| Step | Hyp | Ref
| Expression |
| 1 | | fourierdlem75.xre |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ ℝ) |
| 2 | 1 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝑋 ∈ ℝ) |
| 3 | | fourierdlem75.v |
. . . . . . . . . 10
⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) |
| 4 | | fourierdlem75.m |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℕ) |
| 5 | | fourierdlem75.p |
. . . . . . . . . . . 12
⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
| 6 | 5 | fourierdlem2 46124 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ ℕ → (𝑉 ∈ (𝑃‘𝑀) ↔ (𝑉 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉‘𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))))) |
| 7 | 4, 6 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑉 ∈ (𝑃‘𝑀) ↔ (𝑉 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉‘𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))))) |
| 8 | 3, 7 | mpbid 232 |
. . . . . . . . 9
⊢ (𝜑 → (𝑉 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑉‘0) = (-π + 𝑋) ∧ (𝑉‘𝑀) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1))))) |
| 9 | 8 | simpld 494 |
. . . . . . . 8
⊢ (𝜑 → 𝑉 ∈ (ℝ ↑m
(0...𝑀))) |
| 10 | | elmapi 8889 |
. . . . . . . 8
⊢ (𝑉 ∈ (ℝ
↑m (0...𝑀))
→ 𝑉:(0...𝑀)⟶ℝ) |
| 11 | 9, 10 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑉:(0...𝑀)⟶ℝ) |
| 12 | 11 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶ℝ) |
| 13 | | fzofzp1 13803 |
. . . . . . 7
⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀)) |
| 14 | 13 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀)) |
| 15 | 12, 14 | ffvelcdmd 7105 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ) |
| 16 | 15 | adantr 480 |
. . . 4
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑉‘(𝑖 + 1)) ∈ ℝ) |
| 17 | | eqcom 2744 |
. . . . . . 7
⊢ ((𝑉‘𝑖) = 𝑋 ↔ 𝑋 = (𝑉‘𝑖)) |
| 18 | 17 | biimpi 216 |
. . . . . 6
⊢ ((𝑉‘𝑖) = 𝑋 → 𝑋 = (𝑉‘𝑖)) |
| 19 | 18 | adantl 481 |
. . . . 5
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝑋 = (𝑉‘𝑖)) |
| 20 | 8 | simprrd 774 |
. . . . . . 7
⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1))) |
| 21 | 20 | r19.21bi 3251 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) < (𝑉‘(𝑖 + 1))) |
| 22 | 21 | adantr 480 |
. . . . 5
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑉‘𝑖) < (𝑉‘(𝑖 + 1))) |
| 23 | 19, 22 | eqbrtrd 5165 |
. . . 4
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝑋 < (𝑉‘(𝑖 + 1))) |
| 24 | | fourierdlem75.f |
. . . . . . 7
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
| 25 | 24 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℝ) |
| 26 | | ioossre 13448 |
. . . . . . 7
⊢ (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ |
| 27 | 26 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ) |
| 28 | 25, 27 | fssresd 6775 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):(𝑋(,)(𝑉‘(𝑖 + 1)))⟶ℝ) |
| 29 | 28 | adantr 480 |
. . . 4
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):(𝑋(,)(𝑉‘(𝑖 + 1)))⟶ℝ) |
| 30 | | limcresi 25920 |
. . . . . . . 8
⊢ ((𝐹 ↾ (𝑋(,)+∞)) limℂ 𝑋) ⊆ (((𝐹 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) limℂ 𝑋) |
| 31 | | fourierdlem75.y |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) limℂ 𝑋)) |
| 32 | 30, 31 | sselid 3981 |
. . . . . . 7
⊢ (𝜑 → 𝑌 ∈ (((𝐹 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) limℂ 𝑋)) |
| 33 | 32 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ (((𝐹 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) limℂ 𝑋)) |
| 34 | | pnfxr 11315 |
. . . . . . . . . 10
⊢ +∞
∈ ℝ* |
| 35 | 34 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → +∞ ∈
ℝ*) |
| 36 | 15 | rexrd 11311 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈
ℝ*) |
| 37 | 15 | ltpnfd 13163 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) < +∞) |
| 38 | 36, 35, 37 | xrltled 13192 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ≤ +∞) |
| 39 | | iooss2 13423 |
. . . . . . . . 9
⊢
((+∞ ∈ ℝ* ∧ (𝑉‘(𝑖 + 1)) ≤ +∞) → (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ (𝑋(,)+∞)) |
| 40 | 35, 38, 39 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ (𝑋(,)+∞)) |
| 41 | 40 | resabs1d 6026 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) = (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) |
| 42 | 41 | oveq1d 7446 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐹 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) limℂ 𝑋) = ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) limℂ 𝑋)) |
| 43 | 33, 42 | eleqtrd 2843 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) limℂ 𝑋)) |
| 44 | 43 | adantr 480 |
. . . 4
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) limℂ 𝑋)) |
| 45 | | eqid 2737 |
. . . 4
⊢ (ℝ
D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) |
| 46 | | ax-resscn 11212 |
. . . . . . . . . . 11
⊢ ℝ
⊆ ℂ |
| 47 | 46 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℝ ⊆
ℂ) |
| 48 | 24, 47 | fssd 6753 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:ℝ⟶ℂ) |
| 49 | | ssid 4006 |
. . . . . . . . . . 11
⊢ ℝ
⊆ ℝ |
| 50 | 49 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℝ ⊆
ℝ) |
| 51 | 26 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ) |
| 52 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
| 53 | | tgioo4 24826 |
. . . . . . . . . . 11
⊢
(topGen‘ran (,)) = ((TopOpen‘ℂfld)
↾t ℝ) |
| 54 | 52, 53 | dvres 25946 |
. . . . . . . . . 10
⊢
(((ℝ ⊆ ℂ ∧ 𝐹:ℝ⟶ℂ) ∧ (ℝ
⊆ ℝ ∧ (𝑋(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ)) → (ℝ D
(𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran
(,)))‘(𝑋(,)(𝑉‘(𝑖 + 1)))))) |
| 55 | 47, 48, 50, 51, 54 | syl22anc 839 |
. . . . . . . . 9
⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran
(,)))‘(𝑋(,)(𝑉‘(𝑖 + 1)))))) |
| 56 | | fourierdlem75.g |
. . . . . . . . . . 11
⊢ 𝐺 = (ℝ D 𝐹) |
| 57 | 56 | eqcomi 2746 |
. . . . . . . . . 10
⊢ (ℝ
D 𝐹) = 𝐺 |
| 58 | | ioontr 45524 |
. . . . . . . . . 10
⊢
((int‘(topGen‘ran (,)))‘(𝑋(,)(𝑉‘(𝑖 + 1)))) = (𝑋(,)(𝑉‘(𝑖 + 1))) |
| 59 | 57, 58 | reseq12i 5995 |
. . . . . . . . 9
⊢ ((ℝ
D 𝐹) ↾
((int‘(topGen‘ran (,)))‘(𝑋(,)(𝑉‘(𝑖 + 1))))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) |
| 60 | 55, 59 | eqtrdi 2793 |
. . . . . . . 8
⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) |
| 61 | 60 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑉‘𝑖) = 𝑋) → (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) |
| 62 | 61 | dmeqd 5916 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑉‘𝑖) = 𝑋) → dom (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = dom (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) |
| 63 | 62 | adantlr 715 |
. . . . 5
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → dom (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = dom (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) |
| 64 | | fourierdlem75.gcn |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ) |
| 65 | 64 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝐺 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ) |
| 66 | | oveq1 7438 |
. . . . . . . . . . 11
⊢ ((𝑉‘𝑖) = 𝑋 → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) = (𝑋(,)(𝑉‘(𝑖 + 1)))) |
| 67 | 66 | reseq2d 5997 |
. . . . . . . . . 10
⊢ ((𝑉‘𝑖) = 𝑋 → (𝐺 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) |
| 68 | 67 | feq1d 6720 |
. . . . . . . . 9
⊢ ((𝑉‘𝑖) = 𝑋 → ((𝐺 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ ↔ (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)) |
| 69 | 68 | adantl 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → ((𝐺 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ ↔ (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)) |
| 70 | 65, 69 | mpbid 232 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ) |
| 71 | 66 | adantl 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) = (𝑋(,)(𝑉‘(𝑖 + 1)))) |
| 72 | 71 | feq2d 6722 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → ((𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ ↔ (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):(𝑋(,)(𝑉‘(𝑖 + 1)))⟶ℂ)) |
| 73 | 70, 72 | mpbid 232 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):(𝑋(,)(𝑉‘(𝑖 + 1)))⟶ℂ) |
| 74 | | fdm 6745 |
. . . . . 6
⊢ ((𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))):(𝑋(,)(𝑉‘(𝑖 + 1)))⟶ℂ → dom (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) = (𝑋(,)(𝑉‘(𝑖 + 1)))) |
| 75 | 73, 74 | syl 17 |
. . . . 5
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → dom (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) = (𝑋(,)(𝑉‘(𝑖 + 1)))) |
| 76 | 63, 75 | eqtrd 2777 |
. . . 4
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → dom (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = (𝑋(,)(𝑉‘(𝑖 + 1)))) |
| 77 | | limcresi 25920 |
. . . . . . . 8
⊢ ((𝐺 ↾ (𝑋(,)+∞)) limℂ 𝑋) ⊆ (((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) limℂ 𝑋) |
| 78 | | fourierdlem75.e |
. . . . . . . 8
⊢ (𝜑 → 𝐸 ∈ ((𝐺 ↾ (𝑋(,)+∞)) limℂ 𝑋)) |
| 79 | 77, 78 | sselid 3981 |
. . . . . . 7
⊢ (𝜑 → 𝐸 ∈ (((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) limℂ 𝑋)) |
| 80 | 79 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐸 ∈ (((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) limℂ 𝑋)) |
| 81 | 40 | resabs1d 6026 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) |
| 82 | 60 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) = (𝐺 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) |
| 83 | 81, 82 | eqtr4d 2780 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) = (ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))))) |
| 84 | 83 | oveq1d 7446 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝐺 ↾ (𝑋(,)+∞)) ↾ (𝑋(,)(𝑉‘(𝑖 + 1)))) limℂ 𝑋) = ((ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) limℂ 𝑋)) |
| 85 | 80, 84 | eleqtrd 2843 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐸 ∈ ((ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) limℂ 𝑋)) |
| 86 | 85 | adantr 480 |
. . . 4
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝐸 ∈ ((ℝ D (𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))) limℂ 𝑋)) |
| 87 | | eqid 2737 |
. . . 4
⊢ (𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠)) = (𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠)) |
| 88 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑥 = 𝑠 → (𝑋 + 𝑥) = (𝑋 + 𝑠)) |
| 89 | 88 | fveq2d 6910 |
. . . . . 6
⊢ (𝑥 = 𝑠 → ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑥)) = ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠))) |
| 90 | 89 | oveq1d 7446 |
. . . . 5
⊢ (𝑥 = 𝑠 → (((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑥)) − 𝑌) = (((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌)) |
| 91 | 90 | cbvmptv 5255 |
. . . 4
⊢ (𝑥 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ (((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑥)) − 𝑌)) = (𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ (((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌)) |
| 92 | | id 22 |
. . . . 5
⊢ (𝑥 = 𝑠 → 𝑥 = 𝑠) |
| 93 | 92 | cbvmptv 5255 |
. . . 4
⊢ (𝑥 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ 𝑥) = (𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ 𝑠) |
| 94 | 2, 16, 23, 29, 44, 45, 76, 86, 87, 91, 93 | fourierdlem61 46182 |
. . 3
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝐸 ∈ ((𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠)) limℂ 0)) |
| 95 | | fourierdlem75.a |
. . . . 5
⊢ 𝐴 = if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) |
| 96 | | iftrue 4531 |
. . . . 5
⊢ ((𝑉‘𝑖) = 𝑋 → if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) = 𝐸) |
| 97 | 95, 96 | eqtrid 2789 |
. . . 4
⊢ ((𝑉‘𝑖) = 𝑋 → 𝐴 = 𝐸) |
| 98 | 97 | adantl 481 |
. . 3
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝐴 = 𝐸) |
| 99 | | fourierdlem75.h |
. . . . . . 7
⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) |
| 100 | 99 | reseq1i 5993 |
. . . . . 6
⊢ (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
| 101 | 100 | a1i 11 |
. . . . 5
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) |
| 102 | | ioossicc 13473 |
. . . . . . . 8
⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) |
| 103 | | pire 26500 |
. . . . . . . . . . . 12
⊢ π
∈ ℝ |
| 104 | 103 | renegcli 11570 |
. . . . . . . . . . 11
⊢ -π
∈ ℝ |
| 105 | 104 | rexri 11319 |
. . . . . . . . . 10
⊢ -π
∈ ℝ* |
| 106 | 105 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -π ∈
ℝ*) |
| 107 | 103 | rexri 11319 |
. . . . . . . . . 10
⊢ π
∈ ℝ* |
| 108 | 107 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → π ∈
ℝ*) |
| 109 | 104 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → -π ∈
ℝ) |
| 110 | 103 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → π ∈
ℝ) |
| 111 | 104 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → -π ∈
ℝ) |
| 112 | 111, 1 | readdcld 11290 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (-π + 𝑋) ∈ ℝ) |
| 113 | 103 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → π ∈
ℝ) |
| 114 | 113, 1 | readdcld 11290 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (π + 𝑋) ∈ ℝ) |
| 115 | 112, 114 | iccssred 13474 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((-π + 𝑋)[,](π + 𝑋)) ⊆ ℝ) |
| 116 | 115 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((-π + 𝑋)[,](π + 𝑋)) ⊆ ℝ) |
| 117 | 5, 4, 3 | fourierdlem15 46137 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑉:(0...𝑀)⟶((-π + 𝑋)[,](π + 𝑋))) |
| 118 | 117 | ffvelcdmda 7104 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑉‘𝑖) ∈ ((-π + 𝑋)[,](π + 𝑋))) |
| 119 | 116, 118 | sseldd 3984 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑉‘𝑖) ∈ ℝ) |
| 120 | 1 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ) |
| 121 | 119, 120 | resubcld 11691 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ) |
| 122 | 111 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → -π ∈
ℂ) |
| 123 | 1 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑋 ∈ ℂ) |
| 124 | 122, 123 | pncand 11621 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((-π + 𝑋) − 𝑋) = -π) |
| 125 | 124 | eqcomd 2743 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → -π = ((-π + 𝑋) − 𝑋)) |
| 126 | 125 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → -π = ((-π + 𝑋) − 𝑋)) |
| 127 | 112 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (-π + 𝑋) ∈ ℝ) |
| 128 | 114 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (π + 𝑋) ∈ ℝ) |
| 129 | | elicc2 13452 |
. . . . . . . . . . . . . . . . 17
⊢ (((-π
+ 𝑋) ∈ ℝ ∧
(π + 𝑋) ∈ ℝ)
→ ((𝑉‘𝑖) ∈ ((-π + 𝑋)[,](π + 𝑋)) ↔ ((𝑉‘𝑖) ∈ ℝ ∧ (-π + 𝑋) ≤ (𝑉‘𝑖) ∧ (𝑉‘𝑖) ≤ (π + 𝑋)))) |
| 130 | 127, 128,
129 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) ∈ ((-π + 𝑋)[,](π + 𝑋)) ↔ ((𝑉‘𝑖) ∈ ℝ ∧ (-π + 𝑋) ≤ (𝑉‘𝑖) ∧ (𝑉‘𝑖) ≤ (π + 𝑋)))) |
| 131 | 118, 130 | mpbid 232 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) ∈ ℝ ∧ (-π + 𝑋) ≤ (𝑉‘𝑖) ∧ (𝑉‘𝑖) ≤ (π + 𝑋))) |
| 132 | 131 | simp2d 1144 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (-π + 𝑋) ≤ (𝑉‘𝑖)) |
| 133 | 127, 119,
120, 132 | lesub1dd 11879 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((-π + 𝑋) − 𝑋) ≤ ((𝑉‘𝑖) − 𝑋)) |
| 134 | 126, 133 | eqbrtrd 5165 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → -π ≤ ((𝑉‘𝑖) − 𝑋)) |
| 135 | 131 | simp3d 1145 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑉‘𝑖) ≤ (π + 𝑋)) |
| 136 | 119, 128,
120, 135 | lesub1dd 11879 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) − 𝑋) ≤ ((π + 𝑋) − 𝑋)) |
| 137 | 110 | recnd 11289 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → π ∈
ℂ) |
| 138 | 123 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℂ) |
| 139 | 137, 138 | pncand 11621 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((π + 𝑋) − 𝑋) = π) |
| 140 | 136, 139 | breqtrd 5169 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) − 𝑋) ≤ π) |
| 141 | 109, 110,
121, 134, 140 | eliccd 45517 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ (-π[,]π)) |
| 142 | | fourierdlem75.q |
. . . . . . . . . . 11
⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) |
| 143 | 141, 142 | fmptd 7134 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄:(0...𝑀)⟶(-π[,]π)) |
| 144 | 143 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(-π[,]π)) |
| 145 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀)) |
| 146 | 106, 108,
144, 145 | fourierdlem8 46130 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆
(-π[,]π)) |
| 147 | 102, 146 | sstrid 3995 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆
(-π[,]π)) |
| 148 | 147 | resmptd 6058 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))) |
| 149 | 148 | adantr 480 |
. . . . 5
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))) |
| 150 | | elfzofz 13715 |
. . . . . . . 8
⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀)) |
| 151 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀)) |
| 152 | 142 | fvmpt2 7027 |
. . . . . . . . . . 11
⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑉‘𝑖) − 𝑋) ∈ (-π[,]π)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋)) |
| 153 | 151, 141,
152 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋)) |
| 154 | 153 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋)) |
| 155 | | oveq1 7438 |
. . . . . . . . . 10
⊢ ((𝑉‘𝑖) = 𝑋 → ((𝑉‘𝑖) − 𝑋) = (𝑋 − 𝑋)) |
| 156 | 155 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → ((𝑉‘𝑖) − 𝑋) = (𝑋 − 𝑋)) |
| 157 | 123 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝑋 ∈ ℂ) |
| 158 | 157 | subidd 11608 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑋 − 𝑋) = 0) |
| 159 | 154, 156,
158 | 3eqtrd 2781 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑄‘𝑖) = 0) |
| 160 | 150, 159 | sylanl2 681 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑄‘𝑖) = 0) |
| 161 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → (𝑉‘𝑖) = (𝑉‘𝑗)) |
| 162 | 161 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑗 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑗) − 𝑋)) |
| 163 | 162 | cbvmptv 5255 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋)) |
| 164 | 142, 163 | eqtri 2765 |
. . . . . . . . . 10
⊢ 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋)) |
| 165 | 164 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))) |
| 166 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑗 = (𝑖 + 1) → (𝑉‘𝑗) = (𝑉‘(𝑖 + 1))) |
| 167 | 166 | oveq1d 7446 |
. . . . . . . . . 10
⊢ (𝑗 = (𝑖 + 1) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋)) |
| 168 | 167 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋)) |
| 169 | 1 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ) |
| 170 | 15, 169 | resubcld 11691 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ) |
| 171 | 165, 168,
14, 170 | fvmptd 7023 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋)) |
| 172 | 171 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋)) |
| 173 | 160, 172 | oveq12d 7449 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = (0(,)((𝑉‘(𝑖 + 1)) − 𝑋))) |
| 174 | | simplr 769 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = 0) → 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
| 175 | | fourierdlem75.o |
. . . . . . . . . . . . 13
⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
| 176 | 4 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 = 0) → 𝑀 ∈ ℕ) |
| 177 | 111, 113,
1, 5, 175, 4, 3, 142 | fourierdlem14 46136 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑄 ∈ (𝑂‘𝑀)) |
| 178 | 177 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 = 0) → 𝑄 ∈ (𝑂‘𝑀)) |
| 179 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 = 0) → 𝑠 = 0) |
| 180 | | fourierdlem75.x |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑋 ∈ ran 𝑉) |
| 181 | | ffn 6736 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑉:(0...𝑀)⟶((-π + 𝑋)[,](π + 𝑋)) → 𝑉 Fn (0...𝑀)) |
| 182 | | fvelrnb 6969 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑉 Fn (0...𝑀) → (𝑋 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (0...𝑀)(𝑉‘𝑖) = 𝑋)) |
| 183 | 117, 181,
182 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑋 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (0...𝑀)(𝑉‘𝑖) = 𝑋)) |
| 184 | 180, 183 | mpbid 232 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑉‘𝑖) = 𝑋) |
| 185 | 159 | ex 412 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) = 𝑋 → (𝑄‘𝑖) = 0)) |
| 186 | 185 | reximdva 3168 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑉‘𝑖) = 𝑋 → ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = 0)) |
| 187 | 184, 186 | mpd 15 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = 0) |
| 188 | 121, 142 | fmptd 7134 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ) |
| 189 | | ffn 6736 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀)) |
| 190 | | fvelrnb 6969 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑄 Fn (0...𝑀) → (0 ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = 0)) |
| 191 | 188, 189,
190 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (0 ∈ ran 𝑄 ↔ ∃𝑖 ∈ (0...𝑀)(𝑄‘𝑖) = 0)) |
| 192 | 187, 191 | mpbird 257 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 0 ∈ ran 𝑄) |
| 193 | 192 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 = 0) → 0 ∈ ran 𝑄) |
| 194 | 179, 193 | eqeltrd 2841 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 = 0) → 𝑠 ∈ ran 𝑄) |
| 195 | 175, 176,
178, 194 | fourierdlem12 46134 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 = 0) ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
| 196 | 195 | an32s 652 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 = 0) → ¬ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
| 197 | 196 | adantlr 715 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑠 = 0) → ¬ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
| 198 | 174, 197 | pm2.65da 817 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 = 0) |
| 199 | 198 | adantlr 715 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 = 0) |
| 200 | 199 | iffalsed 4536 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) |
| 201 | 160 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 0 = (𝑄‘𝑖)) |
| 202 | 201 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 = (𝑄‘𝑖)) |
| 203 | | elioo3g 13416 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ* ∧ 𝑠 ∈ ℝ*)
∧ ((𝑄‘𝑖) < 𝑠 ∧ 𝑠 < (𝑄‘(𝑖 + 1))))) |
| 204 | 203 | biimpi 216 |
. . . . . . . . . . . . 13
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (((𝑄‘𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ* ∧ 𝑠 ∈ ℝ*)
∧ ((𝑄‘𝑖) < 𝑠 ∧ 𝑠 < (𝑄‘(𝑖 + 1))))) |
| 205 | 204 | simprld 772 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝑄‘𝑖) < 𝑠) |
| 206 | 205 | adantl 481 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑠) |
| 207 | 202, 206 | eqbrtrd 5165 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 < 𝑠) |
| 208 | 207 | iftrued 4533 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑌) |
| 209 | 208 | oveq2d 7447 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) = ((𝐹‘(𝑋 + 𝑠)) − 𝑌)) |
| 210 | 209 | oveq1d 7446 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) = (((𝐹‘(𝑋 + 𝑠)) − 𝑌) / 𝑠)) |
| 211 | 1 | rexrd 11311 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑋 ∈
ℝ*) |
| 212 | 211 | ad3antrrr 730 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈
ℝ*) |
| 213 | 36 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘(𝑖 + 1)) ∈
ℝ*) |
| 214 | 169 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ) |
| 215 | | elioore 13417 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℝ) |
| 216 | 215 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ) |
| 217 | 214, 216 | readdcld 11290 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ) |
| 218 | 216, 207 | elrpd 13074 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ+) |
| 219 | 214, 218 | ltaddrpd 13110 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 < (𝑋 + 𝑠)) |
| 220 | 215 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ) |
| 221 | 188 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ) |
| 222 | 221, 14 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
| 223 | 222 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
| 224 | 1 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ) |
| 225 | 204 | simprrd 774 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 < (𝑄‘(𝑖 + 1))) |
| 226 | 225 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1))) |
| 227 | 220, 223,
224, 226 | ltadd2dd 11420 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑋 + (𝑄‘(𝑖 + 1)))) |
| 228 | 171 | oveq2d 7447 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋))) |
| 229 | 123 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℂ) |
| 230 | 15 | recnd 11289 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℂ) |
| 231 | 229, 230 | pncan3d 11623 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + ((𝑉‘(𝑖 + 1)) − 𝑋)) = (𝑉‘(𝑖 + 1))) |
| 232 | 228, 231 | eqtrd 2777 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1))) |
| 233 | 232 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄‘(𝑖 + 1))) = (𝑉‘(𝑖 + 1))) |
| 234 | 227, 233 | breqtrd 5169 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑉‘(𝑖 + 1))) |
| 235 | 234 | adantlr 715 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) < (𝑉‘(𝑖 + 1))) |
| 236 | 212, 213,
217, 219, 235 | eliood 45511 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ (𝑋(,)(𝑉‘(𝑖 + 1)))) |
| 237 | | fvres 6925 |
. . . . . . . . . . 11
⊢ ((𝑋 + 𝑠) ∈ (𝑋(,)(𝑉‘(𝑖 + 1))) → ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠))) |
| 238 | 236, 237 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠))) |
| 239 | 238 | eqcomd 2743 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) = ((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠))) |
| 240 | 239 | oveq1d 7446 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − 𝑌) = (((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌)) |
| 241 | 240 | oveq1d 7446 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (((𝐹‘(𝑋 + 𝑠)) − 𝑌) / 𝑠) = ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠)) |
| 242 | 200, 210,
241 | 3eqtrd 2781 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) = ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠)) |
| 243 | 173, 242 | mpteq12dva 5231 |
. . . . 5
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) = (𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠))) |
| 244 | 101, 149,
243 | 3eqtrd 2781 |
. . . 4
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠))) |
| 245 | 244, 160 | oveq12d 7449 |
. . 3
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)) = ((𝑠 ∈ (0(,)((𝑉‘(𝑖 + 1)) − 𝑋)) ↦ ((((𝐹 ↾ (𝑋(,)(𝑉‘(𝑖 + 1))))‘(𝑋 + 𝑠)) − 𝑌) / 𝑠)) limℂ 0)) |
| 246 | 94, 98, 245 | 3eltr4d 2856 |
. 2
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) = 𝑋) → 𝐴 ∈ ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
| 247 | | eqid 2737 |
. . . . 5
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) |
| 248 | | eqid 2737 |
. . . . 5
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠) |
| 249 | | eqid 2737 |
. . . . 5
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) |
| 250 | 24 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐹:ℝ⟶ℝ) |
| 251 | 1 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ) |
| 252 | 215 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ) |
| 253 | 251, 252 | readdcld 11290 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ) |
| 254 | 250, 253 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ) |
| 255 | 254 | recnd 11289 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ) |
| 256 | 255 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ) |
| 257 | 256 | 3adantl3 1169 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ) |
| 258 | | limccl 25910 |
. . . . . . . . . 10
⊢ ((𝐹 ↾ (𝑋(,)+∞)) limℂ 𝑋) ⊆
ℂ |
| 259 | 258, 31 | sselid 3981 |
. . . . . . . . 9
⊢ (𝜑 → 𝑌 ∈ ℂ) |
| 260 | | fourierdlem75.w |
. . . . . . . . . 10
⊢ (𝜑 → 𝑊 ∈ ℝ) |
| 261 | 260 | recnd 11289 |
. . . . . . . . 9
⊢ (𝜑 → 𝑊 ∈ ℂ) |
| 262 | 259, 261 | ifcld 4572 |
. . . . . . . 8
⊢ (𝜑 → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ) |
| 263 | 262 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ) |
| 264 | 263 | 3ad2antl1 1186 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ) |
| 265 | 257, 264 | subcld 11620 |
. . . . 5
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ) |
| 266 | 215 | recnd 11289 |
. . . . . . 7
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℂ) |
| 267 | 266 | adantl 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℂ) |
| 268 | | velsn 4642 |
. . . . . . . 8
⊢ (𝑠 ∈ {0} ↔ 𝑠 = 0) |
| 269 | 198, 268 | sylnibr 329 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 ∈ {0}) |
| 270 | 269 | 3adantl3 1169 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 ∈ {0}) |
| 271 | 267, 270 | eldifd 3962 |
. . . . 5
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ (ℂ ∖
{0})) |
| 272 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) |
| 273 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑊) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑊) |
| 274 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) |
| 275 | 261 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑊 ∈ ℂ) |
| 276 | | ioossre 13448 |
. . . . . . . . . . . 12
⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ |
| 277 | 276 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ) |
| 278 | 150, 119 | sylan2 593 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℝ) |
| 279 | 278 | rexrd 11311 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈
ℝ*) |
| 280 | 279 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) ∈
ℝ*) |
| 281 | 36 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘(𝑖 + 1)) ∈
ℝ*) |
| 282 | 253 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ℝ) |
| 283 | | iccssre 13469 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((-π
∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆
ℝ) |
| 284 | 104, 103,
283 | mp2an 692 |
. . . . . . . . . . . . . . . . . 18
⊢
(-π[,]π) ⊆ ℝ |
| 285 | 284, 46 | sstri 3993 |
. . . . . . . . . . . . . . . . 17
⊢
(-π[,]π) ⊆ ℂ |
| 286 | 153, 141 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ (-π[,]π)) |
| 287 | 150, 286 | sylan2 593 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (-π[,]π)) |
| 288 | 285, 287 | sselid 3981 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ) |
| 289 | 229, 288 | addcomd 11463 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑄‘𝑖)) = ((𝑄‘𝑖) + 𝑋)) |
| 290 | 150 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀)) |
| 291 | 150, 121 | sylan2 593 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ) |
| 292 | 142 | fvmpt2 7027 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑉‘𝑖) − 𝑋) ∈ ℝ) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋)) |
| 293 | 290, 291,
292 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋)) |
| 294 | 293 | oveq1d 7446 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑋) = (((𝑉‘𝑖) − 𝑋) + 𝑋)) |
| 295 | 278 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℂ) |
| 296 | 295, 229 | npcand 11624 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑉‘𝑖) − 𝑋) + 𝑋) = (𝑉‘𝑖)) |
| 297 | 289, 294,
296 | 3eqtrrd 2782 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) = (𝑋 + (𝑄‘𝑖))) |
| 298 | 297 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) = (𝑋 + (𝑄‘𝑖))) |
| 299 | 293, 291 | eqeltrd 2841 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ) |
| 300 | 299 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ) |
| 301 | 205 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑠) |
| 302 | 300, 220,
224, 301 | ltadd2dd 11420 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + (𝑄‘𝑖)) < (𝑋 + 𝑠)) |
| 303 | 298, 302 | eqbrtrd 5165 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑉‘𝑖) < (𝑋 + 𝑠)) |
| 304 | 280, 281,
282, 303, 234 | eliood 45511 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑋 + 𝑠) ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) |
| 305 | | ioossre 13448 |
. . . . . . . . . . . 12
⊢ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ |
| 306 | 305 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1))) ⊆ ℝ) |
| 307 | 300, 301 | gtned 11396 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ (𝑄‘𝑖)) |
| 308 | | fourierdlem75.r |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑉‘𝑖))) |
| 309 | 297 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑉‘𝑖)) = ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑋 + (𝑄‘𝑖)))) |
| 310 | 308, 309 | eleqtrd 2843 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) limℂ (𝑋 + (𝑄‘𝑖)))) |
| 311 | 25, 169, 277, 272, 304, 306, 307, 310, 288 | fourierdlem53 46174 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐹‘(𝑋 + 𝑠))) limℂ (𝑄‘𝑖))) |
| 312 | | ioosscn 13449 |
. . . . . . . . . . . 12
⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ |
| 313 | 312 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ) |
| 314 | 261 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ ℂ) |
| 315 | 273, 313,
314, 288 | constlimc 45639 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑊) limℂ (𝑄‘𝑖))) |
| 316 | 272, 273,
274, 256, 275, 311, 315 | sublimc 45667 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 − 𝑊) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) limℂ (𝑄‘𝑖))) |
| 317 | 316 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑅 − 𝑊) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) limℂ (𝑄‘𝑖))) |
| 318 | | iftrue 4531 |
. . . . . . . . . 10
⊢ ((𝑉‘𝑖) < 𝑋 → if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌) = 𝑊) |
| 319 | 318 | oveq2d 7447 |
. . . . . . . . 9
⊢ ((𝑉‘𝑖) < 𝑋 → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) = (𝑅 − 𝑊)) |
| 320 | 319 | adantl 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) = (𝑅 − 𝑊)) |
| 321 | 215 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ) |
| 322 | | 0red 11264 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ∈
ℝ) |
| 323 | 222 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
| 324 | 225 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < (𝑄‘(𝑖 + 1))) |
| 325 | 171 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋)) |
| 326 | 279 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → (𝑉‘𝑖) ∈
ℝ*) |
| 327 | 36 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → (𝑉‘(𝑖 + 1)) ∈
ℝ*) |
| 328 | 169 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → 𝑋 ∈ ℝ) |
| 329 | | simplr 769 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → (𝑉‘𝑖) < 𝑋) |
| 330 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) |
| 331 | 1 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → 𝑋 ∈ ℝ) |
| 332 | 15 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → (𝑉‘(𝑖 + 1)) ∈ ℝ) |
| 333 | 331, 332 | ltnled 11408 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → (𝑋 < (𝑉‘(𝑖 + 1)) ↔ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋)) |
| 334 | 330, 333 | mpbird 257 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → 𝑋 < (𝑉‘(𝑖 + 1))) |
| 335 | 334 | adantlr 715 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → 𝑋 < (𝑉‘(𝑖 + 1))) |
| 336 | 326, 327,
328, 329, 335 | eliood 45511 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → 𝑋 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) |
| 337 | 5, 4, 3, 180 | fourierdlem12 46134 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) |
| 338 | 337 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ ¬ (𝑉‘(𝑖 + 1)) ≤ 𝑋) → ¬ 𝑋 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) |
| 339 | 336, 338 | condan 818 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑉‘(𝑖 + 1)) ≤ 𝑋) |
| 340 | 15 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑉‘(𝑖 + 1)) ∈ ℝ) |
| 341 | 1 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → 𝑋 ∈ ℝ) |
| 342 | 340, 341 | suble0d 11854 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (((𝑉‘(𝑖 + 1)) − 𝑋) ≤ 0 ↔ (𝑉‘(𝑖 + 1)) ≤ 𝑋)) |
| 343 | 339, 342 | mpbird 257 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → ((𝑉‘(𝑖 + 1)) − 𝑋) ≤ 0) |
| 344 | 325, 343 | eqbrtrd 5165 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑄‘(𝑖 + 1)) ≤ 0) |
| 345 | 344 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ≤ 0) |
| 346 | 321, 323,
322, 324, 345 | ltletrd 11421 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 < 0) |
| 347 | 321, 322,
346 | ltnsymd 11410 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 0 < 𝑠) |
| 348 | 347 | iffalsed 4536 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑊) |
| 349 | 348 | oveq2d 7447 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) = ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) |
| 350 | 349 | mpteq2dva 5242 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊))) |
| 351 | 350 | oveq1d 7446 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) limℂ (𝑄‘𝑖)) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑊)) limℂ (𝑄‘𝑖))) |
| 352 | 317, 320,
351 | 3eltr4d 2856 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ (𝑉‘𝑖) < 𝑋) → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) limℂ (𝑄‘𝑖))) |
| 353 | 352 | 3adantl3 1169 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ (𝑉‘𝑖) < 𝑋) → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) limℂ (𝑄‘𝑖))) |
| 354 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑌) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑌) |
| 355 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌)) |
| 356 | 259 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑌 ∈ ℂ) |
| 357 | 259 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ ℂ) |
| 358 | 354, 313,
357, 288 | constlimc 45639 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑌 ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑌) limℂ (𝑄‘𝑖))) |
| 359 | 272, 354,
355, 256, 356, 311, 358 | sublimc 45667 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑅 − 𝑌) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌)) limℂ (𝑄‘𝑖))) |
| 360 | 359 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → (𝑅 − 𝑌) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌)) limℂ (𝑄‘𝑖))) |
| 361 | | iffalse 4534 |
. . . . . . . . . 10
⊢ (¬
(𝑉‘𝑖) < 𝑋 → if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌) = 𝑌) |
| 362 | 361 | oveq2d 7447 |
. . . . . . . . 9
⊢ (¬
(𝑉‘𝑖) < 𝑋 → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) = (𝑅 − 𝑌)) |
| 363 | 362 | adantl 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) = (𝑅 − 𝑌)) |
| 364 | | 0red 11264 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ∈
ℝ) |
| 365 | 299 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ) |
| 366 | 215 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ) |
| 367 | 1 | ad2antrr 726 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → 𝑋 ∈ ℝ) |
| 368 | 278 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → (𝑉‘𝑖) ∈ ℝ) |
| 369 | | simpr 484 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → ¬ (𝑉‘𝑖) < 𝑋) |
| 370 | 367, 368,
369 | nltled 11411 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → 𝑋 ≤ (𝑉‘𝑖)) |
| 371 | 368, 367 | subge0d 11853 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → (0 ≤ ((𝑉‘𝑖) − 𝑋) ↔ 𝑋 ≤ (𝑉‘𝑖))) |
| 372 | 370, 371 | mpbird 257 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → 0 ≤ ((𝑉‘𝑖) − 𝑋)) |
| 373 | 293 | eqcomd 2743 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) = (𝑄‘𝑖)) |
| 374 | 373 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → ((𝑉‘𝑖) − 𝑋) = (𝑄‘𝑖)) |
| 375 | 372, 374 | breqtrd 5169 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → 0 ≤ (𝑄‘𝑖)) |
| 376 | 375 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ≤ (𝑄‘𝑖)) |
| 377 | 205 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < 𝑠) |
| 378 | 364, 365,
366, 376, 377 | lelttrd 11419 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 < 𝑠) |
| 379 | 378 | iftrued 4533 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑌) |
| 380 | 379 | oveq2d 7447 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) = ((𝐹‘(𝑋 + 𝑠)) − 𝑌)) |
| 381 | 380 | mpteq2dva 5242 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌))) |
| 382 | 381 | oveq1d 7446 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) limℂ (𝑄‘𝑖)) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝑌)) limℂ (𝑄‘𝑖))) |
| 383 | 360, 363,
382 | 3eltr4d 2856 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) < 𝑋) → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) limℂ (𝑄‘𝑖))) |
| 384 | 383 | 3adantl3 1169 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ ¬ (𝑉‘𝑖) < 𝑋) → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) limℂ (𝑄‘𝑖))) |
| 385 | 353, 384 | pm2.61dan 813 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → (𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) limℂ (𝑄‘𝑖))) |
| 386 | 313, 248,
288 | idlimc 45641 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠) limℂ (𝑄‘𝑖))) |
| 387 | 386 | 3adant3 1133 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → (𝑄‘𝑖) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ 𝑠) limℂ (𝑄‘𝑖))) |
| 388 | 293 | 3adant3 1133 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋)) |
| 389 | 295 | 3adant3 1133 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → (𝑉‘𝑖) ∈ ℂ) |
| 390 | 229 | 3adant3 1133 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → 𝑋 ∈ ℂ) |
| 391 | | neqne 2948 |
. . . . . . . 8
⊢ (¬
(𝑉‘𝑖) = 𝑋 → (𝑉‘𝑖) ≠ 𝑋) |
| 392 | 391 | 3ad2ant3 1136 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → (𝑉‘𝑖) ≠ 𝑋) |
| 393 | 389, 390,
392 | subne0d 11629 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → ((𝑉‘𝑖) − 𝑋) ≠ 0) |
| 394 | 388, 393 | eqnetrd 3008 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → (𝑄‘𝑖) ≠ 0) |
| 395 | 198 | 3adantl3 1169 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ¬ 𝑠 = 0) |
| 396 | 395 | neqned 2947 |
. . . . 5
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ 0) |
| 397 | 247, 248,
249, 265, 271, 385, 387, 394, 396 | divlimc 45671 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖)) ∈ ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) limℂ (𝑄‘𝑖))) |
| 398 | | iffalse 4534 |
. . . . . 6
⊢ (¬
(𝑉‘𝑖) = 𝑋 → if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) = ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) |
| 399 | 95, 398 | eqtrid 2789 |
. . . . 5
⊢ (¬
(𝑉‘𝑖) = 𝑋 → 𝐴 = ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) |
| 400 | 399 | 3ad2ant3 1136 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → 𝐴 = ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) |
| 401 | | ioossre 13448 |
. . . . . . . . . . . . 13
⊢ (𝑋(,)+∞) ⊆
ℝ |
| 402 | 401 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑋(,)+∞) ⊆
ℝ) |
| 403 | 24, 402 | fssresd 6775 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹 ↾ (𝑋(,)+∞)):(𝑋(,)+∞)⟶ℝ) |
| 404 | 401, 47 | sstrid 3995 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑋(,)+∞) ⊆
ℂ) |
| 405 | 34 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → +∞ ∈
ℝ*) |
| 406 | 1 | ltpnfd 13163 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑋 < +∞) |
| 407 | 52, 405, 1, 406 | lptioo1cn 45661 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 ∈
((limPt‘(TopOpen‘ℂfld))‘(𝑋(,)+∞))) |
| 408 | 403, 404,
407, 31 | limcrecl 45644 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑌 ∈ ℝ) |
| 409 | 24, 1, 408, 260, 99 | fourierdlem9 46131 |
. . . . . . . . 9
⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℝ) |
| 410 | 409 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐻:(-π[,]π)⟶ℝ) |
| 411 | 410, 147 | feqresmpt 6978 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠))) |
| 412 | 147 | sselda 3983 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ (-π[,]π)) |
| 413 | | 0cnd 11254 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 0 ∈
ℂ) |
| 414 | 262 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ) |
| 415 | 256, 414 | subcld 11620 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ) |
| 416 | 266 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℂ) |
| 417 | 198 | neqned 2947 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ≠ 0) |
| 418 | 415, 416,
417 | divcld 12043 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) ∈ ℂ) |
| 419 | 413, 418 | ifcld 4572 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) ∈ ℂ) |
| 420 | 99 | fvmpt2 7027 |
. . . . . . . . . 10
⊢ ((𝑠 ∈ (-π[,]π) ∧
if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) ∈ ℂ) → (𝐻‘𝑠) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) |
| 421 | 412, 419,
420 | syl2anc 584 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻‘𝑠) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) |
| 422 | 198 | iffalsed 4536 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) |
| 423 | 421, 422 | eqtrd 2777 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) |
| 424 | 423 | mpteq2dva 5242 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) |
| 425 | 411, 424 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) |
| 426 | 425 | 3adant3 1133 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) |
| 427 | 426 | oveq1d 7446 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)) = ((𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) limℂ (𝑄‘𝑖))) |
| 428 | 397, 400,
427 | 3eltr4d 2856 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ ¬ (𝑉‘𝑖) = 𝑋) → 𝐴 ∈ ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
| 429 | 428 | 3expa 1119 |
. 2
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ¬ (𝑉‘𝑖) = 𝑋) → 𝐴 ∈ ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
| 430 | 246, 429 | pm2.61dan 813 |
1
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |