) ) |
57 |
|
eqid |
⊢ ( 𝑧 ∈ ( ( ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ 𝐽 ) ) ) + ( ( 𝑉 ‘ ( 𝐽 + 1 ) ) − ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) ) ) (,) ( ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) + ( ( 𝑉 ‘ ( 𝐽 + 1 ) ) − ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) ) ) ) ↦ ( ( 𝐹 ↾ ( ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ 𝐽 ) ) ) (,) ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) ) ) ‘ ( 𝑧 − ( ( 𝑉 ‘ ( 𝐽 + 1 ) ) − ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) ) ) ) ) = ( 𝑧 ∈ ( ( ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ 𝐽 ) ) ) + ( ( 𝑉 ‘ ( 𝐽 + 1 ) ) − ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) ) ) (,) ( ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) + ( ( 𝑉 ‘ ( 𝐽 + 1 ) ) − ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) ) ) ) ↦ ( ( 𝐹 ↾ ( ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ 𝐽 ) ) ) (,) ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) ) ) ‘ ( 𝑧 − ( ( 𝑉 ‘ ( 𝐽 + 1 ) ) − ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) ) ) ) ) |
58 |
|
fveq2 |
⊢ ( 𝑖 = 𝑡 → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 𝑡 ) ) |
59 |
58
|
breq1d |
⊢ ( 𝑖 = 𝑡 → ( ( 𝑄 ‘ 𝑖 ) ≤ ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) ↔ ( 𝑄 ‘ 𝑡 ) ≤ ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) ) ) |
60 |
59
|
cbvrabv |
⊢ { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } = { 𝑡 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑡 ) ≤ ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } |
61 |
|
fveq2 |
⊢ ( 𝑤 = 𝑥 → ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑤 ) = ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) |
62 |
61
|
fveq2d |
⊢ ( 𝑤 = 𝑥 → ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑤 ) ) = ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) ) |
63 |
62
|
eqcomd |
⊢ ( 𝑤 = 𝑥 → ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) = ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑤 ) ) ) |
64 |
63
|
breq2d |
⊢ ( 𝑤 = 𝑥 → ( ( 𝑄 ‘ 𝑡 ) ≤ ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) ↔ ( 𝑄 ‘ 𝑡 ) ≤ ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑤 ) ) ) ) |
65 |
64
|
rabbidv |
⊢ ( 𝑤 = 𝑥 → { 𝑡 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑡 ) ≤ ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } = { 𝑡 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑡 ) ≤ ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑤 ) ) } ) |
66 |
60 65
|
eqtr2id |
⊢ ( 𝑤 = 𝑥 → { 𝑡 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑡 ) ≤ ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑤 ) ) } = { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } ) |
67 |
66
|
supeq1d |
⊢ ( 𝑤 = 𝑥 → sup ( { 𝑡 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑡 ) ≤ ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑤 ) ) } , ℝ , < ) = sup ( { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } , ℝ , < ) ) |
68 |
67
|
cbvmptv |
⊢ ( 𝑤 ∈ ℝ ↦ sup ( { 𝑡 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑡 ) ≤ ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑤 ) ) } , ℝ , < ) ) = ( 𝑥 ∈ ℝ ↦ sup ( { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( ( 𝑢 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑢 = 𝐵 , 𝐴 , 𝑢 ) ) ‘ ( ( 𝑣 ∈ ℝ ↦ ( 𝑣 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑣 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } , ℝ , < ) ) |
69 |
2 3 4 5 14 6 7 8 9 15 21 30 43 50 54 10 55 56 57 68
|
fourierdlem90 |
⊢ ( 𝜑 → ( 𝐹 ↾ ( ( 𝑉 ‘ 𝐽 ) (,) ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) ) ∈ ( ( ( 𝑉 ‘ 𝐽 ) (,) ( 𝑉 ‘ ( 𝐽 + 1 ) ) ) –cn→ ℂ ) ) |