Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem105.p |
⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) |
2 |
|
fourierdlem105.t |
⊢ 𝑇 = ( 𝐵 − 𝐴 ) |
3 |
|
fourierdlem105.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
4 |
|
fourierdlem105.q |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ) |
5 |
|
fourierdlem105.f |
⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℂ ) |
6 |
|
fourierdlem105.6 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
7 |
|
fourierdlem105.fcn |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
8 |
|
fourierdlem105.r |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
9 |
|
fourierdlem105.l |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
10 |
|
fourierdlem105.c |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
11 |
|
fourierdlem105.d |
⊢ ( 𝜑 → 𝐷 ∈ ( 𝐶 (,) +∞ ) ) |
12 |
|
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 ) ) ) } ) |
13 |
|
eqid |
⊢ ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) = ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) |
14 |
|
oveq1 |
⊢ ( 𝑤 = 𝑦 → ( 𝑤 + ( 𝑗 · 𝑇 ) ) = ( 𝑦 + ( 𝑗 · 𝑇 ) ) ) |
15 |
14
|
eleq1d |
⊢ ( 𝑤 = 𝑦 → ( ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 ↔ ( 𝑦 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 ) ) |
16 |
15
|
rexbidv |
⊢ ( 𝑤 = 𝑦 → ( ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ 𝑗 ∈ ℤ ( 𝑦 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 ) ) |
17 |
|
oveq1 |
⊢ ( 𝑗 = 𝑘 → ( 𝑗 · 𝑇 ) = ( 𝑘 · 𝑇 ) ) |
18 |
17
|
oveq2d |
⊢ ( 𝑗 = 𝑘 → ( 𝑦 + ( 𝑗 · 𝑇 ) ) = ( 𝑦 + ( 𝑘 · 𝑇 ) ) ) |
19 |
18
|
eleq1d |
⊢ ( 𝑗 = 𝑘 → ( ( 𝑦 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 ↔ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ) ) |
20 |
19
|
cbvrexvw |
⊢ ( ∃ 𝑗 ∈ ℤ ( 𝑦 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ) |
21 |
16 20
|
bitrdi |
⊢ ( 𝑤 = 𝑦 → ( ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ) ) |
22 |
21
|
cbvrabv |
⊢ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } = { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } |
23 |
22
|
uneq2i |
⊢ ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) = ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) |
24 |
|
isoeq1 |
⊢ ( 𝑔 = 𝑓 → ( 𝑔 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ↔ 𝑓 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) ) |
25 |
24
|
cbviotavw |
⊢ ( ℩ 𝑔 𝑔 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) = ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) ) , ( { 𝐶 , 𝐷 } ∪ { 𝑤 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑗 ∈ ℤ ( 𝑤 + ( 𝑗 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) |
26 |
|
id |
⊢ ( 𝑤 = 𝑥 → 𝑤 = 𝑥 ) |
27 |
|
oveq2 |
⊢ ( 𝑤 = 𝑥 → ( 𝐵 − 𝑤 ) = ( 𝐵 − 𝑥 ) ) |
28 |
27
|
oveq1d |
⊢ ( 𝑤 = 𝑥 → ( ( 𝐵 − 𝑤 ) / 𝑇 ) = ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) |
29 |
28
|
fveq2d |
⊢ ( 𝑤 = 𝑥 → ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) = ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) ) |
30 |
29
|
oveq1d |
⊢ ( 𝑤 = 𝑥 → ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) |
31 |
26 30
|
oveq12d |
⊢ ( 𝑤 = 𝑥 → ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) = ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
32 |
31
|
cbvmptv |
⊢ ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
33 |
|
eqeq1 |
⊢ ( 𝑤 = 𝑦 → ( 𝑤 = 𝐵 ↔ 𝑦 = 𝐵 ) ) |
34 |
|
id |
⊢ ( 𝑤 = 𝑦 → 𝑤 = 𝑦 ) |
35 |
33 34
|
ifbieq2d |
⊢ ( 𝑤 = 𝑦 → if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) = if ( 𝑦 = 𝐵 , 𝐴 , 𝑦 ) ) |
36 |
35
|
cbvmptv |
⊢ ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) = ( 𝑦 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑦 = 𝐵 , 𝐴 , 𝑦 ) ) |
37 |
|
fveq2 |
⊢ ( 𝑧 = 𝑥 → ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑧 ) = ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) |
38 |
37
|
fveq2d |
⊢ ( 𝑧 = 𝑥 → ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑧 ) ) = ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) ) |
39 |
38
|
breq2d |
⊢ ( 𝑧 = 𝑥 → ( ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑧 ) ) ↔ ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) ) ) |
40 |
39
|
rabbidv |
⊢ ( 𝑧 = 𝑥 → { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑧 ) ) } = { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } ) |
41 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ 𝑖 ) ) |
42 |
41
|
breq1d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) ↔ ( 𝑄 ‘ 𝑖 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) ) ) |
43 |
42
|
cbvrabv |
⊢ { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } = { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } |
44 |
40 43
|
eqtrdi |
⊢ ( 𝑧 = 𝑥 → { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑧 ) ) } = { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } ) |
45 |
44
|
supeq1d |
⊢ ( 𝑧 = 𝑥 → sup ( { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑧 ) ) } , ℝ , < ) = sup ( { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } , ℝ , < ) ) |
46 |
45
|
cbvmptv |
⊢ ( 𝑧 ∈ ℝ ↦ sup ( { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑧 ) ) } , ℝ , < ) ) = ( 𝑥 ∈ ℝ ↦ sup ( { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( ( 𝑤 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑤 = 𝐵 , 𝐴 , 𝑤 ) ) ‘ ( ( 𝑤 ∈ ℝ ↦ ( 𝑤 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑤 ) / 𝑇 ) ) · 𝑇 ) ) ) ‘ 𝑥 ) ) } , ℝ , < ) ) |
47 |
1 2 3 4 5 6 7 8 9 10 11 12 13 23 25 32 36 46
|
fourierdlem100 |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐶 [,] 𝐷 ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ 𝐿1 ) |