Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem91.p |
⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) |
2 |
|
fourierdlem91.t |
⊢ 𝑇 = ( 𝐵 − 𝐴 ) |
3 |
|
fourierdlem91.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
4 |
|
fourierdlem91.q |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ) |
5 |
|
fourierdlem91.f |
⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ℂ ) |
6 |
|
fourierdlem91.6 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
7 |
|
fourierdlem91.fcn |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
8 |
|
fourierdlem91.l |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
9 |
|
fourierdlem91.c |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
10 |
|
fourierdlem91.d |
⊢ ( 𝜑 → 𝐷 ∈ ( 𝐶 (,) +∞ ) ) |
11 |
|
fourierdlem91.o |
⊢ 𝑂 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐶 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐷 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) |
12 |
|
fourierdlem91.h |
⊢ 𝐻 = ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) |
13 |
|
fourierdlem91.n |
⊢ 𝑁 = ( ( ♯ ‘ 𝐻 ) − 1 ) |
14 |
|
fourierdlem91.s |
⊢ 𝑆 = ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , 𝐻 ) ) |
15 |
|
fourierdlem91.e |
⊢ 𝐸 = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
16 |
|
fourierdlem91.J |
⊢ 𝑍 = ( 𝑦 ∈ ( 𝐴 (,] 𝐵 ) ↦ if ( 𝑦 = 𝐵 , 𝐴 , 𝑦 ) ) |
17 |
|
fourierdlem91.17 |
⊢ ( 𝜑 → 𝐽 ∈ ( 0 ..^ 𝑁 ) ) |
18 |
|
fourierdlem91.u |
⊢ 𝑈 = ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
19 |
|
fourierdlem91.i |
⊢ 𝐼 = ( 𝑥 ∈ ℝ ↦ sup ( { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑍 ‘ ( 𝐸 ‘ 𝑥 ) ) } , ℝ , < ) ) |
20 |
|
fourierdlem91.w |
⊢ 𝑊 = ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ 𝐿 ) |
21 |
1
|
fourierdlem2 |
⊢ ( 𝑀 ∈ ℕ → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
22 |
3 21
|
syl |
⊢ ( 𝜑 → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
23 |
4 22
|
mpbid |
⊢ ( 𝜑 → ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
24 |
23
|
simpld |
⊢ ( 𝜑 → 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ) |
25 |
|
elmapi |
⊢ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
26 |
24 25
|
syl |
⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
27 |
|
fzossfz |
⊢ ( 0 ..^ 𝑀 ) ⊆ ( 0 ... 𝑀 ) |
28 |
1 3 4 2 15 16 19
|
fourierdlem37 |
⊢ ( 𝜑 → ( 𝐼 : ℝ ⟶ ( 0 ..^ 𝑀 ) ∧ ( 𝑥 ∈ ℝ → sup ( { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑍 ‘ ( 𝐸 ‘ 𝑥 ) ) } , ℝ , < ) ∈ { 𝑖 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑍 ‘ ( 𝐸 ‘ 𝑥 ) ) } ) ) ) |
29 |
28
|
simpld |
⊢ ( 𝜑 → 𝐼 : ℝ ⟶ ( 0 ..^ 𝑀 ) ) |
30 |
|
elioore |
⊢ ( 𝐷 ∈ ( 𝐶 (,) +∞ ) → 𝐷 ∈ ℝ ) |
31 |
10 30
|
syl |
⊢ ( 𝜑 → 𝐷 ∈ ℝ ) |
32 |
|
elioo4g |
⊢ ( 𝐷 ∈ ( 𝐶 (,) +∞ ) ↔ ( ( 𝐶 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐷 ∈ ℝ ) ∧ ( 𝐶 < 𝐷 ∧ 𝐷 < +∞ ) ) ) |
33 |
10 32
|
sylib |
⊢ ( 𝜑 → ( ( 𝐶 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐷 ∈ ℝ ) ∧ ( 𝐶 < 𝐷 ∧ 𝐷 < +∞ ) ) ) |
34 |
33
|
simprd |
⊢ ( 𝜑 → ( 𝐶 < 𝐷 ∧ 𝐷 < +∞ ) ) |
35 |
34
|
simpld |
⊢ ( 𝜑 → 𝐶 < 𝐷 ) |
36 |
|
oveq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 + ( 𝑘 · 𝑇 ) ) = ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) |
37 |
36
|
eleq1d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ↔ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ) ) |
38 |
37
|
rexbidv |
⊢ ( 𝑦 = 𝑥 → ( ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ 𝑘 ∈ ℤ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ) ) |
39 |
38
|
cbvrabv |
⊢ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } = { 𝑥 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } |
40 |
39
|
uneq2i |
⊢ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) = ( { 𝐶 , 𝐷 } ∪ { 𝑥 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) |
41 |
12
|
fveq2i |
⊢ ( ♯ ‘ 𝐻 ) = ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) |
42 |
41
|
oveq1i |
⊢ ( ( ♯ ‘ 𝐻 ) − 1 ) = ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) |
43 |
13 42
|
eqtri |
⊢ 𝑁 = ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) |
44 |
|
isoeq5 |
⊢ ( 𝐻 = ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) → ( 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , 𝐻 ) ↔ 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) ) |
45 |
12 44
|
ax-mp |
⊢ ( 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , 𝐻 ) ↔ 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) |
46 |
45
|
iotabii |
⊢ ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , 𝐻 ) ) = ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) |
47 |
14 46
|
eqtri |
⊢ 𝑆 = ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) |
48 |
2 1 3 4 9 31 35 11 40 43 47
|
fourierdlem54 |
⊢ ( 𝜑 → ( ( 𝑁 ∈ ℕ ∧ 𝑆 ∈ ( 𝑂 ‘ 𝑁 ) ) ∧ 𝑆 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) ) |
49 |
48
|
simpld |
⊢ ( 𝜑 → ( 𝑁 ∈ ℕ ∧ 𝑆 ∈ ( 𝑂 ‘ 𝑁 ) ) ) |
50 |
49
|
simprd |
⊢ ( 𝜑 → 𝑆 ∈ ( 𝑂 ‘ 𝑁 ) ) |
51 |
49
|
simpld |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
52 |
11
|
fourierdlem2 |
⊢ ( 𝑁 ∈ ℕ → ( 𝑆 ∈ ( 𝑂 ‘ 𝑁 ) ↔ ( 𝑆 ∈ ( ℝ ↑m ( 0 ... 𝑁 ) ) ∧ ( ( ( 𝑆 ‘ 0 ) = 𝐶 ∧ ( 𝑆 ‘ 𝑁 ) = 𝐷 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑁 ) ( 𝑆 ‘ 𝑖 ) < ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
53 |
51 52
|
syl |
⊢ ( 𝜑 → ( 𝑆 ∈ ( 𝑂 ‘ 𝑁 ) ↔ ( 𝑆 ∈ ( ℝ ↑m ( 0 ... 𝑁 ) ) ∧ ( ( ( 𝑆 ‘ 0 ) = 𝐶 ∧ ( 𝑆 ‘ 𝑁 ) = 𝐷 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑁 ) ( 𝑆 ‘ 𝑖 ) < ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
54 |
50 53
|
mpbid |
⊢ ( 𝜑 → ( 𝑆 ∈ ( ℝ ↑m ( 0 ... 𝑁 ) ) ∧ ( ( ( 𝑆 ‘ 0 ) = 𝐶 ∧ ( 𝑆 ‘ 𝑁 ) = 𝐷 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑁 ) ( 𝑆 ‘ 𝑖 ) < ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) ) ) |
55 |
54
|
simpld |
⊢ ( 𝜑 → 𝑆 ∈ ( ℝ ↑m ( 0 ... 𝑁 ) ) ) |
56 |
|
elmapi |
⊢ ( 𝑆 ∈ ( ℝ ↑m ( 0 ... 𝑁 ) ) → 𝑆 : ( 0 ... 𝑁 ) ⟶ ℝ ) |
57 |
55 56
|
syl |
⊢ ( 𝜑 → 𝑆 : ( 0 ... 𝑁 ) ⟶ ℝ ) |
58 |
|
elfzofz |
⊢ ( 𝐽 ∈ ( 0 ..^ 𝑁 ) → 𝐽 ∈ ( 0 ... 𝑁 ) ) |
59 |
17 58
|
syl |
⊢ ( 𝜑 → 𝐽 ∈ ( 0 ... 𝑁 ) ) |
60 |
57 59
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐽 ) ∈ ℝ ) |
61 |
29 60
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) |
62 |
27 61
|
sseldi |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ... 𝑀 ) ) |
63 |
26 62
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ∈ ℝ ) |
64 |
63
|
rexrd |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ∈ ℝ* ) |
65 |
64
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) → ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ∈ ℝ* ) |
66 |
|
fzofzp1 |
⊢ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) → ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ∈ ( 0 ... 𝑀 ) ) |
67 |
61 66
|
syl |
⊢ ( 𝜑 → ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ∈ ( 0 ... 𝑀 ) ) |
68 |
26 67
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ∈ ℝ ) |
69 |
68
|
rexrd |
⊢ ( 𝜑 → ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ∈ ℝ* ) |
70 |
69
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) → ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ∈ ℝ* ) |
71 |
1 3 4
|
fourierdlem11 |
⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) ) |
72 |
71
|
simp1d |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
73 |
72
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
74 |
71
|
simp2d |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
75 |
|
iocssre |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) → ( 𝐴 (,] 𝐵 ) ⊆ ℝ ) |
76 |
73 74 75
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 (,] 𝐵 ) ⊆ ℝ ) |
77 |
71
|
simp3d |
⊢ ( 𝜑 → 𝐴 < 𝐵 ) |
78 |
72 74 77 2 15
|
fourierdlem4 |
⊢ ( 𝜑 → 𝐸 : ℝ ⟶ ( 𝐴 (,] 𝐵 ) ) |
79 |
|
fzofzp1 |
⊢ ( 𝐽 ∈ ( 0 ..^ 𝑁 ) → ( 𝐽 + 1 ) ∈ ( 0 ... 𝑁 ) ) |
80 |
17 79
|
syl |
⊢ ( 𝜑 → ( 𝐽 + 1 ) ∈ ( 0 ... 𝑁 ) ) |
81 |
57 80
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐽 + 1 ) ) ∈ ℝ ) |
82 |
78 81
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∈ ( 𝐴 (,] 𝐵 ) ) |
83 |
76 82
|
sseldd |
⊢ ( 𝜑 → ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∈ ℝ ) |
84 |
83
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) → ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∈ ℝ ) |
85 |
72 74
|
iccssred |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
86 |
72 74 77 16
|
fourierdlem17 |
⊢ ( 𝜑 → 𝑍 : ( 𝐴 (,] 𝐵 ) ⟶ ( 𝐴 [,] 𝐵 ) ) |
87 |
78 60
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 𝐴 (,] 𝐵 ) ) |
88 |
86 87
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ∈ ( 𝐴 [,] 𝐵 ) ) |
89 |
85 88
|
sseldd |
⊢ ( 𝜑 → ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ∈ ℝ ) |
90 |
54
|
simprrd |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 0 ..^ 𝑁 ) ( 𝑆 ‘ 𝑖 ) < ( 𝑆 ‘ ( 𝑖 + 1 ) ) ) |
91 |
|
fveq2 |
⊢ ( 𝑖 = 𝐽 → ( 𝑆 ‘ 𝑖 ) = ( 𝑆 ‘ 𝐽 ) ) |
92 |
|
oveq1 |
⊢ ( 𝑖 = 𝐽 → ( 𝑖 + 1 ) = ( 𝐽 + 1 ) ) |
93 |
92
|
fveq2d |
⊢ ( 𝑖 = 𝐽 → ( 𝑆 ‘ ( 𝑖 + 1 ) ) = ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) |
94 |
91 93
|
breq12d |
⊢ ( 𝑖 = 𝐽 → ( ( 𝑆 ‘ 𝑖 ) < ( 𝑆 ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑆 ‘ 𝐽 ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
95 |
94
|
rspccva |
⊢ ( ( ∀ 𝑖 ∈ ( 0 ..^ 𝑁 ) ( 𝑆 ‘ 𝑖 ) < ( 𝑆 ‘ ( 𝑖 + 1 ) ) ∧ 𝐽 ∈ ( 0 ..^ 𝑁 ) ) → ( 𝑆 ‘ 𝐽 ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) |
96 |
90 17 95
|
syl2anc |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐽 ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) |
97 |
60 81
|
posdifd |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐽 ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ↔ 0 < ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) ) |
98 |
96 97
|
mpbid |
⊢ ( 𝜑 → 0 < ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) |
99 |
|
eleq1 |
⊢ ( 𝑗 = 𝐽 → ( 𝑗 ∈ ( 0 ..^ 𝑁 ) ↔ 𝐽 ∈ ( 0 ..^ 𝑁 ) ) ) |
100 |
99
|
anbi2d |
⊢ ( 𝑗 = 𝐽 → ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) ↔ ( 𝜑 ∧ 𝐽 ∈ ( 0 ..^ 𝑁 ) ) ) ) |
101 |
|
oveq1 |
⊢ ( 𝑗 = 𝐽 → ( 𝑗 + 1 ) = ( 𝐽 + 1 ) ) |
102 |
101
|
fveq2d |
⊢ ( 𝑗 = 𝐽 → ( 𝑆 ‘ ( 𝑗 + 1 ) ) = ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) |
103 |
102
|
fveq2d |
⊢ ( 𝑗 = 𝐽 → ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) = ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
104 |
|
fveq2 |
⊢ ( 𝑗 = 𝐽 → ( 𝑆 ‘ 𝑗 ) = ( 𝑆 ‘ 𝐽 ) ) |
105 |
104
|
fveq2d |
⊢ ( 𝑗 = 𝐽 → ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) = ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) |
106 |
105
|
fveq2d |
⊢ ( 𝑗 = 𝐽 → ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) = ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ) |
107 |
103 106
|
oveq12d |
⊢ ( 𝑗 = 𝐽 → ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) − ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) ) = ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ) ) |
108 |
102 104
|
oveq12d |
⊢ ( 𝑗 = 𝐽 → ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝑆 ‘ 𝑗 ) ) = ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) |
109 |
107 108
|
eqeq12d |
⊢ ( 𝑗 = 𝐽 → ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) − ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) ) = ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝑆 ‘ 𝑗 ) ) ↔ ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ) = ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) ) |
110 |
100 109
|
imbi12d |
⊢ ( 𝑗 = 𝐽 → ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) − ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) ) = ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝑆 ‘ 𝑗 ) ) ) ↔ ( ( 𝜑 ∧ 𝐽 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ) = ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) ) ) |
111 |
2
|
oveq2i |
⊢ ( 𝑘 · 𝑇 ) = ( 𝑘 · ( 𝐵 − 𝐴 ) ) |
112 |
111
|
oveq2i |
⊢ ( 𝑦 + ( 𝑘 · 𝑇 ) ) = ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) |
113 |
112
|
eleq1i |
⊢ ( ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ↔ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 ) |
114 |
113
|
rexbii |
⊢ ( ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 ) |
115 |
114
|
rgenw |
⊢ ∀ 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ( ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 ↔ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 ) |
116 |
|
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 } ) |
117 |
115 116
|
mpbi |
⊢ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } = { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } |
118 |
117
|
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 } ) |
119 |
118
|
fveq2i |
⊢ ( ♯ ‘ ( { 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 } ) ) |
120 |
119
|
oveq1i |
⊢ ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) − 1 ) = ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) ) − 1 ) |
121 |
43 120
|
eqtri |
⊢ 𝑁 = ( ( ♯ ‘ ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) ) − 1 ) |
122 |
|
isoeq5 |
⊢ ( ( { 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 } ) → ( 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ↔ 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) ) ) ) |
123 |
118 122
|
ax-mp |
⊢ ( 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ↔ 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) ) ) |
124 |
123
|
iotabii |
⊢ ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) ) ) = ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) ) ) |
125 |
47 124
|
eqtri |
⊢ 𝑆 = ( ℩ 𝑓 𝑓 Isom < , < ( ( 0 ... 𝑁 ) , ( { 𝐶 , 𝐷 } ∪ { 𝑦 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑦 + ( 𝑘 · ( 𝐵 − 𝐴 ) ) ) ∈ ran 𝑄 } ) ) ) |
126 |
|
eqid |
⊢ ( ( 𝑆 ‘ 𝑗 ) + ( 𝐵 − ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) ) = ( ( 𝑆 ‘ 𝑗 ) + ( 𝐵 − ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) ) |
127 |
1 2 3 4 9 10 11 121 125 15 16 126
|
fourierdlem65 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) − ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) ) = ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝑆 ‘ 𝑗 ) ) ) |
128 |
110 127
|
vtoclg |
⊢ ( 𝐽 ∈ ( 0 ..^ 𝑁 ) → ( ( 𝜑 ∧ 𝐽 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ) = ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) ) |
129 |
128
|
anabsi7 |
⊢ ( ( 𝜑 ∧ 𝐽 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ) = ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) |
130 |
17 129
|
mpdan |
⊢ ( 𝜑 → ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ) = ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) |
131 |
98 130
|
breqtrrd |
⊢ ( 𝜑 → 0 < ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ) ) |
132 |
89 83
|
posdifd |
⊢ ( 𝜑 → ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) < ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ↔ 0 < ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ) ) ) |
133 |
131 132
|
mpbird |
⊢ ( 𝜑 → ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) < ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
134 |
106 103
|
oveq12d |
⊢ ( 𝑗 = 𝐽 → ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) = ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) |
135 |
104
|
fveq2d |
⊢ ( 𝑗 = 𝐽 → ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) |
136 |
135
|
fveq2d |
⊢ ( 𝑗 = 𝐽 → ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) ) = ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ) |
137 |
135
|
oveq1d |
⊢ ( 𝑗 = 𝐽 → ( ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) + 1 ) = ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) |
138 |
137
|
fveq2d |
⊢ ( 𝑗 = 𝐽 → ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) + 1 ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) |
139 |
136 138
|
oveq12d |
⊢ ( 𝑗 = 𝐽 → ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) + 1 ) ) ) = ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) |
140 |
134 139
|
sseq12d |
⊢ ( 𝑗 = 𝐽 → ( ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ⊆ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) + 1 ) ) ) ↔ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ⊆ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ) |
141 |
100 140
|
imbi12d |
⊢ ( 𝑗 = 𝐽 → ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ⊆ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) + 1 ) ) ) ) ↔ ( ( 𝜑 ∧ 𝐽 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ⊆ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ) ) |
142 |
12 40
|
eqtri |
⊢ 𝐻 = ( { 𝐶 , 𝐷 } ∪ { 𝑥 ∈ ( 𝐶 [,] 𝐷 ) ∣ ∃ 𝑘 ∈ ℤ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ ran 𝑄 } ) |
143 |
|
eqid |
⊢ ( ( 𝑆 ‘ 𝑗 ) + if ( ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝑆 ‘ 𝑗 ) ) < ( ( 𝑄 ‘ 1 ) − 𝐴 ) , ( ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝑆 ‘ 𝑗 ) ) / 2 ) , ( ( ( 𝑄 ‘ 1 ) − 𝐴 ) / 2 ) ) ) = ( ( 𝑆 ‘ 𝑗 ) + if ( ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝑆 ‘ 𝑗 ) ) < ( ( 𝑄 ‘ 1 ) − 𝐴 ) , ( ( ( 𝑆 ‘ ( 𝑗 + 1 ) ) − ( 𝑆 ‘ 𝑗 ) ) / 2 ) , ( ( ( 𝑄 ‘ 1 ) − 𝐴 ) / 2 ) ) ) |
144 |
2 1 3 4 9 31 35 11 142 13 14 15 16 143 19
|
fourierdlem79 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝑗 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝑗 + 1 ) ) ) ) ⊆ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝑗 ) ) + 1 ) ) ) ) |
145 |
141 144
|
vtoclg |
⊢ ( 𝐽 ∈ ( 0 ..^ 𝑁 ) → ( ( 𝜑 ∧ 𝐽 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ⊆ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ) |
146 |
145
|
anabsi7 |
⊢ ( ( 𝜑 ∧ 𝐽 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ⊆ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) |
147 |
17 146
|
mpdan |
⊢ ( 𝜑 → ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ⊆ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) |
148 |
63 68 89 83 133 147
|
fourierdlem10 |
⊢ ( 𝜑 → ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ≤ ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ∧ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ≤ ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) |
149 |
148
|
simpld |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ≤ ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ) |
150 |
63 89 83 149 133
|
lelttrd |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) < ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
151 |
150
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) → ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) < ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
152 |
68
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) → ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ∈ ℝ ) |
153 |
148
|
simprd |
⊢ ( 𝜑 → ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ≤ ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) |
154 |
153
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) → ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ≤ ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) |
155 |
|
neqne |
⊢ ( ¬ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) → ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ≠ ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) |
156 |
155
|
necomd |
⊢ ( ¬ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) → ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ≠ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
157 |
156
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) → ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ≠ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
158 |
84 152 154 157
|
leneltd |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) → ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) < ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) |
159 |
65 70 84 151 158
|
eliood |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) → ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∈ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) |
160 |
|
fvres |
⊢ ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∈ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) = ( 𝐹 ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) |
161 |
159 160
|
syl |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) = ( 𝐹 ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) |
162 |
161
|
eqcomd |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) → ( 𝐹 ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) |
163 |
162
|
ifeq2da |
⊢ ( 𝜑 → if ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) , ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) , ( 𝐹 ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) = if ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) , ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) , ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) ) |
164 |
|
fdm |
⊢ ( 𝐹 : ℝ ⟶ ℂ → dom 𝐹 = ℝ ) |
165 |
5 164
|
syl |
⊢ ( 𝜑 → dom 𝐹 = ℝ ) |
166 |
165
|
feq2d |
⊢ ( 𝜑 → ( 𝐹 : dom 𝐹 ⟶ ℂ ↔ 𝐹 : ℝ ⟶ ℂ ) ) |
167 |
5 166
|
mpbird |
⊢ ( 𝜑 → 𝐹 : dom 𝐹 ⟶ ℂ ) |
168 |
|
ioosscn |
⊢ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ⊆ ℂ |
169 |
168
|
a1i |
⊢ ( 𝜑 → ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ⊆ ℂ ) |
170 |
|
ioossre |
⊢ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ⊆ ℝ |
171 |
170 165
|
sseqtrrid |
⊢ ( 𝜑 → ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ⊆ dom 𝐹 ) |
172 |
81 83
|
resubcld |
⊢ ( 𝜑 → ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ∈ ℝ ) |
173 |
18 172
|
eqeltrid |
⊢ ( 𝜑 → 𝑈 ∈ ℝ ) |
174 |
173
|
recnd |
⊢ ( 𝜑 → 𝑈 ∈ ℂ ) |
175 |
|
eqid |
⊢ { 𝑥 ∈ ℂ ∣ ∃ 𝑦 ∈ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) 𝑥 = ( 𝑦 + 𝑈 ) } = { 𝑥 ∈ ℂ ∣ ∃ 𝑦 ∈ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) 𝑥 = ( 𝑦 + 𝑈 ) } |
176 |
89 83 173
|
iooshift |
⊢ ( 𝜑 → ( ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) + 𝑈 ) (,) ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) + 𝑈 ) ) = { 𝑥 ∈ ℂ ∣ ∃ 𝑦 ∈ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) 𝑥 = ( 𝑦 + 𝑈 ) } ) |
177 |
|
ioossre |
⊢ ( ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) + 𝑈 ) (,) ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) + 𝑈 ) ) ⊆ ℝ |
178 |
177 165
|
sseqtrrid |
⊢ ( 𝜑 → ( ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) + 𝑈 ) (,) ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) + 𝑈 ) ) ⊆ dom 𝐹 ) |
179 |
176 178
|
eqsstrrd |
⊢ ( 𝜑 → { 𝑥 ∈ ℂ ∣ ∃ 𝑦 ∈ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) 𝑥 = ( 𝑦 + 𝑈 ) } ⊆ dom 𝐹 ) |
180 |
|
elioore |
⊢ ( 𝑦 ∈ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) → 𝑦 ∈ ℝ ) |
181 |
74 72
|
resubcld |
⊢ ( 𝜑 → ( 𝐵 − 𝐴 ) ∈ ℝ ) |
182 |
2 181
|
eqeltrid |
⊢ ( 𝜑 → 𝑇 ∈ ℝ ) |
183 |
182
|
recnd |
⊢ ( 𝜑 → 𝑇 ∈ ℂ ) |
184 |
72 74
|
posdifd |
⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ 0 < ( 𝐵 − 𝐴 ) ) ) |
185 |
77 184
|
mpbid |
⊢ ( 𝜑 → 0 < ( 𝐵 − 𝐴 ) ) |
186 |
185 2
|
breqtrrdi |
⊢ ( 𝜑 → 0 < 𝑇 ) |
187 |
186
|
gt0ne0d |
⊢ ( 𝜑 → 𝑇 ≠ 0 ) |
188 |
174 183 187
|
divcan1d |
⊢ ( 𝜑 → ( ( 𝑈 / 𝑇 ) · 𝑇 ) = 𝑈 ) |
189 |
188
|
eqcomd |
⊢ ( 𝜑 → 𝑈 = ( ( 𝑈 / 𝑇 ) · 𝑇 ) ) |
190 |
189
|
oveq2d |
⊢ ( 𝜑 → ( 𝑦 + 𝑈 ) = ( 𝑦 + ( ( 𝑈 / 𝑇 ) · 𝑇 ) ) ) |
191 |
190
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑦 + 𝑈 ) = ( 𝑦 + ( ( 𝑈 / 𝑇 ) · 𝑇 ) ) ) |
192 |
191
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑦 + 𝑈 ) ) = ( 𝐹 ‘ ( 𝑦 + ( ( 𝑈 / 𝑇 ) · 𝑇 ) ) ) ) |
193 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐹 : ℝ ⟶ ℂ ) |
194 |
182
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑇 ∈ ℝ ) |
195 |
83
|
recnd |
⊢ ( 𝜑 → ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∈ ℂ ) |
196 |
81
|
recnd |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐽 + 1 ) ) ∈ ℂ ) |
197 |
195 196
|
negsubdi2d |
⊢ ( 𝜑 → - ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) |
198 |
197
|
eqcomd |
⊢ ( 𝜑 → ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) = - ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
199 |
198
|
oveq1d |
⊢ ( 𝜑 → ( ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) / 𝑇 ) = ( - ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) |
200 |
18
|
oveq1i |
⊢ ( 𝑈 / 𝑇 ) = ( ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) / 𝑇 ) |
201 |
200
|
a1i |
⊢ ( 𝜑 → ( 𝑈 / 𝑇 ) = ( ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) / 𝑇 ) ) |
202 |
15
|
a1i |
⊢ ( 𝜑 → 𝐸 = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) ) |
203 |
|
id |
⊢ ( 𝑥 = ( 𝑆 ‘ ( 𝐽 + 1 ) ) → 𝑥 = ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) |
204 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑆 ‘ ( 𝐽 + 1 ) ) → ( 𝐵 − 𝑥 ) = ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
205 |
204
|
oveq1d |
⊢ ( 𝑥 = ( 𝑆 ‘ ( 𝐽 + 1 ) ) → ( ( 𝐵 − 𝑥 ) / 𝑇 ) = ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) |
206 |
205
|
fveq2d |
⊢ ( 𝑥 = ( 𝑆 ‘ ( 𝐽 + 1 ) ) → ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) = ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) ) |
207 |
206
|
oveq1d |
⊢ ( 𝑥 = ( 𝑆 ‘ ( 𝐽 + 1 ) ) → ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) = ( ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) · 𝑇 ) ) |
208 |
203 207
|
oveq12d |
⊢ ( 𝑥 = ( 𝑆 ‘ ( 𝐽 + 1 ) ) → ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) = ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) + ( ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) · 𝑇 ) ) ) |
209 |
208
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 = ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) = ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) + ( ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) · 𝑇 ) ) ) |
210 |
74 81
|
resubcld |
⊢ ( 𝜑 → ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∈ ℝ ) |
211 |
210 182 187
|
redivcld |
⊢ ( 𝜑 → ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ∈ ℝ ) |
212 |
211
|
flcld |
⊢ ( 𝜑 → ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) ∈ ℤ ) |
213 |
212
|
zred |
⊢ ( 𝜑 → ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) ∈ ℝ ) |
214 |
213 182
|
remulcld |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) · 𝑇 ) ∈ ℝ ) |
215 |
81 214
|
readdcld |
⊢ ( 𝜑 → ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) + ( ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) · 𝑇 ) ) ∈ ℝ ) |
216 |
202 209 81 215
|
fvmptd |
⊢ ( 𝜑 → ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) + ( ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) · 𝑇 ) ) ) |
217 |
216
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) + ( ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) · 𝑇 ) ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
218 |
212
|
zcnd |
⊢ ( 𝜑 → ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) ∈ ℂ ) |
219 |
218 183
|
mulcld |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) · 𝑇 ) ∈ ℂ ) |
220 |
196 219
|
pncan2d |
⊢ ( 𝜑 → ( ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) + ( ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) · 𝑇 ) ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) · 𝑇 ) ) |
221 |
217 220
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) · 𝑇 ) ) |
222 |
221 219
|
eqeltrd |
⊢ ( 𝜑 → ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∈ ℂ ) |
223 |
222 183 187
|
divnegd |
⊢ ( 𝜑 → - ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) = ( - ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) |
224 |
199 201 223
|
3eqtr4d |
⊢ ( 𝜑 → ( 𝑈 / 𝑇 ) = - ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) |
225 |
221
|
oveq1d |
⊢ ( 𝜑 → ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) = ( ( ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) · 𝑇 ) / 𝑇 ) ) |
226 |
218 183 187
|
divcan4d |
⊢ ( 𝜑 → ( ( ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) · 𝑇 ) / 𝑇 ) = ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) ) |
227 |
225 226
|
eqtrd |
⊢ ( 𝜑 → ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) = ( ⌊ ‘ ( ( 𝐵 − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ) ) |
228 |
227 212
|
eqeltrd |
⊢ ( 𝜑 → ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ∈ ℤ ) |
229 |
228
|
znegcld |
⊢ ( 𝜑 → - ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) / 𝑇 ) ∈ ℤ ) |
230 |
224 229
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑈 / 𝑇 ) ∈ ℤ ) |
231 |
230
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑈 / 𝑇 ) ∈ ℤ ) |
232 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ℝ ) |
233 |
6
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑥 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑥 + 𝑇 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
234 |
193 194 231 232 233
|
fperiodmul |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑦 + ( ( 𝑈 / 𝑇 ) · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑦 ) ) |
235 |
192 234
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐹 ‘ ( 𝑦 + 𝑈 ) ) = ( 𝐹 ‘ 𝑦 ) ) |
236 |
180 235
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) → ( 𝐹 ‘ ( 𝑦 + 𝑈 ) ) = ( 𝐹 ‘ 𝑦 ) ) |
237 |
23
|
simprrd |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
238 |
|
fveq2 |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ) |
239 |
|
oveq1 |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( 𝑖 + 1 ) = ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) |
240 |
239
|
fveq2d |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) |
241 |
238 240
|
breq12d |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) < ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) |
242 |
241
|
rspccva |
⊢ ( ( ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) < ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) |
243 |
237 61 242
|
syl2anc |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) < ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) |
244 |
61
|
ancli |
⊢ ( 𝜑 → ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) ) |
245 |
|
eleq1 |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↔ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) ) |
246 |
245
|
anbi2d |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ↔ ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) ) ) |
247 |
238 240
|
oveq12d |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) |
248 |
247
|
reseq2d |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ) |
249 |
247
|
oveq1d |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) = ( ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) –cn→ ℂ ) ) |
250 |
248 249
|
eleq12d |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ↔ ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) –cn→ ℂ ) ) ) |
251 |
246 250
|
imbi12d |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) ↔ ( ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) –cn→ ℂ ) ) ) ) |
252 |
251 7
|
vtoclg |
⊢ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) → ( ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) –cn→ ℂ ) ) ) |
253 |
61 244 252
|
sylc |
⊢ ( 𝜑 → ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) –cn→ ℂ ) ) |
254 |
|
nfv |
⊢ Ⅎ 𝑖 ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) |
255 |
|
nfmpt1 |
⊢ Ⅎ 𝑖 ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↦ 𝐿 ) |
256 |
20 255
|
nfcxfr |
⊢ Ⅎ 𝑖 𝑊 |
257 |
|
nfcv |
⊢ Ⅎ 𝑖 ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) |
258 |
256 257
|
nffv |
⊢ Ⅎ 𝑖 ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) |
259 |
258
|
nfel1 |
⊢ Ⅎ 𝑖 ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) |
260 |
254 259
|
nfim |
⊢ Ⅎ 𝑖 ( ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) |
261 |
246
|
biimpar |
⊢ ( ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∧ ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) ) → ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ) |
262 |
261
|
3adant2 |
⊢ ( ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∧ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) ) → ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ) |
263 |
262 8
|
syl |
⊢ ( ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∧ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
264 |
|
fveq2 |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ) |
265 |
264
|
eqcomd |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) = ( 𝑊 ‘ 𝑖 ) ) |
266 |
265
|
adantr |
⊢ ( ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∧ ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) ) → ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) = ( 𝑊 ‘ 𝑖 ) ) |
267 |
261
|
simprd |
⊢ ( ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∧ ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) ) → 𝑖 ∈ ( 0 ..^ 𝑀 ) ) |
268 |
|
elex |
⊢ ( 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → 𝐿 ∈ V ) |
269 |
261 8 268
|
3syl |
⊢ ( ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∧ ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) ) → 𝐿 ∈ V ) |
270 |
20
|
fvmpt2 |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝐿 ∈ V ) → ( 𝑊 ‘ 𝑖 ) = 𝐿 ) |
271 |
267 269 270
|
syl2anc |
⊢ ( ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∧ ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) ) → ( 𝑊 ‘ 𝑖 ) = 𝐿 ) |
272 |
266 271
|
eqtrd |
⊢ ( ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∧ ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) ) → ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) = 𝐿 ) |
273 |
272
|
3adant2 |
⊢ ( ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∧ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) ) → ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) = 𝐿 ) |
274 |
248 240
|
oveq12d |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) |
275 |
274
|
eqcomd |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
276 |
275
|
3ad2ant1 |
⊢ ( ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∧ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
277 |
263 273 276
|
3eltr4d |
⊢ ( ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∧ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) ) → ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) |
278 |
277
|
3exp |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ) ) |
279 |
8
|
2a1i |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( ( ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) → ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
280 |
278 279
|
impbid |
⊢ ( 𝑖 = ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↔ ( ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ) ) |
281 |
260 280 8
|
vtoclg1f |
⊢ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) → ( ( 𝜑 ∧ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ) |
282 |
61 244 281
|
sylc |
⊢ ( 𝜑 → ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) |
283 |
|
eqid |
⊢ if ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) , ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) , ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) = if ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) , ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) , ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) |
284 |
|
eqid |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ∪ { ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) } ) ) = ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ∪ { ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) } ) ) |
285 |
63 68 243 253 282 89 83 133 147 283 284
|
fourierdlem33 |
⊢ ( 𝜑 → if ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) , ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) , ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) ∈ ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ↾ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) limℂ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) |
286 |
147
|
resabs1d |
⊢ ( 𝜑 → ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ↾ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) = ( 𝐹 ↾ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) ) |
287 |
286
|
oveq1d |
⊢ ( 𝜑 → ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ↾ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) limℂ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) = ( ( 𝐹 ↾ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) limℂ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) |
288 |
285 287
|
eleqtrd |
⊢ ( 𝜑 → if ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) , ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) , ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) limℂ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) |
289 |
167 169 171 174 175 179 236 288
|
limcperiod |
⊢ ( 𝜑 → if ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) , ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) , ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) ∈ ( ( 𝐹 ↾ { 𝑥 ∈ ℂ ∣ ∃ 𝑦 ∈ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) 𝑥 = ( 𝑦 + 𝑈 ) } ) limℂ ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) + 𝑈 ) ) ) |
290 |
18
|
oveq2i |
⊢ ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) + 𝑈 ) = ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) + ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) |
291 |
195 196
|
pncan3d |
⊢ ( 𝜑 → ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) + ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) = ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) |
292 |
290 291
|
syl5eq |
⊢ ( 𝜑 → ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) + 𝑈 ) = ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) |
293 |
292
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝐹 ↾ { 𝑥 ∈ ℂ ∣ ∃ 𝑦 ∈ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) 𝑥 = ( 𝑦 + 𝑈 ) } ) limℂ ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) + 𝑈 ) ) = ( ( 𝐹 ↾ { 𝑥 ∈ ℂ ∣ ∃ 𝑦 ∈ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) 𝑥 = ( 𝑦 + 𝑈 ) } ) limℂ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
294 |
289 293
|
eleqtrd |
⊢ ( 𝜑 → if ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) , ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) , ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) ∈ ( ( 𝐹 ↾ { 𝑥 ∈ ℂ ∣ ∃ 𝑦 ∈ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) 𝑥 = ( 𝑦 + 𝑈 ) } ) limℂ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
295 |
18
|
oveq2i |
⊢ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) + 𝑈 ) = ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) + ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) |
296 |
295
|
a1i |
⊢ ( 𝜑 → ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) + 𝑈 ) = ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) + ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) ) |
297 |
9 31
|
iccssred |
⊢ ( 𝜑 → ( 𝐶 [,] 𝐷 ) ⊆ ℝ ) |
298 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
299 |
297 298
|
sstrdi |
⊢ ( 𝜑 → ( 𝐶 [,] 𝐷 ) ⊆ ℂ ) |
300 |
11 51 50
|
fourierdlem15 |
⊢ ( 𝜑 → 𝑆 : ( 0 ... 𝑁 ) ⟶ ( 𝐶 [,] 𝐷 ) ) |
301 |
300 59
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐽 ) ∈ ( 𝐶 [,] 𝐷 ) ) |
302 |
299 301
|
sseldd |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐽 ) ∈ ℂ ) |
303 |
196 302
|
subcld |
⊢ ( 𝜑 → ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ∈ ℂ ) |
304 |
89
|
recnd |
⊢ ( 𝜑 → ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ∈ ℂ ) |
305 |
195 303 304
|
subsub23d |
⊢ ( 𝜑 → ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) = ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ↔ ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ) = ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) ) |
306 |
130 305
|
mpbird |
⊢ ( 𝜑 → ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) = ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) ) |
307 |
306
|
eqcomd |
⊢ ( 𝜑 → ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) = ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) ) |
308 |
307
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) + ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) = ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) + ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) ) |
309 |
195 303
|
subcld |
⊢ ( 𝜑 → ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) ∈ ℂ ) |
310 |
309 196 195
|
addsub12d |
⊢ ( 𝜑 → ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) + ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) = ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) + ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) ) |
311 |
195 303 195
|
sub32d |
⊢ ( 𝜑 → ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) = ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) ) |
312 |
195
|
subidd |
⊢ ( 𝜑 → ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) = 0 ) |
313 |
312
|
oveq1d |
⊢ ( 𝜑 → ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) = ( 0 − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) ) |
314 |
|
df-neg |
⊢ - ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) = ( 0 − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) |
315 |
196 302
|
negsubdi2d |
⊢ ( 𝜑 → - ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) = ( ( 𝑆 ‘ 𝐽 ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
316 |
314 315
|
eqtr3id |
⊢ ( 𝜑 → ( 0 − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) = ( ( 𝑆 ‘ 𝐽 ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
317 |
311 313 316
|
3eqtrd |
⊢ ( 𝜑 → ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) = ( ( 𝑆 ‘ 𝐽 ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
318 |
317
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) + ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) = ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) + ( ( 𝑆 ‘ 𝐽 ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) |
319 |
196 302
|
pncan3d |
⊢ ( 𝜑 → ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) + ( ( 𝑆 ‘ 𝐽 ) − ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) = ( 𝑆 ‘ 𝐽 ) ) |
320 |
310 318 319
|
3eqtrd |
⊢ ( 𝜑 → ( ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) − ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝑆 ‘ 𝐽 ) ) ) + ( ( 𝑆 ‘ ( 𝐽 + 1 ) ) − ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) = ( 𝑆 ‘ 𝐽 ) ) |
321 |
296 308 320
|
3eqtrd |
⊢ ( 𝜑 → ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) + 𝑈 ) = ( 𝑆 ‘ 𝐽 ) ) |
322 |
321 292
|
oveq12d |
⊢ ( 𝜑 → ( ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) + 𝑈 ) (,) ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) + 𝑈 ) ) = ( ( 𝑆 ‘ 𝐽 ) (,) ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
323 |
176 322
|
eqtr3d |
⊢ ( 𝜑 → { 𝑥 ∈ ℂ ∣ ∃ 𝑦 ∈ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) 𝑥 = ( 𝑦 + 𝑈 ) } = ( ( 𝑆 ‘ 𝐽 ) (,) ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
324 |
323
|
reseq2d |
⊢ ( 𝜑 → ( 𝐹 ↾ { 𝑥 ∈ ℂ ∣ ∃ 𝑦 ∈ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) 𝑥 = ( 𝑦 + 𝑈 ) } ) = ( 𝐹 ↾ ( ( 𝑆 ‘ 𝐽 ) (,) ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) |
325 |
324
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐹 ↾ { 𝑥 ∈ ℂ ∣ ∃ 𝑦 ∈ ( ( 𝑍 ‘ ( 𝐸 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) 𝑥 = ( 𝑦 + 𝑈 ) } ) limℂ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝐽 ) (,) ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) limℂ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
326 |
294 325
|
eleqtrd |
⊢ ( 𝜑 → if ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) , ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) , ( ( 𝐹 ↾ ( ( 𝑄 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) (,) ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) ) ) ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝐽 ) (,) ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) limℂ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
327 |
163 326
|
eqeltrd |
⊢ ( 𝜑 → if ( ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) = ( 𝑄 ‘ ( ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) + 1 ) ) , ( 𝑊 ‘ ( 𝐼 ‘ ( 𝑆 ‘ 𝐽 ) ) ) , ( 𝐹 ‘ ( 𝐸 ‘ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑆 ‘ 𝐽 ) (,) ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) limℂ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |