| 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 46136 |
. 2
⊢ (𝜑 → 𝑄 ∈ (𝑂‘𝑀)) |
| 10 | | fourierdlem84.f |
. . . . . . 7
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
| 11 | 10 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℝ) |
| 12 | 5 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℝ) |
| 13 | 3 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ) |
| 14 | 4 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ) |
| 15 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ (𝐴[,]𝐵)) |
| 16 | | eliccre 45518 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ) |
| 17 | 13, 14, 15, 16 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ) |
| 18 | 12, 17 | readdcld 11290 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝑋 + 𝑠) ∈ ℝ) |
| 19 | 11, 18 | ffvelcdmd 7105 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ) |
| 20 | | fourierdlem84.d |
. . . . . . . 8
⊢ (𝜑 → 𝐷 ∈ (ℝ–cn→ℝ)) |
| 21 | | cncff 24919 |
. . . . . . . 8
⊢ (𝐷 ∈ (ℝ–cn→ℝ) → 𝐷:ℝ⟶ℝ) |
| 22 | 20, 21 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐷:ℝ⟶ℝ) |
| 23 | 22 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐷:ℝ⟶ℝ) |
| 24 | 23, 17 | ffvelcdmd 7105 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐷‘𝑠) ∈ ℝ) |
| 25 | 19, 24 | remulcld 11291 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠)) ∈ ℝ) |
| 26 | 25 | recnd 11289 |
. . 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 7134 |
. 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 5996 |
. . . 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 13473 |
. . . . . 6
⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) |
| 32 | 3 | rexrd 11311 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 33 | 32 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈
ℝ*) |
| 34 | 4 | rexrd 11311 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 35 | 34 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐵 ∈
ℝ*) |
| 36 | 1, 2, 9 | fourierdlem15 46137 |
. . . . . . . 8
⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) |
| 37 | 36 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) |
| 38 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀)) |
| 39 | 33, 35, 37, 38 | fourierdlem8 46130 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵)) |
| 40 | 31, 39 | sstrid 3995 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵)) |
| 41 | 40 | resmptd 6058 |
. . . 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 2777 |
. . 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 11290 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐴 + 𝑋) ∈ ℝ) |
| 44 | 4, 5 | readdcld 11290 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐵 + 𝑋) ∈ ℝ) |
| 45 | 43, 44 | iccssred 13474 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐴 + 𝑋)[,](𝐵 + 𝑋)) ⊆ ℝ) |
| 46 | 45 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐴 + 𝑋)[,](𝐵 + 𝑋)) ⊆ ℝ) |
| 47 | 6, 2, 7 | fourierdlem15 46137 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑉:(0...𝑀)⟶((𝐴 + 𝑋)[,](𝐵 + 𝑋))) |
| 48 | 47 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶((𝐴 + 𝑋)[,](𝐵 + 𝑋))) |
| 49 | | elfzofz 13715 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀)) |
| 50 | 49 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀)) |
| 51 | 48, 50 | ffvelcdmd 7105 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ((𝐴 + 𝑋)[,](𝐵 + 𝑋))) |
| 52 | 46, 51 | sseldd 3984 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℝ) |
| 53 | 52 | rexrd 11311 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈
ℝ*) |
| 54 | 53 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) ∈
ℝ*) |
| 55 | | fzofzp1 13803 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀)) |
| 56 | 55 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀)) |
| 57 | 48, 56 | ffvelcdmd 7105 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ((𝐴 + 𝑋)[,](𝐵 + 𝑋))) |
| 58 | 46, 57 | sseldd 3984 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ) |
| 59 | 58 | rexrd 11311 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈
ℝ*) |
| 60 | 59 | adantr 480 |
. . . . . . . . 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 13417 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℝ) |
| 63 | 62 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ) |
| 64 | 61, 63 | readdcld 11290 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ) |
| 65 | 5 | recnd 11289 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑋 ∈ ℂ) |
| 66 | 65 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ) |
| 67 | 3, 4 | iccssred 13474 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
| 68 | 67 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐴[,]𝐵) ⊆ ℝ) |
| 69 | 37, 50 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (𝐴[,]𝐵)) |
| 70 | 68, 69 | sseldd 3984 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ) |
| 71 | 70 | recnd 11289 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ) |
| 72 | 66, 71 | addcomd 11463 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘𝑖)) = ((𝑄‘𝑖) + 𝑋)) |
| 73 | 5 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ) |
| 74 | 52, 73 | resubcld 11691 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ) |
| 75 | 8 | fvmpt2 7027 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑉‘𝑖) − 𝑋) ∈ ℝ) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋)) |
| 76 | 50, 74, 75 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋)) |
| 77 | 76 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑋) = (((𝑉‘𝑖) − 𝑋) + 𝑋)) |
| 78 | 52 | recnd 11289 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℂ) |
| 79 | 78, 66 | npcand 11624 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑉‘𝑖) − 𝑋) + 𝑋) = (𝑉‘𝑖)) |
| 80 | 72, 77, 79 | 3eqtrrd 2782 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) = (𝑋 + (𝑄‘𝑖))) |
| 81 | 80 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) = (𝑋 + (𝑄‘𝑖))) |
| 82 | 70 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ) |
| 83 | 70 | rexrd 11311 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈
ℝ*) |
| 84 | 83 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈
ℝ*) |
| 85 | 37, 68 | fssd 6753 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ) |
| 86 | 85, 56 | ffvelcdmd 7105 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
| 87 | 86 | rexrd 11311 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈
ℝ*) |
| 88 | 87 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈
ℝ*) |
| 89 | | simpr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
| 90 | | ioogtlb 45508 |
. . . . . . . . . . . 12
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ* ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑠) |
| 91 | 84, 88, 89, 90 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑠) |
| 92 | 82, 63, 61, 91 | ltadd2dd 11420 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄‘𝑖)) < (𝑋 + 𝑠)) |
| 93 | 81, 92 | eqbrtrd 5165 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) < (𝑋 + 𝑠)) |
| 94 | 86 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
| 95 | | iooltub 45523 |
. . . . . . . . . . . 12
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ* ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1))) |
| 96 | 84, 88, 89, 95 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1))) |
| 97 | 63, 94, 61, 96 | ltadd2dd 11420 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑋 + (𝑄‘(𝑖 + 1)))) |
| 98 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 𝑗 → (𝑉‘𝑖) = (𝑉‘𝑗)) |
| 99 | 98 | oveq1d 7446 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝑗 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑗) − 𝑋)) |
| 100 | 99 | cbvmptv 5255 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋)) |
| 101 | 8, 100 | eqtri 2765 |
. . . . . . . . . . . . . . 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 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = (𝑖 + 1) → (𝑉‘𝑗) = (𝑉‘(𝑖 + 1))) |
| 104 | 103 | oveq1d 7446 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = (𝑖 + 1) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋)) |
| 105 | 104 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋)) |
| 106 | 58, 73 | resubcld 11691 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ) |
| 107 | 102, 105,
56, 106 | fvmptd 7023 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋)) |
| 108 | 107 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋))) |
| 109 | 58 | recnd 11289 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℂ) |
| 110 | 66, 109 | pncan3d 11623 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋)) = (𝑉‘(𝑖 + 1))) |
| 111 | 108, 110 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1))) |
| 112 | 111 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1))) |
| 113 | 97, 112 | breqtrd 5169 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑉‘(𝑖 + 1))) |
| 114 | 54, 60, 64, 93, 113 | eliood 45511 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) |
| 115 | | fvres 6925 |
. . . . . . . 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 2743 |
. . . . . 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 5242 |
. . . . 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 13449 |
. . . . . . 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 13449 |
. . . . . . 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 46145 |
. . . . 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 2841 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
| 126 | | eqid 2737 |
. . . . 5
⊢ (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) = (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) |
| 127 | | ax-resscn 11212 |
. . . . . . . 8
⊢ ℝ
⊆ ℂ |
| 128 | | ssid 4006 |
. . . . . . . 8
⊢ ℂ
⊆ ℂ |
| 129 | | cncfss 24925 |
. . . . . . . 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 6977 |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 = (𝑠 ∈ ℝ ↦ (𝐷‘𝑠))) |
| 132 | 131 | eqcomd 2743 |
. . . . . . . 8
⊢ (𝜑 → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) = 𝐷) |
| 133 | 132, 20 | eqeltrd 2841 |
. . . . . . 7
⊢ (𝜑 → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ∈ (ℝ–cn→ℝ)) |
| 134 | 130, 133 | sselid 3981 |
. . . . . 6
⊢ (𝜑 → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ∈ (ℝ–cn→ℂ)) |
| 135 | 134 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ∈ (ℝ–cn→ℂ)) |
| 136 | 40, 68 | sstrd 3994 |
. . . . 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 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐷:ℝ⟶ℝ) |
| 139 | 62 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ) |
| 140 | 138, 139 | ffvelcdmd 7105 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐷‘𝑠) ∈ ℝ) |
| 141 | 140 | recnd 11289 |
. . . . . 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 45886 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
| 144 | 125, 143 | mulcncf 25480 |
. . 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 2841 |
. 2
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
| 146 | | eqid 2737 |
. . . 4
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) |
| 147 | | eqid 2737 |
. . . 4
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐷‘𝑠)) |
| 148 | | eqid 2737 |
. . . 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 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐹:ℝ⟶ℝ) |
| 150 | 5 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ) |
| 151 | 150, 139 | readdcld 11290 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ) |
| 152 | 149, 151 | ffvelcdmd 7105 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ) |
| 153 | 152 | recnd 11289 |
. . . . 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 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℝ) |
| 156 | | ioossre 13448 |
. . . . . 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 11396 |
. . . . 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 7447 |
. . . . . 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 2843 |
. . . . 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 46174 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) limℂ (𝑄‘𝑖))) |
| 163 | | limcresi 25920 |
. . . . . 6
⊢ ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) limℂ (𝑄‘𝑖)) ⊆ (((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)) |
| 164 | 130, 20 | sselid 3981 |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 ∈ (ℝ–cn→ℂ)) |
| 165 | 164 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐷 ∈ (ℝ–cn→ℂ)) |
| 166 | 165, 70 | cnlimci 25924 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘𝑖)) ∈ (𝐷 limℂ (𝑄‘𝑖))) |
| 167 | 131 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → (𝐷 limℂ (𝑄‘𝑖)) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) limℂ (𝑄‘𝑖))) |
| 168 | 167 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷 limℂ (𝑄‘𝑖)) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) limℂ (𝑄‘𝑖))) |
| 169 | 166, 168 | eleqtrd 2843 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘𝑖)) ∈ ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) limℂ (𝑄‘𝑖))) |
| 170 | 163, 169 | sselid 3981 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘𝑖)) ∈ (((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
| 171 | 136 | resmptd 6058 |
. . . . . 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 7446 |
. . . . 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 2843 |
. . . 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 45631 |
. . 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 5993 |
. . . . 5
⊢ (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
| 176 | 175, 41 | eqtr2id 2790 |
. . . 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 7446 |
. . 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 2843 |
. 2
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 · (𝐷‘(𝑄‘𝑖))) ∈ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
| 179 | 63, 96 | ltned 11397 |
. . . . 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 2743 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) = (𝑋 + (𝑄‘(𝑖 + 1)))) |
| 182 | 181 | oveq2d 7447 |
. . . . . 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 2843 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑋 + (𝑄‘(𝑖 + 1))))) |
| 184 | 86 | recnd 11289 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ) |
| 185 | 155, 73, 136, 146, 114, 157, 179, 183, 184 | fourierdlem53 46174 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) limℂ (𝑄‘(𝑖 + 1)))) |
| 186 | | limcresi 25920 |
. . . . . 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 25924 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘(𝑖 + 1))) ∈ (𝐷 limℂ (𝑄‘(𝑖 + 1)))) |
| 188 | 131 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → (𝐷 limℂ (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) limℂ (𝑄‘(𝑖 + 1)))) |
| 189 | 188 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷 limℂ (𝑄‘(𝑖 + 1))) = ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) limℂ (𝑄‘(𝑖 + 1)))) |
| 190 | 187, 189 | eleqtrd 2843 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐷‘(𝑄‘(𝑖 + 1))) ∈ ((𝑠 ∈ ℝ ↦ (𝐷‘𝑠)) limℂ (𝑄‘(𝑖 + 1)))) |
| 191 | 186, 190 | sselid 3981 |
. . . . 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 7446 |
. . . . 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 2843 |
. . . 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 45631 |
. . 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 7446 |
. . 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 2843 |
. 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 46190 |
1
⊢ (𝜑 → 𝐺 ∈
𝐿1) |