Step | Hyp | Ref
| Expression |
1 | | fourierdlem84.o |
. 2
⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
2 | | fourierdlem84.m |
. 2
⊢ (𝜑 → 𝑀 ∈ ℕ) |
3 | | fourierdlem84.1 |
. . 3
⊢ (𝜑 → 𝐴 ∈ ℝ) |
4 | | fourierdlem84.2 |
. . 3
⊢ (𝜑 → 𝐵 ∈ ℝ) |
5 | | fourierdlem84.xre |
. . 3
⊢ (𝜑 → 𝑋 ∈ ℝ) |
6 | | fourierdlem84.p |
. . 3
⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑋) ∧ (𝑝‘𝑚) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
7 | | fourierdlem84.v |
. . 3
⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) |
8 | | fourierdlem84.q |
. . 3
⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) |
9 | 3, 4, 5, 6, 1, 2, 7, 8 | fourierdlem14 43204 |
. 2
⊢ (𝜑 → 𝑄 ∈ (𝑂‘𝑀)) |
10 | | fourierdlem84.f |
. . . . . . 7
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
11 | 10 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℝ) |
12 | 5 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℝ) |
13 | 3 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ) |
14 | 4 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ) |
15 | | simpr 488 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ (𝐴[,]𝐵)) |
16 | | eliccre 42583 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ) |
17 | 13, 14, 15, 16 | syl3anc 1372 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ) |
18 | 12, 17 | readdcld 10748 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝑋 + 𝑠) ∈ ℝ) |
19 | 11, 18 | ffvelrnd 6862 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ) |
20 | | fourierdlem84.d |
. . . . . . . 8
⊢ (𝜑 → 𝐷 ∈ (ℝ–cn→ℝ)) |
21 | | cncff 23645 |
. . . . . . . 8
⊢ (𝐷 ∈ (ℝ–cn→ℝ) → 𝐷:ℝ⟶ℝ) |
22 | 20, 21 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐷:ℝ⟶ℝ) |
23 | 22 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐷:ℝ⟶ℝ) |
24 | 23, 17 | ffvelrnd 6862 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐷‘𝑠) ∈ ℝ) |
25 | 19, 24 | remulcld 10749 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠)) ∈ ℝ) |
26 | 25 | recnd 10747 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠)) ∈ ℂ) |
27 | | fourierdlem84.g |
. . 3
⊢ 𝐺 = (𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) |
28 | 26, 27 | fmptd 6888 |
. 2
⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ) |
29 | 27 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐺 = (𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠)))) |
30 | 29 | reseq1d 5824 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) |
31 | | ioossicc 12907 |
. . . . . 6
⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) |
32 | 3 | rexrd 10769 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
33 | 32 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈
ℝ*) |
34 | 4 | rexrd 10769 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
35 | 34 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐵 ∈
ℝ*) |
36 | 1, 2, 9 | fourierdlem15 43205 |
. . . . . . . 8
⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) |
37 | 36 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) |
38 | | simpr 488 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀)) |
39 | 33, 35, 37, 38 | fourierdlem8 43198 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵)) |
40 | 31, 39 | sstrid 3888 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵)) |
41 | 40 | resmptd 5882 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠)))) |
42 | 30, 41 | eqtrd 2773 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠)))) |
43 | 3, 5 | readdcld 10748 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐴 + 𝑋) ∈ ℝ) |
44 | 4, 5 | readdcld 10748 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐵 + 𝑋) ∈ ℝ) |
45 | 43, 44 | iccssred 12908 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐴 + 𝑋)[,](𝐵 + 𝑋)) ⊆ ℝ) |
46 | 45 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐴 + 𝑋)[,](𝐵 + 𝑋)) ⊆ ℝ) |
47 | 6, 2, 7 | fourierdlem15 43205 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑉:(0...𝑀)⟶((𝐴 + 𝑋)[,](𝐵 + 𝑋))) |
48 | 47 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶((𝐴 + 𝑋)[,](𝐵 + 𝑋))) |
49 | | elfzofz 13144 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀)) |
50 | 49 | adantl 485 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀)) |
51 | 48, 50 | ffvelrnd 6862 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ((𝐴 + 𝑋)[,](𝐵 + 𝑋))) |
52 | 46, 51 | sseldd 3878 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℝ) |
53 | 52 | rexrd 10769 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈
ℝ*) |
54 | 53 | adantr 484 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) ∈
ℝ*) |
55 | | fzofzp1 13225 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀)) |
56 | 55 | adantl 485 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀)) |
57 | 48, 56 | ffvelrnd 6862 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ((𝐴 + 𝑋)[,](𝐵 + 𝑋))) |
58 | 46, 57 | sseldd 3878 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ) |
59 | 58 | rexrd 10769 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈
ℝ*) |
60 | 59 | adantr 484 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘(𝑖 + 1)) ∈
ℝ*) |
61 | 5 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ) |
62 | | elioore 12851 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℝ) |
63 | 62 | adantl 485 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ) |
64 | 61, 63 | readdcld 10748 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ) |
65 | 5 | recnd 10747 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑋 ∈ ℂ) |
66 | 65 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ) |
67 | 3, 4 | iccssred 12908 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
68 | 67 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐴[,]𝐵) ⊆ ℝ) |
69 | 37, 50 | ffvelrnd 6862 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (𝐴[,]𝐵)) |
70 | 68, 69 | sseldd 3878 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ) |
71 | 70 | recnd 10747 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ) |
72 | 66, 71 | addcomd 10920 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘𝑖)) = ((𝑄‘𝑖) + 𝑋)) |
73 | 5 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ) |
74 | 52, 73 | resubcld 11146 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ) |
75 | 8 | fvmpt2 6786 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑉‘𝑖) − 𝑋) ∈ ℝ) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋)) |
76 | 50, 74, 75 | syl2anc 587 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋)) |
77 | 76 | oveq1d 7185 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑋) = (((𝑉‘𝑖) − 𝑋) + 𝑋)) |
78 | 52 | recnd 10747 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℂ) |
79 | 78, 66 | npcand 11079 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑉‘𝑖) − 𝑋) + 𝑋) = (𝑉‘𝑖)) |
80 | 72, 77, 79 | 3eqtrrd 2778 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) = (𝑋 + (𝑄‘𝑖))) |
81 | 80 | adantr 484 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) = (𝑋 + (𝑄‘𝑖))) |
82 | 70 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ) |
83 | 70 | rexrd 10769 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈
ℝ*) |
84 | 83 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈
ℝ*) |
85 | 37, 68 | fssd 6522 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ) |
86 | 85, 56 | ffvelrnd 6862 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
87 | 86 | rexrd 10769 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈
ℝ*) |
88 | 87 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈
ℝ*) |
89 | | simpr 488 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
90 | | ioogtlb 42573 |
. . . . . . . . . . . 12
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ* ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑠) |
91 | 84, 88, 89, 90 | syl3anc 1372 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑠) |
92 | 82, 63, 61, 91 | ltadd2dd 10877 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄‘𝑖)) < (𝑋 + 𝑠)) |
93 | 81, 92 | eqbrtrd 5052 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) < (𝑋 + 𝑠)) |
94 | 86 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
95 | | iooltub 42588 |
. . . . . . . . . . . 12
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ* ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1))) |
96 | 84, 88, 89, 95 | syl3anc 1372 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1))) |
97 | 63, 94, 61, 96 | ltadd2dd 10877 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑋 + (𝑄‘(𝑖 + 1)))) |
98 | | fveq2 6674 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 𝑗 → (𝑉‘𝑖) = (𝑉‘𝑗)) |
99 | 98 | oveq1d 7185 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝑗 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑗) − 𝑋)) |
100 | 99 | cbvmptv 5133 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋)) |
101 | 8, 100 | eqtri 2761 |
. . . . . . . . . . . . . . 15
⊢ 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋)) |
102 | 101 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))) |
103 | | fveq2 6674 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = (𝑖 + 1) → (𝑉‘𝑗) = (𝑉‘(𝑖 + 1))) |
104 | 103 | oveq1d 7185 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = (𝑖 + 1) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋)) |
105 | 104 | adantl 485 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋)) |
106 | 58, 73 | resubcld 11146 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ) |
107 | 102, 105,
56, 106 | fvmptd 6782 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋)) |
108 | 107 | oveq2d 7186 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋))) |
109 | 58 | recnd 10747 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℂ) |
110 | 66, 109 | pncan3d 11078 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋)) = (𝑉‘(𝑖 + 1))) |
111 | 108, 110 | eqtrd 2773 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1))) |
112 | 111 | adantr 484 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1))) |
113 | 97, 112 | breqtrd 5056 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑉‘(𝑖 + 1))) |
114 | 54, 60, 64, 93, 113 | eliood 42576 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) |
115 | | fvres 6693 |
. . . . . . . 8
⊢ ((𝑋 + 𝑠) ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠))) |
116 | 114, 115 | syl 17 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠))) |
117 | 116 | eqcomd 2744 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) = ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠))) |
118 | 117 | mpteq2dva 5125 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)))) |
119 | | ioosscn 12883 |
. . . . . . 7
⊢ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℂ |
120 | 119 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℂ) |
121 | | fourierdlem84.fcn |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ)) |
122 | | ioosscn 12883 |
. . . . . . 7
⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ |
123 | 122 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ) |
124 | 120, 121,
123, 66, 114 | fourierdlem23 43213 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
125 | 118, 124 | eqeltrd 2833 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
126 | | eqid 2738 |
. . . . 5
⊢ (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) = (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) |
127 | | ax-resscn 10672 |
. . . . . . . 8
⊢ ℝ
⊆ ℂ |
128 | | ssid 3899 |
. . . . . . . 8
⊢ ℂ
⊆ ℂ |
129 | | cncfss 23651 |
. . . . . . . 8
⊢ ((ℝ
⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ–cn→ℝ) ⊆ (ℝ–cn→ℂ)) |
130 | 127, 128,
129 | mp2an 692 |
. . . . . . 7
⊢
(ℝ–cn→ℝ)
⊆ (ℝ–cn→ℂ) |
131 | 22 | feqmptd 6737 |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 = (𝑠 ∈ ℝ ↦ (𝐷‘𝑠))) |
132 | 131 | eqcomd 2744 |
. . . . . . . 8
⊢ (𝜑 → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) = 𝐷) |
133 | 132, 20 | eqeltrd 2833 |
. . . . . . 7
⊢ (𝜑 → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ∈ (ℝ–cn→ℝ)) |
134 | 130, 133 | sseldi 3875 |
. . . . . 6
⊢ (𝜑 → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ∈ (ℝ–cn→ℂ)) |
135 | 134 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ∈ (ℝ–cn→ℂ)) |
136 | 40, 68 | sstrd 3887 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ) |
137 | 128 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ℂ ⊆
ℂ) |
138 | 22 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐷:ℝ⟶ℝ) |
139 | 62 | adantl 485 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ) |
140 | 138, 139 | ffvelrnd 6862 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐷‘𝑠) ∈ ℝ) |
141 | 140 | recnd 10747 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐷‘𝑠) ∈ ℂ) |
142 | 141 | adantlr 715 |
. . . . 5
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐷‘𝑠) ∈ ℂ) |
143 | 126, 135,
136, 137, 142 | cncfmptssg 42954 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
144 | 125, 143 | mulcncf 24198 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
145 | 42, 144 | eqeltrd 2833 |
. 2
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
146 | | eqid 2738 |
. . . 4
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) |
147 | | eqid 2738 |
. . . 4
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) |
148 | | eqid 2738 |
. . . 4
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) |
149 | 10 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐹:ℝ⟶ℝ) |
150 | 5 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ) |
151 | 150, 139 | readdcld 10748 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ) |
152 | 149, 151 | ffvelrnd 6862 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ) |
153 | 152 | recnd 10747 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ) |
154 | 153 | adantlr 715 |
. . . 4
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ) |
155 | 10 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℝ) |
156 | | ioossre 12882 |
. . . . . 6
⊢ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ |
157 | 156 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ) |
158 | 82, 91 | gtned 10853 |
. . . . 5
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ (𝑄‘𝑖)) |
159 | | fourierdlem84.r |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑉‘𝑖))) |
160 | 80 | oveq2d 7186 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑉‘𝑖)) = ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑋 + (𝑄‘𝑖)))) |
161 | 159, 160 | eleqtrd 2835 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑋 + (𝑄‘𝑖)))) |
162 | 155, 73, 136, 146, 114, 157, 158, 161, 71 | fourierdlem53 43242 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) limℂ (𝑄‘𝑖))) |
163 | | limcresi 24637 |
. . . . . 6
⊢ ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) limℂ (𝑄‘𝑖)) ⊆ (((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)) |
164 | 130, 20 | sseldi 3875 |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 ∈ (ℝ–cn→ℂ)) |
165 | 164 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐷 ∈ (ℝ–cn→ℂ)) |
166 | 165, 70 | cnlimci 24641 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘𝑖)) ∈ (𝐷 limℂ (𝑄‘𝑖))) |
167 | 131 | oveq1d 7185 |
. . . . . . . 8
⊢ (𝜑 → (𝐷 limℂ (𝑄‘𝑖)) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) limℂ (𝑄‘𝑖))) |
168 | 167 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷 limℂ (𝑄‘𝑖)) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) limℂ (𝑄‘𝑖))) |
169 | 166, 168 | eleqtrd 2835 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘𝑖)) ∈ ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) limℂ (𝑄‘𝑖))) |
170 | 163, 169 | sseldi 3875 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘𝑖)) ∈ (((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
171 | 136 | resmptd 5882 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠))) |
172 | 171 | oveq1d 7185 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) limℂ (𝑄‘𝑖))) |
173 | 170, 172 | eleqtrd 2835 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘𝑖)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) limℂ (𝑄‘𝑖))) |
174 | 146, 147,
148, 154, 142, 162, 173 | mullimc 42699 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 · (𝐷‘(𝑄‘𝑖))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) limℂ (𝑄‘𝑖))) |
175 | 27 | reseq1i 5821 |
. . . . 5
⊢ (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
176 | 175, 41 | eqtr2id 2786 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) = (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) |
177 | 176 | oveq1d 7185 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) limℂ (𝑄‘𝑖)) = ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
178 | 174, 177 | eleqtrd 2835 |
. 2
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 · (𝐷‘(𝑄‘𝑖))) ∈ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
179 | 63, 96 | ltned 10854 |
. . . . 5
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ (𝑄‘(𝑖 + 1))) |
180 | | fourierdlem84.l |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑉‘(𝑖 + 1)))) |
181 | 111 | eqcomd 2744 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) = (𝑋 + (𝑄‘(𝑖 + 1)))) |
182 | 181 | oveq2d 7186 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑉‘(𝑖 + 1))) = ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑋 + (𝑄‘(𝑖 + 1))))) |
183 | 180, 182 | eleqtrd 2835 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑋 + (𝑄‘(𝑖 + 1))))) |
184 | 86 | recnd 10747 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ) |
185 | 155, 73, 136, 146, 114, 157, 179, 183, 184 | fourierdlem53 43242 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) limℂ (𝑄‘(𝑖 + 1)))) |
186 | | limcresi 24637 |
. . . . . 6
⊢ ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) limℂ (𝑄‘(𝑖 + 1))) ⊆ (((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1))) |
187 | 165, 86 | cnlimci 24641 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘(𝑖 + 1))) ∈ (𝐷 limℂ (𝑄‘(𝑖 + 1)))) |
188 | 131 | oveq1d 7185 |
. . . . . . . 8
⊢ (𝜑 → (𝐷 limℂ (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) limℂ (𝑄‘(𝑖 + 1)))) |
189 | 188 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷 limℂ (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) limℂ (𝑄‘(𝑖 + 1)))) |
190 | 187, 189 | eleqtrd 2835 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘(𝑖 + 1))) ∈ ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) limℂ (𝑄‘(𝑖 + 1)))) |
191 | 186, 190 | sseldi 3875 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘(𝑖 + 1))) ∈ (((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) |
192 | 171 | oveq1d 7185 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) limℂ (𝑄‘(𝑖 + 1)))) |
193 | 191, 192 | eleqtrd 2835 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘(𝑖 + 1))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) limℂ (𝑄‘(𝑖 + 1)))) |
194 | 146, 147,
148, 154, 142, 185, 193 | mullimc 42699 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐿 · (𝐷‘(𝑄‘(𝑖 + 1)))) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) limℂ (𝑄‘(𝑖 + 1)))) |
195 | 176 | oveq1d 7185 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) limℂ (𝑄‘(𝑖 + 1))) = ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) |
196 | 194, 195 | eleqtrd 2835 |
. 2
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐿 · (𝐷‘(𝑄‘(𝑖 + 1)))) ∈ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) |
197 | 1, 2, 9, 28, 145, 178, 196 | fourierdlem69 43258 |
1
⊢ (𝜑 → 𝐺 ∈
𝐿1) |