| Step | Hyp | Ref
| Expression |
| 1 | | fourierdlem107.t |
. . . . . . . . . . . . . . . . 17
⊢ 𝑇 = (𝐵 − 𝐴) |
| 2 | 1 | oveq2i 7442 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 − 𝑋) + 𝑇) = ((𝐴 − 𝑋) + (𝐵 − 𝐴)) |
| 3 | | fourierdlem107.a |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 4 | 3 | recnd 11289 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 5 | | fourierdlem107.x |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑋 ∈
ℝ+) |
| 6 | 5 | rpred 13077 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑋 ∈ ℝ) |
| 7 | 6 | recnd 11289 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑋 ∈ ℂ) |
| 8 | | fourierdlem107.b |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 9 | 8 | recnd 11289 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 10 | 4, 7, 9, 4 | subadd4b 45294 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝐴 − 𝑋) + (𝐵 − 𝐴)) = ((𝐴 − 𝐴) + (𝐵 − 𝑋))) |
| 11 | 2, 10 | eqtrid 2789 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐴 − 𝑋) + 𝑇) = ((𝐴 − 𝐴) + (𝐵 − 𝑋))) |
| 12 | 4 | subidd 11608 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐴 − 𝐴) = 0) |
| 13 | 12 | oveq1d 7446 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐴 − 𝐴) + (𝐵 − 𝑋)) = (0 + (𝐵 − 𝑋))) |
| 14 | 8, 6 | resubcld 11691 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ) |
| 15 | 14 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℂ) |
| 16 | 15 | addlidd 11462 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (0 + (𝐵 − 𝑋)) = (𝐵 − 𝑋)) |
| 17 | 11, 13, 16 | 3eqtrd 2781 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐴 − 𝑋) + 𝑇) = (𝐵 − 𝑋)) |
| 18 | 1 | oveq2i 7442 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 + 𝑇) = (𝐴 + (𝐵 − 𝐴)) |
| 19 | 4, 9 | pncan3d 11623 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
| 20 | 18, 19 | eqtrid 2789 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐴 + 𝑇) = 𝐵) |
| 21 | 17, 20 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇)) = ((𝐵 − 𝑋)[,]𝐵)) |
| 22 | 21 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐵 − 𝑋)[,]𝐵) = (((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇))) |
| 23 | 22 | itgeq1d 45972 |
. . . . . . . . . . 11
⊢ (𝜑 → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫(((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇))(𝐹‘𝑥) d𝑥) |
| 24 | 3, 6 | resubcld 11691 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ) |
| 25 | | fourierdlem107.o |
. . . . . . . . . . . . 13
⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
| 26 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = 𝑗 → (𝑝‘𝑖) = (𝑝‘𝑗)) |
| 27 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1)) |
| 28 | 27 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = 𝑗 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑗 + 1))) |
| 29 | 26, 28 | breq12d 5156 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 𝑗 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))) |
| 30 | 29 | cbvralvw 3237 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑖 ∈
(0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1))) |
| 31 | 30 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ ℕ →
(∀𝑖 ∈
(0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))) |
| 32 | 31 | anbi2d 630 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ ℕ → ((((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1))))) |
| 33 | 32 | rabbidv 3444 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ℕ → {𝑝 ∈ (ℝ
↑m (0...𝑚))
∣ (((𝑝‘0) =
(𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} = {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))}) |
| 34 | 33 | mpteq2ia 5245 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ
↑m (0...𝑚))
∣ (((𝑝‘0) =
(𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))}) |
| 35 | 25, 34 | eqtri 2765 |
. . . . . . . . . . . 12
⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))}) |
| 36 | | fourierdlem107.p |
. . . . . . . . . . . . . . 15
⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
| 37 | | fourierdlem107.m |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 ∈ ℕ) |
| 38 | | fourierdlem107.q |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) |
| 39 | 3, 5 | ltsubrpd 13109 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴 − 𝑋) < 𝐴) |
| 40 | | fourierdlem107.h |
. . . . . . . . . . . . . . 15
⊢ 𝐻 = ({(𝐴 − 𝑋), 𝐴} ∪ {𝑦 ∈ ((𝐴 − 𝑋)[,]𝐴) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) |
| 41 | | fourierdlem107.n |
. . . . . . . . . . . . . . 15
⊢ 𝑁 = ((♯‘𝐻) − 1) |
| 42 | | fourierdlem107.s |
. . . . . . . . . . . . . . 15
⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻)) |
| 43 | 1, 36, 37, 38, 24, 3, 39, 25, 40, 41, 42 | fourierdlem54 46175 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻))) |
| 44 | 43 | simpld 494 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁))) |
| 45 | 44 | simpld 494 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 46 | 8, 3 | resubcld 11691 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
| 47 | 1, 46 | eqeltrid 2845 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑇 ∈ ℝ) |
| 48 | 44 | simprd 495 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁)) |
| 49 | 24 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐴 − 𝑋) ∈ ℝ) |
| 50 | 3 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝐴 ∈ ℝ) |
| 51 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) |
| 52 | | eliccre 45518 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ) |
| 53 | 49, 50, 51, 52 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ) |
| 54 | | fourierdlem107.fper |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
| 55 | 53, 54 | syldan 591 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
| 56 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑗 → (𝑆‘𝑖) = (𝑆‘𝑗)) |
| 57 | 56 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → ((𝑆‘𝑖) + 𝑇) = ((𝑆‘𝑗) + 𝑇)) |
| 58 | 57 | cbvmptv 5255 |
. . . . . . . . . . . 12
⊢ (𝑖 ∈ (0...𝑁) ↦ ((𝑆‘𝑖) + 𝑇)) = (𝑗 ∈ (0...𝑁) ↦ ((𝑆‘𝑗) + 𝑇)) |
| 59 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ
↑m (0...𝑚))
∣ (((𝑝‘0) =
((𝐴 − 𝑋) + 𝑇) ∧ (𝑝‘𝑚) = (𝐴 + 𝑇)) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))}) = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = ((𝐴 − 𝑋) + 𝑇) ∧ (𝑝‘𝑚) = (𝐴 + 𝑇)) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))}) |
| 60 | | fourierdlem107.f |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹:ℝ⟶ℂ) |
| 61 | 37 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑀 ∈ ℕ) |
| 62 | 38 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑄 ∈ (𝑃‘𝑀)) |
| 63 | 60 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℂ) |
| 64 | 54 | adantlr 715 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
| 65 | | fourierdlem107.fcn |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
| 66 | 65 | adantlr 715 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
| 67 | 24 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴 − 𝑋) ∈ ℝ) |
| 68 | 67 | rexrd 11311 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴 − 𝑋) ∈
ℝ*) |
| 69 | | pnfxr 11315 |
. . . . . . . . . . . . . . 15
⊢ +∞
∈ ℝ* |
| 70 | 69 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → +∞ ∈
ℝ*) |
| 71 | 3 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 ∈ ℝ) |
| 72 | 39 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴 − 𝑋) < 𝐴) |
| 73 | 3 | ltpnfd 13163 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐴 < +∞) |
| 74 | 73 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 < +∞) |
| 75 | 68, 70, 71, 72, 74 | eliood 45511 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 ∈ ((𝐴 − 𝑋)(,)+∞)) |
| 76 | | fourierdlem107.e |
. . . . . . . . . . . . 13
⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) |
| 77 | | fourierdlem107.z |
. . . . . . . . . . . . 13
⊢ 𝑍 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦)) |
| 78 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0..^𝑁)) |
| 79 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))) = ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))) |
| 80 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1))))) = (𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1))))) |
| 81 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (((𝑍‘(𝐸‘(𝑆‘𝑗))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))(,)((𝐸‘(𝑆‘(𝑗 + 1))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))) ↦ ((𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))‘(𝑦 − ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1))))))) = (𝑦 ∈ (((𝑍‘(𝐸‘(𝑆‘𝑗))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))(,)((𝐸‘(𝑆‘(𝑗 + 1))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))) ↦ ((𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))‘(𝑦 − ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1))))))) |
| 82 | | fourierdlem107.i |
. . . . . . . . . . . . 13
⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝑍‘(𝐸‘𝑥))}, ℝ, < )) |
| 83 | 36, 1, 61, 62, 63, 64, 66, 67, 75, 25, 40, 41, 42, 76, 77, 78, 79, 80, 81, 82 | fourierdlem90 46211 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)) |
| 84 | | fourierdlem107.r |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
| 85 | 84 | adantlr 715 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
| 86 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (𝑖 ∈ (0..^𝑀) ↦ 𝑅) = (𝑖 ∈ (0..^𝑀) ↦ 𝑅) |
| 87 | 36, 1, 61, 62, 63, 64, 66, 85, 67, 75, 25, 40, 41, 42, 76, 77, 78, 79, 82, 86 | fourierdlem89 46210 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → if((𝑍‘(𝐸‘(𝑆‘𝑗))) = (𝑄‘(𝐼‘(𝑆‘𝑗))), ((𝑖 ∈ (0..^𝑀) ↦ 𝑅)‘(𝐼‘(𝑆‘𝑗))), (𝐹‘(𝑍‘(𝐸‘(𝑆‘𝑗))))) ∈ ((𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) limℂ (𝑆‘𝑗))) |
| 88 | | fourierdlem107.l |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) |
| 89 | 88 | adantlr 715 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) |
| 90 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (𝑖 ∈ (0..^𝑀) ↦ 𝐿) = (𝑖 ∈ (0..^𝑀) ↦ 𝐿) |
| 91 | 36, 1, 61, 62, 63, 64, 66, 89, 67, 75, 25, 40, 41, 42, 76, 77, 78, 79, 82, 90 | fourierdlem91 46212 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → if((𝐸‘(𝑆‘(𝑗 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)), ((𝑖 ∈ (0..^𝑀) ↦ 𝐿)‘(𝐼‘(𝑆‘𝑗))), (𝐹‘(𝐸‘(𝑆‘(𝑗 + 1))))) ∈ ((𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) limℂ (𝑆‘(𝑗 + 1)))) |
| 92 | 24, 3, 35, 45, 47, 48, 55, 58, 59, 60, 83, 87, 91 | fourierdlem92 46213 |
. . . . . . . . . . 11
⊢ (𝜑 → ∫(((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) |
| 93 | 23, 92 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (𝜑 → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) |
| 94 | 60 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝐹:ℝ⟶ℂ) |
| 95 | 14 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → (𝐵 − 𝑋) ∈ ℝ) |
| 96 | 8 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝐵 ∈ ℝ) |
| 97 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) |
| 98 | | eliccre 45518 |
. . . . . . . . . . . . 13
⊢ (((𝐵 − 𝑋) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝑥 ∈ ℝ) |
| 99 | 95, 96, 97, 98 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝑥 ∈ ℝ) |
| 100 | 94, 99 | ffvelcdmd 7105 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ) |
| 101 | 14 | rexrd 11311 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐵 − 𝑋) ∈
ℝ*) |
| 102 | 69 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → +∞ ∈
ℝ*) |
| 103 | 8, 5 | ltsubrpd 13109 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐵 − 𝑋) < 𝐵) |
| 104 | 8 | ltpnfd 13163 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐵 < +∞) |
| 105 | 101, 102,
8, 103, 104 | eliood 45511 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 ∈ ((𝐵 − 𝑋)(,)+∞)) |
| 106 | 36, 1, 37, 38, 60, 54, 65, 84, 88, 14, 105 | fourierdlem105 46226 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵) ↦ (𝐹‘𝑥)) ∈
𝐿1) |
| 107 | 100, 106 | itgcl 25819 |
. . . . . . . . . 10
⊢ (𝜑 → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ) |
| 108 | 93, 107 | eqeltrrd 2842 |
. . . . . . . . 9
⊢ (𝜑 → ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ) |
| 109 | 108 | subidd 11608 |
. . . . . . . 8
⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = 0) |
| 110 | 109 | eqcomd 2743 |
. . . . . . 7
⊢ (𝜑 → 0 = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) |
| 111 | 110 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 0 = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) |
| 112 | 24 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐴 − 𝑋) ∈ ℝ) |
| 113 | 3 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ∈ ℝ) |
| 114 | 14 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ∈ ℝ) |
| 115 | 36, 37, 38 | fourierdlem11 46133 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)) |
| 116 | 115 | simp3d 1145 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 < 𝐵) |
| 117 | 3, 8, 116 | ltled 11409 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| 118 | 117 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ≤ 𝐵) |
| 119 | 3, 8, 6 | lesub1d 11870 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝑋) ≤ (𝐵 − 𝑋))) |
| 120 | 119 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝑋) ≤ (𝐵 − 𝑋))) |
| 121 | 118, 120 | mpbid 232 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐴 − 𝑋) ≤ (𝐵 − 𝑋)) |
| 122 | 8 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐵 ∈ ℝ) |
| 123 | 6 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑋 ∈ ℝ) |
| 124 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑇 < 𝑋) |
| 125 | 1, 124 | eqbrtrrid 5179 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝐴) < 𝑋) |
| 126 | 122, 113,
123, 125 | ltsub23d 11868 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) < 𝐴) |
| 127 | 114, 113,
126 | ltled 11409 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ≤ 𝐴) |
| 128 | 112, 113,
114, 121, 127 | eliccd 45517 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ∈ ((𝐴 − 𝑋)[,]𝐴)) |
| 129 | 60 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝐹:ℝ⟶ℂ) |
| 130 | 129, 53 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐹‘𝑥) ∈ ℂ) |
| 131 | 130 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐹‘𝑥) ∈ ℂ) |
| 132 | 24 | rexrd 11311 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴 − 𝑋) ∈
ℝ*) |
| 133 | 3, 8, 6, 116 | ltsub1dd 11875 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴 − 𝑋) < (𝐵 − 𝑋)) |
| 134 | 14 | ltpnfd 13163 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐵 − 𝑋) < +∞) |
| 135 | 132, 102,
14, 133, 134 | eliood 45511 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐵 − 𝑋) ∈ ((𝐴 − 𝑋)(,)+∞)) |
| 136 | 36, 1, 37, 38, 60, 54, 65, 84, 88, 24, 135 | fourierdlem105 46226 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↦ (𝐹‘𝑥)) ∈
𝐿1) |
| 137 | 136 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↦ (𝐹‘𝑥)) ∈
𝐿1) |
| 138 | 37 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑀 ∈ ℕ) |
| 139 | 38 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑄 ∈ (𝑃‘𝑀)) |
| 140 | 60 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐹:ℝ⟶ℂ) |
| 141 | 54 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
| 142 | 65 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
| 143 | 84 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
| 144 | 88 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) |
| 145 | 101 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ∈
ℝ*) |
| 146 | 69 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → +∞ ∈
ℝ*) |
| 147 | 113 | ltpnfd 13163 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 < +∞) |
| 148 | 145, 146,
113, 126, 147 | eliood 45511 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ∈ ((𝐵 − 𝑋)(,)+∞)) |
| 149 | 36, 1, 138, 139, 140, 141, 142, 143, 144, 114, 148 | fourierdlem105 46226 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴) ↦ (𝐹‘𝑥)) ∈
𝐿1) |
| 150 | 112, 113,
128, 131, 137, 149 | itgspliticc 25872 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 = (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) |
| 151 | 150 | oveq1d 7446 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = ((∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) |
| 152 | 60 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐹:ℝ⟶ℂ) |
| 153 | 24 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈ ℝ) |
| 154 | 14 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ) |
| 155 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) |
| 156 | | eliccre 45518 |
. . . . . . . . . . 11
⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ) |
| 157 | 153, 154,
155, 156 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ) |
| 158 | 152, 157 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘𝑥) ∈ ℂ) |
| 159 | 158, 136 | itgcl 25819 |
. . . . . . . 8
⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 ∈ ℂ) |
| 160 | 159 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 ∈ ℂ) |
| 161 | 60 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝐹:ℝ⟶ℂ) |
| 162 | 14 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → (𝐵 − 𝑋) ∈ ℝ) |
| 163 | 3 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝐴 ∈ ℝ) |
| 164 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) |
| 165 | | eliccre 45518 |
. . . . . . . . . . 11
⊢ (((𝐵 − 𝑋) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ) |
| 166 | 162, 163,
164, 165 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ) |
| 167 | 161, 166 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → (𝐹‘𝑥) ∈ ℂ) |
| 168 | 167 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → (𝐹‘𝑥) ∈ ℂ) |
| 169 | 168, 149 | itgcl 25819 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ) |
| 170 | 108 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ) |
| 171 | 160, 169,
170 | addsubassd 11640 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ((∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))) |
| 172 | 111, 151,
171 | 3eqtrd 2781 |
. . . . 5
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 0 = (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))) |
| 173 | 172 | oveq2d 7447 |
. . . 4
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − 0) = (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)))) |
| 174 | 160 | subid1d 11609 |
. . . 4
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − 0) = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) |
| 175 | 159 | subidd 11608 |
. . . . . . 7
⊢ (𝜑 → (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) = 0) |
| 176 | 175 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 → ((∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) = (0 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))) |
| 177 | 176 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ((∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) = (0 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))) |
| 178 | 169, 170 | subcld 11620 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) ∈ ℂ) |
| 179 | 160, 160,
178 | subsub4d 11651 |
. . . . 5
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ((∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) = (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)))) |
| 180 | | df-neg 11495 |
. . . . . 6
⊢
-(∫((𝐵 −
𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = (0 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) |
| 181 | 169, 170 | negsubdi2d 11636 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → -(∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) |
| 182 | 180, 181 | eqtr3id 2791 |
. . . . 5
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (0 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) |
| 183 | 177, 179,
182 | 3eqtr3d 2785 |
. . . 4
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))) = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) |
| 184 | 173, 174,
183 | 3eqtr3d 2785 |
. . 3
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) |
| 185 | 107 | subidd 11608 |
. . . . . . . 8
⊢ (𝜑 → (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = 0) |
| 186 | 185 | eqcomd 2743 |
. . . . . . 7
⊢ (𝜑 → 0 = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 187 | 186 | oveq2d 7447 |
. . . . . 6
⊢ (𝜑 → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + 0) = (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))) |
| 188 | 187 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + 0) = (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))) |
| 189 | 169 | addridd 11461 |
. . . . 5
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + 0) = ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) |
| 190 | 114, 122,
113, 127, 118 | eliccd 45517 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ∈ ((𝐵 − 𝑋)[,]𝐵)) |
| 191 | 100 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ) |
| 192 | 3, 8 | iccssred 13474 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
| 193 | 60, 192 | feqresmpt 6978 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥))) |
| 194 | 60, 192 | fssresd 6775 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℂ) |
| 195 | | ioossicc 13473 |
. . . . . . . . . . . . . . 15
⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) |
| 196 | 3 | rexrd 11311 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 197 | 196 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈
ℝ*) |
| 198 | 8 | rexrd 11311 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 199 | 198 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐵 ∈
ℝ*) |
| 200 | 36, 37, 38 | fourierdlem15 46137 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) |
| 201 | 200 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) |
| 202 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀)) |
| 203 | 197, 199,
201, 202 | fourierdlem8 46130 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵)) |
| 204 | 195, 203 | sstrid 3995 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵)) |
| 205 | 204 | resabs1d 6026 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) |
| 206 | 205, 65 | eqeltrd 2841 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
| 207 | 205 | eqcomd 2743 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) |
| 208 | 207 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)) = (((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
| 209 | 84, 208 | eleqtrd 2843 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ (((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
| 210 | 207 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1))) = (((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) |
| 211 | 88, 210 | eleqtrd 2843 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ (((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) |
| 212 | 36, 37, 38, 194, 206, 209, 211 | fourierdlem69 46190 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈
𝐿1) |
| 213 | 193, 212 | eqeltrrd 2842 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈
𝐿1) |
| 214 | 213 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈
𝐿1) |
| 215 | 114, 122,
190, 191, 149, 214 | itgspliticc 25872 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 216 | 215 | oveq2d 7447 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))) |
| 217 | 216 | oveq2d 7447 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) = (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)))) |
| 218 | 107 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ) |
| 219 | 215, 218 | eqeltrrd 2842 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) ∈ ℂ) |
| 220 | 169, 218,
219 | addsub12d 11643 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))) = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)))) |
| 221 | 60 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℂ) |
| 222 | 3 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ) |
| 223 | 8 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ) |
| 224 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵)) |
| 225 | | eliccre 45518 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ) |
| 226 | 222, 223,
224, 225 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ) |
| 227 | 221, 226 | ffvelcdmd 7105 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ) |
| 228 | 227, 213 | itgcl 25819 |
. . . . . . . . . . 11
⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ) |
| 229 | 228 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ) |
| 230 | 169, 169,
229 | subsub4d 11651 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ((∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) = (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))) |
| 231 | 230 | eqcomd 2743 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) = ((∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 232 | 231 | oveq2d 7447 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))) = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 + ((∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))) |
| 233 | 169 | subidd 11608 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = 0) |
| 234 | 233 | oveq1d 7446 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ((∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) = (0 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 235 | | df-neg 11495 |
. . . . . . . . 9
⊢
-∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = (0 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) |
| 236 | 234, 235 | eqtr4di 2795 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ((∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) = -∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) |
| 237 | 236 | oveq2d 7447 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 + ((∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 + -∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 238 | 218, 229 | negsubd 11626 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 + -∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 239 | 232, 237,
238 | 3eqtrd 2781 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))) = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 240 | 217, 220,
239 | 3eqtrd 2781 |
. . . . 5
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 241 | 188, 189,
240 | 3eqtr3d 2785 |
. . . 4
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 242 | 241 | oveq2d 7447 |
. . 3
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))) |
| 243 | 108, 107,
228 | subsubd 11648 |
. . . . 5
⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) = ((∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 244 | 93 | oveq2d 7447 |
. . . . . . 7
⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) |
| 245 | 244, 109 | eqtrd 2777 |
. . . . . 6
⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = 0) |
| 246 | 245 | oveq1d 7446 |
. . . . 5
⊢ (𝜑 → ((∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) = (0 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 247 | 228 | addlidd 11462 |
. . . . 5
⊢ (𝜑 → (0 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) |
| 248 | 243, 246,
247 | 3eqtrd 2781 |
. . . 4
⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) |
| 249 | 248 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) |
| 250 | 184, 242,
249 | 3eqtrd 2781 |
. 2
⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) |
| 251 | 24 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐴 − 𝑋) ∈ ℝ) |
| 252 | 14 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐵 − 𝑋) ∈ ℝ) |
| 253 | 3 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐴 ∈ ℝ) |
| 254 | 24, 3, 39 | ltled 11409 |
. . . . . . 7
⊢ (𝜑 → (𝐴 − 𝑋) ≤ 𝐴) |
| 255 | 254 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐴 − 𝑋) ≤ 𝐴) |
| 256 | 6 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝑋 ∈ ℝ) |
| 257 | 8 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐵 ∈ ℝ) |
| 258 | | id 22 |
. . . . . . . . 9
⊢ (𝑋 ≤ 𝑇 → 𝑋 ≤ 𝑇) |
| 259 | 258, 1 | breqtrdi 5184 |
. . . . . . . 8
⊢ (𝑋 ≤ 𝑇 → 𝑋 ≤ (𝐵 − 𝐴)) |
| 260 | 259 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝑋 ≤ (𝐵 − 𝐴)) |
| 261 | 256, 257,
253, 260 | lesubd 11867 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐴 ≤ (𝐵 − 𝑋)) |
| 262 | 251, 252,
253, 255, 261 | eliccd 45517 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐴 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) |
| 263 | 158 | adantlr 715 |
. . . . 5
⊢ (((𝜑 ∧ 𝑋 ≤ 𝑇) ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘𝑥) ∈ ℂ) |
| 264 | 132, 102,
3, 39, 73 | eliood 45511 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ((𝐴 − 𝑋)(,)+∞)) |
| 265 | 36, 1, 37, 38, 60, 54, 65, 84, 88, 24, 264 | fourierdlem105 46226 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴) ↦ (𝐹‘𝑥)) ∈
𝐿1) |
| 266 | 265 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴) ↦ (𝐹‘𝑥)) ∈
𝐿1) |
| 267 | 3 | leidd 11829 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ≤ 𝐴) |
| 268 | 5 | rpge0d 13081 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ 𝑋) |
| 269 | 8, 6 | subge02d 11855 |
. . . . . . . . 9
⊢ (𝜑 → (0 ≤ 𝑋 ↔ (𝐵 − 𝑋) ≤ 𝐵)) |
| 270 | 268, 269 | mpbid 232 |
. . . . . . . 8
⊢ (𝜑 → (𝐵 − 𝑋) ≤ 𝐵) |
| 271 | | iccss 13455 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐴 ∧ (𝐵 − 𝑋) ≤ 𝐵)) → (𝐴[,](𝐵 − 𝑋)) ⊆ (𝐴[,]𝐵)) |
| 272 | 3, 8, 267, 270, 271 | syl22anc 839 |
. . . . . . 7
⊢ (𝜑 → (𝐴[,](𝐵 − 𝑋)) ⊆ (𝐴[,]𝐵)) |
| 273 | | iccmbl 25601 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ) → (𝐴[,](𝐵 − 𝑋)) ∈ dom vol) |
| 274 | 3, 14, 273 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → (𝐴[,](𝐵 − 𝑋)) ∈ dom vol) |
| 275 | 272, 274,
227, 213 | iblss 25840 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (𝐴[,](𝐵 − 𝑋)) ↦ (𝐹‘𝑥)) ∈
𝐿1) |
| 276 | 275 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝑥 ∈ (𝐴[,](𝐵 − 𝑋)) ↦ (𝐹‘𝑥)) ∈
𝐿1) |
| 277 | 251, 252,
262, 263, 266, 276 | itgspliticc 25872 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥)) |
| 278 | 268 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 0 ≤ 𝑋) |
| 279 | 269 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (0 ≤ 𝑋 ↔ (𝐵 − 𝑋) ≤ 𝐵)) |
| 280 | 278, 279 | mpbid 232 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐵 − 𝑋) ≤ 𝐵) |
| 281 | 253, 257,
252, 261, 280 | eliccd 45517 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐵 − 𝑋) ∈ (𝐴[,]𝐵)) |
| 282 | 227 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑋 ≤ 𝑇) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ) |
| 283 | 8 | leidd 11829 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 ≤ 𝐵) |
| 284 | 283 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐵 ≤ 𝐵) |
| 285 | | iccss 13455 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ (𝐵 − 𝑋) ∧ 𝐵 ≤ 𝐵)) → ((𝐵 − 𝑋)[,]𝐵) ⊆ (𝐴[,]𝐵)) |
| 286 | 253, 257,
261, 284, 285 | syl22anc 839 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((𝐵 − 𝑋)[,]𝐵) ⊆ (𝐴[,]𝐵)) |
| 287 | | iccmbl 25601 |
. . . . . . . . . . 11
⊢ (((𝐵 − 𝑋) ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 − 𝑋)[,]𝐵) ∈ dom vol) |
| 288 | 14, 8, 287 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐵 − 𝑋)[,]𝐵) ∈ dom vol) |
| 289 | 288 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((𝐵 − 𝑋)[,]𝐵) ∈ dom vol) |
| 290 | 213 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈
𝐿1) |
| 291 | 286, 289,
282, 290 | iblss 25840 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵) ↦ (𝐹‘𝑥)) ∈
𝐿1) |
| 292 | 253, 257,
281, 282, 276, 291 | itgspliticc 25872 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 293 | 292 | oveq1d 7446 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = ((∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 294 | 60 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝐹:ℝ⟶ℂ) |
| 295 | 3 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝐴 ∈ ℝ) |
| 296 | 14 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ) |
| 297 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) |
| 298 | | eliccre 45518 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ) |
| 299 | 295, 296,
297, 298 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ) |
| 300 | 294, 299 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → (𝐹‘𝑥) ∈ ℂ) |
| 301 | 300, 275 | itgcl 25819 |
. . . . . . . 8
⊢ (𝜑 → ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 ∈ ℂ) |
| 302 | 301, 107,
107 | addsubassd 11640 |
. . . . . . 7
⊢ (𝜑 → ((∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))) |
| 303 | 302 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))) |
| 304 | 185 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝜑 → (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) = (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + 0)) |
| 305 | 301 | addridd 11461 |
. . . . . . . 8
⊢ (𝜑 → (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + 0) = ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) |
| 306 | 304, 305 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) = ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) |
| 307 | 306 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) = ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) |
| 308 | 293, 303,
307 | 3eqtrrd 2782 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 309 | 308 | oveq2d 7447 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))) |
| 310 | 93 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) |
| 311 | 107 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ) |
| 312 | 310, 311 | eqeltrrd 2842 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ) |
| 313 | 282, 290 | itgcl 25819 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ) |
| 314 | 312, 313,
311 | addsub12d 11643 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) = (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))) |
| 315 | 313, 312,
311 | addsubassd 11640 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))) |
| 316 | 314, 315 | eqtr4d 2780 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) = ((∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 317 | 277, 309,
316 | 3eqtrd 2781 |
. . 3
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ((∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) |
| 318 | 310 | oveq2d 7447 |
. . 3
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = ((∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) |
| 319 | 313, 312 | pncand 11621 |
. . 3
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) |
| 320 | 317, 318,
319 | 3eqtrd 2781 |
. 2
⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) |
| 321 | 250, 320,
47, 6 | ltlecasei 11369 |
1
⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) |