| Step | Hyp | Ref
| Expression |
| 1 | | fourierdlem89.q |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) |
| 2 | | fourierdlem89.m |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ ℕ) |
| 3 | | fourierdlem89.p |
. . . . . . . . . . . . . 14
⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
| 4 | 3 | fourierdlem2 46124 |
. . . . . . . . . . . . 13
⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))) |
| 5 | 2, 4 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))) |
| 6 | 1, 5 | mpbid 232 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑄 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))) |
| 7 | 6 | simpld 494 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ∈ (ℝ ↑m
(0...𝑀))) |
| 8 | | elmapi 8889 |
. . . . . . . . . 10
⊢ (𝑄 ∈ (ℝ
↑m (0...𝑀))
→ 𝑄:(0...𝑀)⟶ℝ) |
| 9 | 7, 8 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ) |
| 10 | | fzossfz 13718 |
. . . . . . . . . 10
⊢
(0..^𝑀) ⊆
(0...𝑀) |
| 11 | | fourierdlem89.t |
. . . . . . . . . . . . 13
⊢ 𝑇 = (𝐵 − 𝐴) |
| 12 | | fourierdlem89.e |
. . . . . . . . . . . . 13
⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) |
| 13 | | fourierdlem89.z |
. . . . . . . . . . . . 13
⊢ 𝑍 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦)) |
| 14 | | fourierdlem89.20 |
. . . . . . . . . . . . 13
⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝑍‘(𝐸‘𝑥))}, ℝ, < )) |
| 15 | 3, 2, 1, 11, 12, 13, 14 | fourierdlem37 46159 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐼:ℝ⟶(0..^𝑀) ∧ (𝑥 ∈ ℝ → sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝑍‘(𝐸‘𝑥))}, ℝ, < ) ∈ {𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝑍‘(𝐸‘𝑥))}))) |
| 16 | 15 | simpld 494 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐼:ℝ⟶(0..^𝑀)) |
| 17 | | fourierdlem89.c |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐶 ∈ ℝ) |
| 18 | | fourierdlem89.d |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞)) |
| 19 | | elioore 13417 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐷 ∈ (𝐶(,)+∞) → 𝐷 ∈ ℝ) |
| 20 | 18, 19 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐷 ∈ ℝ) |
| 21 | | elioo4g 13447 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐷 ∈ (𝐶(,)+∞) ↔ ((𝐶 ∈ ℝ* ∧ +∞
∈ ℝ* ∧ 𝐷 ∈ ℝ) ∧ (𝐶 < 𝐷 ∧ 𝐷 < +∞))) |
| 22 | 18, 21 | sylib 218 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝐶 ∈ ℝ* ∧ +∞
∈ ℝ* ∧ 𝐷 ∈ ℝ) ∧ (𝐶 < 𝐷 ∧ 𝐷 < +∞))) |
| 23 | 22 | simprd 495 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐶 < 𝐷 ∧ 𝐷 < +∞)) |
| 24 | 23 | simpld 494 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐶 < 𝐷) |
| 25 | | fourierdlem89.o |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
| 26 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = 𝑥 → (𝑦 + (𝑘 · 𝑇)) = (𝑥 + (𝑘 · 𝑇))) |
| 27 | 26 | eleq1d 2826 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑥 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄)) |
| 28 | 27 | rexbidv 3179 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄)) |
| 29 | 28 | cbvrabv 3447 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} |
| 30 | 29 | uneq2i 4165 |
. . . . . . . . . . . . . . . . . 18
⊢ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) |
| 31 | | fourierdlem89.n |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑁 = ((♯‘𝐻) − 1) |
| 32 | | fourierdlem89.12 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) |
| 33 | 32 | fveq2i 6909 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(♯‘𝐻) =
(♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) |
| 34 | 33 | oveq1i 7441 |
. . . . . . . . . . . . . . . . . . 19
⊢
((♯‘𝐻)
− 1) = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1) |
| 35 | 31, 34 | eqtri 2765 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑁 = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1) |
| 36 | | fourierdlem89.s |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻)) |
| 37 | | isoeq5 7341 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) → (𝑓 Isom < , < ((0...𝑁), 𝐻) ↔ 𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})))) |
| 38 | 32, 37 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑓 Isom < , < ((0...𝑁), 𝐻) ↔ 𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))) |
| 39 | 38 | iotabii 6546 |
. . . . . . . . . . . . . . . . . . 19
⊢
(℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻)) = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))) |
| 40 | 36, 39 | eqtri 2765 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))) |
| 41 | 11, 3, 2, 1, 17, 20, 24, 25, 30, 35, 40 | fourierdlem54 46175 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})))) |
| 42 | 41 | simpld 494 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁))) |
| 43 | 42 | simprd 495 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁)) |
| 44 | 42 | simpld 494 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 45 | 25 | fourierdlem2 46124 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈ ℕ → (𝑆 ∈ (𝑂‘𝑁) ↔ (𝑆 ∈ (ℝ ↑m
(0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))) |
| 46 | 44, 45 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑆 ∈ (𝑂‘𝑁) ↔ (𝑆 ∈ (ℝ ↑m
(0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))) |
| 47 | 43, 46 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑆 ∈ (ℝ ↑m
(0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))) |
| 48 | 47 | simpld 494 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑆 ∈ (ℝ ↑m
(0...𝑁))) |
| 49 | | elmapi 8889 |
. . . . . . . . . . . . 13
⊢ (𝑆 ∈ (ℝ
↑m (0...𝑁))
→ 𝑆:(0...𝑁)⟶ℝ) |
| 50 | 48, 49 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑆:(0...𝑁)⟶ℝ) |
| 51 | | fourierdlem89.j |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐽 ∈ (0..^𝑁)) |
| 52 | | elfzofz 13715 |
. . . . . . . . . . . . 13
⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ (0...𝑁)) |
| 53 | 51, 52 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐽 ∈ (0...𝑁)) |
| 54 | 50, 53 | ffvelcdmd 7105 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑆‘𝐽) ∈ ℝ) |
| 55 | 16, 54 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) |
| 56 | 10, 55 | sselid 3981 |
. . . . . . . . 9
⊢ (𝜑 → (𝐼‘(𝑆‘𝐽)) ∈ (0...𝑀)) |
| 57 | 9, 56 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈ ℝ) |
| 58 | 57 | rexrd 11311 |
. . . . . . 7
⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈
ℝ*) |
| 59 | 58 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈
ℝ*) |
| 60 | | fzofzp1 13803 |
. . . . . . . . . 10
⊢ ((𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀) → ((𝐼‘(𝑆‘𝐽)) + 1) ∈ (0...𝑀)) |
| 61 | 55, 60 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐼‘(𝑆‘𝐽)) + 1) ∈ (0...𝑀)) |
| 62 | 9, 61 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ (𝜑 → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ∈ ℝ) |
| 63 | 62 | rexrd 11311 |
. . . . . . 7
⊢ (𝜑 → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ∈
ℝ*) |
| 64 | 63 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)) ∈
ℝ*) |
| 65 | 3, 2, 1 | fourierdlem11 46133 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)) |
| 66 | 65 | simp1d 1143 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 67 | 65 | simp2d 1144 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 68 | 66, 67 | iccssred 13474 |
. . . . . . . 8
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
| 69 | 65 | simp3d 1145 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 < 𝐵) |
| 70 | 66, 67, 69, 13 | fourierdlem17 46139 |
. . . . . . . . 9
⊢ (𝜑 → 𝑍:(𝐴(,]𝐵)⟶(𝐴[,]𝐵)) |
| 71 | 66, 67, 69, 11, 12 | fourierdlem4 46126 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵)) |
| 72 | 71, 54 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ (𝜑 → (𝐸‘(𝑆‘𝐽)) ∈ (𝐴(,]𝐵)) |
| 73 | 70, 72 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ (𝐴[,]𝐵)) |
| 74 | 68, 73 | sseldd 3984 |
. . . . . . 7
⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ℝ) |
| 75 | 74 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ℝ) |
| 76 | 57 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑄‘(𝐼‘(𝑆‘𝐽))) ∈ ℝ) |
| 77 | 66 | rexrd 11311 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 78 | | iocssre 13467 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈ ℝ)
→ (𝐴(,]𝐵) ⊆
ℝ) |
| 79 | 77, 67, 78 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ) |
| 80 | | fzofzp1 13803 |
. . . . . . . . . . . . . 14
⊢ (𝐽 ∈ (0..^𝑁) → (𝐽 + 1) ∈ (0...𝑁)) |
| 81 | 51, 80 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐽 + 1) ∈ (0...𝑁)) |
| 82 | 50, 81 | ffvelcdmd 7105 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℝ) |
| 83 | 71, 82 | ffvelcdmd 7105 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ (𝐴(,]𝐵)) |
| 84 | 79, 83 | sseldd 3984 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ) |
| 85 | 47 | simprrd 774 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))) |
| 86 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = 𝐽 → (𝑆‘𝑖) = (𝑆‘𝐽)) |
| 87 | | oveq1 7438 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝐽 → (𝑖 + 1) = (𝐽 + 1)) |
| 88 | 87 | fveq2d 6910 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = 𝐽 → (𝑆‘(𝑖 + 1)) = (𝑆‘(𝐽 + 1))) |
| 89 | 86, 88 | breq12d 5156 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 = 𝐽 → ((𝑆‘𝑖) < (𝑆‘(𝑖 + 1)) ↔ (𝑆‘𝐽) < (𝑆‘(𝐽 + 1)))) |
| 90 | 89 | rspccva 3621 |
. . . . . . . . . . . . . 14
⊢
((∀𝑖 ∈
(0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)) ∧ 𝐽 ∈ (0..^𝑁)) → (𝑆‘𝐽) < (𝑆‘(𝐽 + 1))) |
| 91 | 85, 51, 90 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑆‘𝐽) < (𝑆‘(𝐽 + 1))) |
| 92 | 54, 82 | posdifd 11850 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑆‘𝐽) < (𝑆‘(𝐽 + 1)) ↔ 0 < ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))) |
| 93 | 91, 92 | mpbid 232 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 < ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) |
| 94 | 51 | ancli 548 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝜑 ∧ 𝐽 ∈ (0..^𝑁))) |
| 95 | | eleq1 2829 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝐽 → (𝑗 ∈ (0..^𝑁) ↔ 𝐽 ∈ (0..^𝑁))) |
| 96 | 95 | anbi2d 630 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝐽 → ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ↔ (𝜑 ∧ 𝐽 ∈ (0..^𝑁)))) |
| 97 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝐽 → (𝑗 + 1) = (𝐽 + 1)) |
| 98 | 97 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝐽 → (𝑆‘(𝑗 + 1)) = (𝑆‘(𝐽 + 1))) |
| 99 | 98 | fveq2d 6910 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝐽 → (𝐸‘(𝑆‘(𝑗 + 1))) = (𝐸‘(𝑆‘(𝐽 + 1)))) |
| 100 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝐽 → (𝑆‘𝑗) = (𝑆‘𝐽)) |
| 101 | 100 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝐽 → (𝐸‘(𝑆‘𝑗)) = (𝐸‘(𝑆‘𝐽))) |
| 102 | 101 | fveq2d 6910 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝐽 → (𝑍‘(𝐸‘(𝑆‘𝑗))) = (𝑍‘(𝐸‘(𝑆‘𝐽)))) |
| 103 | 99, 102 | oveq12d 7449 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝐽 → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝑗)))) = ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽))))) |
| 104 | 98, 100 | oveq12d 7449 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝐽 → ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) |
| 105 | 103, 104 | eqeq12d 2753 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝐽 → (((𝐸‘(𝑆‘(𝑗 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) ↔ ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))) |
| 106 | 96, 105 | imbi12d 344 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝐽 → (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗))) ↔ ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))))) |
| 107 | 11 | oveq2i 7442 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 · 𝑇) = (𝑘 · (𝐵 − 𝐴)) |
| 108 | 107 | oveq2i 7442 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 + (𝑘 · 𝑇)) = (𝑦 + (𝑘 · (𝐵 − 𝐴))) |
| 109 | 108 | eleq1i 2832 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄) |
| 110 | 109 | rexbii 3094 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∃𝑘 ∈
ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄) |
| 111 | 110 | rgenw 3065 |
. . . . . . . . . . . . . . . . . . . 20
⊢
∀𝑦 ∈
(𝐶[,]𝐷)(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄) |
| 112 | | rabbi 3467 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∀𝑦 ∈
(𝐶[,]𝐷)(∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄) ↔ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}) |
| 113 | 111, 112 | mpbi 230 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄} |
| 114 | 113 | uneq2i 4165 |
. . . . . . . . . . . . . . . . . 18
⊢ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}) |
| 115 | 114 | fveq2i 6909 |
. . . . . . . . . . . . . . . . 17
⊢
(♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) = (♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) |
| 116 | 115 | oveq1i 7441 |
. . . . . . . . . . . . . . . 16
⊢
((♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1) = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1) |
| 117 | 35, 116 | eqtri 2765 |
. . . . . . . . . . . . . . 15
⊢ 𝑁 = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1) |
| 118 | | isoeq5 7341 |
. . . . . . . . . . . . . . . . . 18
⊢ (({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}) → (𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))) |
| 119 | 114, 118 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))) |
| 120 | 119 | iotabii 6546 |
. . . . . . . . . . . . . . . 16
⊢
(℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))) = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))) |
| 121 | 40, 120 | eqtri 2765 |
. . . . . . . . . . . . . . 15
⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))) |
| 122 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢ ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) = ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) |
| 123 | 3, 11, 2, 1, 17, 18, 25, 117, 121, 12, 13, 122 | fourierdlem65 46186 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗))) |
| 124 | 106, 123 | vtoclg 3554 |
. . . . . . . . . . . . 13
⊢ (𝐽 ∈ (0..^𝑁) → ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))) |
| 125 | 51, 94, 124 | sylc 65 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) |
| 126 | 93, 125 | breqtrrd 5171 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 < ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽))))) |
| 127 | 74, 84 | posdifd 11850 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽))) < (𝐸‘(𝑆‘(𝐽 + 1))) ↔ 0 < ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))))) |
| 128 | 126, 127 | mpbird 257 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) < (𝐸‘(𝑆‘(𝐽 + 1)))) |
| 129 | | id 22 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝜑) |
| 130 | 102, 99 | oveq12d 7449 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝐽 → ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) = ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) |
| 131 | 100 | fveq2d 6910 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝐽 → (𝐼‘(𝑆‘𝑗)) = (𝐼‘(𝑆‘𝐽))) |
| 132 | 131 | fveq2d 6910 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝐽 → (𝑄‘(𝐼‘(𝑆‘𝑗))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) |
| 133 | 131 | oveq1d 7446 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝐽 → ((𝐼‘(𝑆‘𝑗)) + 1) = ((𝐼‘(𝑆‘𝐽)) + 1)) |
| 134 | 133 | fveq2d 6910 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝐽 → (𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) |
| 135 | 132, 134 | oveq12d 7449 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝐽 → ((𝑄‘(𝐼‘(𝑆‘𝑗)))(,)(𝑄‘((𝐼‘(𝑆‘𝑗)) + 1))) = ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) |
| 136 | 130, 135 | sseq12d 4017 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝐽 → (((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝑗)))(,)(𝑄‘((𝐼‘(𝑆‘𝑗)) + 1))) ↔ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))) |
| 137 | 96, 136 | imbi12d 344 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝐽 → (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝑗)))(,)(𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)))) ↔ ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))))) |
| 138 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢ ((𝑆‘𝑗) + if(((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) < ((𝑄‘1) − 𝐴), (((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) / 2), (((𝑄‘1) − 𝐴) / 2))) = ((𝑆‘𝑗) + if(((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) < ((𝑄‘1) − 𝐴), (((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) / 2), (((𝑄‘1) − 𝐴) / 2))) |
| 139 | 11, 3, 2, 1, 17, 20, 24, 25, 30, 35, 40, 12, 13, 138, 14 | fourierdlem79 46200 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝑗)))(,)(𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)))) |
| 140 | 137, 139 | vtoclg 3554 |
. . . . . . . . . . . 12
⊢ (𝐽 ∈ (0..^𝑁) → ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))) |
| 141 | 140 | anabsi7 671 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐽 ∈ (0..^𝑁)) → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) |
| 142 | 129, 51, 141 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) |
| 143 | 57, 62, 74, 84, 128, 142 | fourierdlem10 46132 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑄‘(𝐼‘(𝑆‘𝐽))) ≤ (𝑍‘(𝐸‘(𝑆‘𝐽))) ∧ (𝐸‘(𝑆‘(𝐽 + 1))) ≤ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) |
| 144 | 143 | simpld 494 |
. . . . . . . 8
⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) ≤ (𝑍‘(𝐸‘(𝑆‘𝐽)))) |
| 145 | 144 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑄‘(𝐼‘(𝑆‘𝐽))) ≤ (𝑍‘(𝐸‘(𝑆‘𝐽)))) |
| 146 | | neqne 2948 |
. . . . . . . 8
⊢ (¬
(𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))) → (𝑍‘(𝐸‘(𝑆‘𝐽))) ≠ (𝑄‘(𝐼‘(𝑆‘𝐽)))) |
| 147 | 146 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑍‘(𝐸‘(𝑆‘𝐽))) ≠ (𝑄‘(𝐼‘(𝑆‘𝐽)))) |
| 148 | 76, 75, 145, 147 | leneltd 11415 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝑍‘(𝐸‘(𝑆‘𝐽)))) |
| 149 | 143 | simprd 495 |
. . . . . . . 8
⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ≤ (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) |
| 150 | 74, 84, 62, 128, 149 | ltletrd 11421 |
. . . . . . 7
⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) |
| 151 | 150 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑍‘(𝐸‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) |
| 152 | 59, 64, 75, 148, 151 | eliood 45511 |
. . . . 5
⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) |
| 153 | | fvres 6925 |
. . . . 5
⊢ ((𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽)))) = (𝐹‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) |
| 154 | 152, 153 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽)))) = (𝐹‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) |
| 155 | 154 | eqcomd 2743 |
. . 3
⊢ ((𝜑 ∧ ¬ (𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) → (𝐹‘(𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) |
| 156 | 155 | ifeq2da 4558 |
. 2
⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), (𝐹‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) = if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽)))))) |
| 157 | | fourierdlem89.f |
. . . . . 6
⊢ (𝜑 → 𝐹:ℝ⟶ℂ) |
| 158 | | fdm 6745 |
. . . . . . . 8
⊢ (𝐹:ℝ⟶ℂ →
dom 𝐹 =
ℝ) |
| 159 | 157, 158 | syl 17 |
. . . . . . 7
⊢ (𝜑 → dom 𝐹 = ℝ) |
| 160 | 159 | feq2d 6722 |
. . . . . 6
⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ↔ 𝐹:ℝ⟶ℂ)) |
| 161 | 157, 160 | mpbird 257 |
. . . . 5
⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ) |
| 162 | | ioosscn 13449 |
. . . . . 6
⊢ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ℂ |
| 163 | 162 | a1i 11 |
. . . . 5
⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ℂ) |
| 164 | | ioossre 13448 |
. . . . . 6
⊢ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ ℝ |
| 165 | 164, 159 | sseqtrrid 4027 |
. . . . 5
⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) ⊆ dom 𝐹) |
| 166 | | fourierdlem89.u |
. . . . . . 7
⊢ 𝑈 = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) |
| 167 | 82, 84 | resubcld 11691 |
. . . . . . 7
⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) ∈ ℝ) |
| 168 | 166, 167 | eqeltrid 2845 |
. . . . . 6
⊢ (𝜑 → 𝑈 ∈ ℝ) |
| 169 | 168 | recnd 11289 |
. . . . 5
⊢ (𝜑 → 𝑈 ∈ ℂ) |
| 170 | | eqid 2737 |
. . . . 5
⊢ {𝑥 ∈ ℂ ∣
∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)} = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)} |
| 171 | 74, 84, 168 | iooshift 45535 |
. . . . . 6
⊢ (𝜑 → (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) |
| 172 | | ioossre 13448 |
. . . . . . 7
⊢ (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ⊆ ℝ |
| 173 | 172, 159 | sseqtrrid 4027 |
. . . . . 6
⊢ (𝜑 → (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ⊆ dom 𝐹) |
| 174 | 171, 173 | eqsstrrd 4019 |
. . . . 5
⊢ (𝜑 → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)} ⊆ dom 𝐹) |
| 175 | | elioore 13417 |
. . . . . 6
⊢ (𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))) → 𝑦 ∈ ℝ) |
| 176 | 67, 66 | resubcld 11691 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
| 177 | 11, 176 | eqeltrid 2845 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑇 ∈ ℝ) |
| 178 | 177 | recnd 11289 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑇 ∈ ℂ) |
| 179 | 66, 67 | posdifd 11850 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
| 180 | 69, 179 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 < (𝐵 − 𝐴)) |
| 181 | 180, 11 | breqtrrdi 5185 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 < 𝑇) |
| 182 | 181 | gt0ne0d 11827 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑇 ≠ 0) |
| 183 | 169, 178,
182 | divcan1d 12044 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑈 / 𝑇) · 𝑇) = 𝑈) |
| 184 | 183 | eqcomd 2743 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑈 = ((𝑈 / 𝑇) · 𝑇)) |
| 185 | 184 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝜑 → (𝑦 + 𝑈) = (𝑦 + ((𝑈 / 𝑇) · 𝑇))) |
| 186 | 185 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + 𝑈) = (𝑦 + ((𝑈 / 𝑇) · 𝑇))) |
| 187 | 186 | fveq2d 6910 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑈)) = (𝐹‘(𝑦 + ((𝑈 / 𝑇) · 𝑇)))) |
| 188 | 157 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐹:ℝ⟶ℂ) |
| 189 | 177 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑇 ∈ ℝ) |
| 190 | 84 | recnd 11289 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℂ) |
| 191 | 82 | recnd 11289 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℂ) |
| 192 | 190, 191 | negsubdi2d 11636 |
. . . . . . . . . . . . 13
⊢ (𝜑 → -((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) |
| 193 | 192 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) = -((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1)))) |
| 194 | 193 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (𝜑 → (((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) / 𝑇) = (-((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇)) |
| 195 | 166 | oveq1i 7441 |
. . . . . . . . . . . 12
⊢ (𝑈 / 𝑇) = (((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) / 𝑇) |
| 196 | 195 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑈 / 𝑇) = (((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) / 𝑇)) |
| 197 | 12 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))) |
| 198 | | id 22 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → 𝑥 = (𝑆‘(𝐽 + 1))) |
| 199 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝐵 − 𝑥) = (𝐵 − (𝑆‘(𝐽 + 1)))) |
| 200 | 199 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) |
| 201 | 200 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇))) |
| 202 | 201 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) |
| 203 | 198, 202 | oveq12d 7449 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))) |
| 204 | 203 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 = (𝑆‘(𝐽 + 1))) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))) |
| 205 | 67, 82 | resubcld 11691 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐵 − (𝑆‘(𝐽 + 1))) ∈ ℝ) |
| 206 | 205, 177,
182 | redivcld 12095 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℝ) |
| 207 | 206 | flcld 13838 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℤ) |
| 208 | 207 | zred 12722 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℝ) |
| 209 | 208, 177 | remulcld 11291 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℝ) |
| 210 | 82, 209 | readdcld 11290 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) ∈ ℝ) |
| 211 | 197, 204,
82, 210 | fvmptd 7023 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))) |
| 212 | 211 | oveq1d 7446 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = (((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) − (𝑆‘(𝐽 + 1)))) |
| 213 | 208 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℂ) |
| 214 | 213, 178 | mulcld 11281 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℂ) |
| 215 | 191, 214 | pncan2d 11622 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) − (𝑆‘(𝐽 + 1))) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) |
| 216 | 212, 215 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) |
| 217 | 216, 214 | eqeltrd 2841 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ∈ ℂ) |
| 218 | 217, 178,
182 | divnegd 12056 |
. . . . . . . . . . 11
⊢ (𝜑 → -(((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (-((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇)) |
| 219 | 194, 196,
218 | 3eqtr4d 2787 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑈 / 𝑇) = -(((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇)) |
| 220 | 216 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) / 𝑇)) |
| 221 | 213, 178,
182 | divcan4d 12049 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) / 𝑇) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇))) |
| 222 | 220, 221 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇))) |
| 223 | 222, 207 | eqeltrd 2841 |
. . . . . . . . . . 11
⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℤ) |
| 224 | 223 | znegcld 12724 |
. . . . . . . . . 10
⊢ (𝜑 → -(((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℤ) |
| 225 | 219, 224 | eqeltrd 2841 |
. . . . . . . . 9
⊢ (𝜑 → (𝑈 / 𝑇) ∈ ℤ) |
| 226 | 225 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑈 / 𝑇) ∈ ℤ) |
| 227 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) |
| 228 | | fourierdlem89.per |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
| 229 | 228 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
| 230 | 188, 189,
226, 227, 229 | fperiodmul 45316 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + ((𝑈 / 𝑇) · 𝑇))) = (𝐹‘𝑦)) |
| 231 | 187, 230 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + 𝑈)) = (𝐹‘𝑦)) |
| 232 | 175, 231 | sylan2 593 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) → (𝐹‘(𝑦 + 𝑈)) = (𝐹‘𝑦)) |
| 233 | 6 | simprrd 774 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) |
| 234 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑄‘𝑖) = (𝑄‘(𝐼‘(𝑆‘𝐽)))) |
| 235 | | oveq1 7438 |
. . . . . . . . . . 11
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑖 + 1) = ((𝐼‘(𝑆‘𝐽)) + 1)) |
| 236 | 235 | fveq2d 6910 |
. . . . . . . . . 10
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑄‘(𝑖 + 1)) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) |
| 237 | 234, 236 | breq12d 5156 |
. . . . . . . . 9
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) |
| 238 | 237 | rspccva 3621 |
. . . . . . . 8
⊢
((∀𝑖 ∈
(0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) |
| 239 | 233, 55, 238 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → (𝑄‘(𝐼‘(𝑆‘𝐽))) < (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))) |
| 240 | 55 | ancli 548 |
. . . . . . . 8
⊢ (𝜑 → (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) |
| 241 | | eleq1 2829 |
. . . . . . . . . . 11
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑖 ∈ (0..^𝑀) ↔ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) |
| 242 | 241 | anbi2d 630 |
. . . . . . . . . 10
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)))) |
| 243 | 234, 236 | oveq12d 7449 |
. . . . . . . . . . . 12
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) |
| 244 | 243 | reseq2d 5997 |
. . . . . . . . . . 11
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))) |
| 245 | 243 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) = (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))–cn→ℂ)) |
| 246 | 244, 245 | eleq12d 2835 |
. . . . . . . . . 10
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) ↔ (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ∈ (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))–cn→ℂ))) |
| 247 | 242, 246 | imbi12d 344 |
. . . . . . . . 9
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) ↔ ((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ∈ (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))–cn→ℂ)))) |
| 248 | | fourierdlem89.fcn |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
| 249 | 247, 248 | vtoclg 3554 |
. . . . . . . 8
⊢ ((𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀) → ((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ∈ (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))–cn→ℂ))) |
| 250 | 55, 240, 249 | sylc 65 |
. . . . . . 7
⊢ (𝜑 → (𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ∈ (((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))–cn→ℂ)) |
| 251 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑖(𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) |
| 252 | | fourierdlem89.21 |
. . . . . . . . . . . . 13
⊢ 𝑉 = (𝑖 ∈ (0..^𝑀) ↦ 𝑅) |
| 253 | | nfmpt1 5250 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑖(𝑖 ∈ (0..^𝑀) ↦ 𝑅) |
| 254 | 252, 253 | nfcxfr 2903 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖𝑉 |
| 255 | | nfcv 2905 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖(𝐼‘(𝑆‘𝐽)) |
| 256 | 254, 255 | nffv 6916 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖(𝑉‘(𝐼‘(𝑆‘𝐽))) |
| 257 | 256 | nfel1 2922 |
. . . . . . . . . 10
⊢
Ⅎ𝑖(𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) limℂ (𝑄‘(𝐼‘(𝑆‘𝐽)))) |
| 258 | 251, 257 | nfim 1896 |
. . . . . . . . 9
⊢
Ⅎ𝑖((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) limℂ (𝑄‘(𝐼‘(𝑆‘𝐽))))) |
| 259 | 242 | biimpar 477 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝜑 ∧ 𝑖 ∈ (0..^𝑀))) |
| 260 | 259 | 3adant2 1132 |
. . . . . . . . . . . . 13
⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝜑 ∧ 𝑖 ∈ (0..^𝑀))) |
| 261 | | fourierdlem89.limc |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
| 262 | 260, 261 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
| 263 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑉‘𝑖) = (𝑉‘(𝐼‘(𝑆‘𝐽)))) |
| 264 | 263 | eqcomd 2743 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (𝑉‘(𝐼‘(𝑆‘𝐽))) = (𝑉‘𝑖)) |
| 265 | 264 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑉‘(𝐼‘(𝑆‘𝐽))) = (𝑉‘𝑖)) |
| 266 | 259 | simprd 495 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → 𝑖 ∈ (0..^𝑀)) |
| 267 | | elex 3501 |
. . . . . . . . . . . . . . . 16
⊢ (𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)) → 𝑅 ∈ V) |
| 268 | 259, 261,
267 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → 𝑅 ∈ V) |
| 269 | 252 | fvmpt2 7027 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑅 ∈ V) → (𝑉‘𝑖) = 𝑅) |
| 270 | 266, 268,
269 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑉‘𝑖) = 𝑅) |
| 271 | 265, 270 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑉‘(𝐼‘(𝑆‘𝐽))) = 𝑅) |
| 272 | 271 | 3adant2 1132 |
. . . . . . . . . . . 12
⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑉‘(𝐼‘(𝑆‘𝐽))) = 𝑅) |
| 273 | 244, 234 | oveq12d 7449 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)) = ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) limℂ (𝑄‘(𝐼‘(𝑆‘𝐽))))) |
| 274 | 273 | eqcomd 2743 |
. . . . . . . . . . . . 13
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) limℂ (𝑄‘(𝐼‘(𝑆‘𝐽)))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
| 275 | 274 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) limℂ (𝑄‘(𝐼‘(𝑆‘𝐽)))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) |
| 276 | 262, 272,
275 | 3eltr4d 2856 |
. . . . . . . . . . 11
⊢ ((𝑖 = (𝐼‘(𝑆‘𝐽)) ∧ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) ∧ (𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀))) → (𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) limℂ (𝑄‘(𝐼‘(𝑆‘𝐽))))) |
| 277 | 276 | 3exp 1120 |
. . . . . . . . . 10
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) → ((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) limℂ (𝑄‘(𝐼‘(𝑆‘𝐽))))))) |
| 278 | 261 | 2a1i 12 |
. . . . . . . . . 10
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) limℂ (𝑄‘(𝐼‘(𝑆‘𝐽))))) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))))) |
| 279 | 277, 278 | impbid 212 |
. . . . . . . . 9
⊢ (𝑖 = (𝐼‘(𝑆‘𝐽)) → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) ↔ ((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) limℂ (𝑄‘(𝐼‘(𝑆‘𝐽))))))) |
| 280 | 258, 279,
261 | vtoclg1f 3570 |
. . . . . . . 8
⊢ ((𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀) → ((𝜑 ∧ (𝐼‘(𝑆‘𝐽)) ∈ (0..^𝑀)) → (𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) limℂ (𝑄‘(𝐼‘(𝑆‘𝐽)))))) |
| 281 | 55, 240, 280 | sylc 65 |
. . . . . . 7
⊢ (𝜑 → (𝑉‘(𝐼‘(𝑆‘𝐽))) ∈ ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) limℂ (𝑄‘(𝐼‘(𝑆‘𝐽))))) |
| 282 | | eqid 2737 |
. . . . . . 7
⊢ if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) = if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) |
| 283 | | eqid 2737 |
. . . . . . 7
⊢
((TopOpen‘ℂfld) ↾t ((𝑄‘(𝐼‘(𝑆‘𝐽)))[,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) =
((TopOpen‘ℂfld) ↾t ((𝑄‘(𝐼‘(𝑆‘𝐽)))[,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) |
| 284 | 57, 62, 239, 250, 281, 74, 84, 128, 142, 282, 283 | fourierdlem32 46154 |
. . . . . 6
⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) ∈ (((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) limℂ (𝑍‘(𝐸‘(𝑆‘𝐽))))) |
| 285 | 142 | resabs1d 6026 |
. . . . . . 7
⊢ (𝜑 → ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) = (𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))) |
| 286 | 285 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 → (((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)))) ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) limℂ (𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) limℂ (𝑍‘(𝐸‘(𝑆‘𝐽))))) |
| 287 | 284, 286 | eleqtrd 2843 |
. . . . 5
⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) ∈ ((𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) limℂ (𝑍‘(𝐸‘(𝑆‘𝐽))))) |
| 288 | 161, 163,
165, 169, 170, 174, 232, 287 | limcperiod 45643 |
. . . 4
⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) ∈ ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) limℂ ((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈))) |
| 289 | 166 | oveq2i 7442 |
. . . . . . 7
⊢ ((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = ((𝑍‘(𝐸‘(𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) |
| 290 | 289 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = ((𝑍‘(𝐸‘(𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))) |
| 291 | 17, 20 | iccssred 13474 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ) |
| 292 | | ax-resscn 11212 |
. . . . . . . . . . . . 13
⊢ ℝ
⊆ ℂ |
| 293 | 291, 292 | sstrdi 3996 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℂ) |
| 294 | 25, 44, 43 | fourierdlem15 46137 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷)) |
| 295 | 294, 53 | ffvelcdmd 7105 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑆‘𝐽) ∈ (𝐶[,]𝐷)) |
| 296 | 293, 295 | sseldd 3984 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑆‘𝐽) ∈ ℂ) |
| 297 | 191, 296 | subcld 11620 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) ∈ ℂ) |
| 298 | 74 | recnd 11289 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) ∈ ℂ) |
| 299 | 190, 297,
298 | subsub23d 45299 |
. . . . . . . . 9
⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (𝑍‘(𝐸‘(𝑆‘𝐽))) ↔ ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑍‘(𝐸‘(𝑆‘𝐽)))) = ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))) |
| 300 | 125, 299 | mpbird 257 |
. . . . . . . 8
⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (𝑍‘(𝐸‘(𝑆‘𝐽)))) |
| 301 | 300 | eqcomd 2743 |
. . . . . . 7
⊢ (𝜑 → (𝑍‘(𝐸‘(𝑆‘𝐽))) = ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))) |
| 302 | 301 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))))) |
| 303 | 190, 297 | subcld 11620 |
. . . . . . . 8
⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) ∈ ℂ) |
| 304 | 303, 191,
190 | addsub12d 11643 |
. . . . . . 7
⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = ((𝑆‘(𝐽 + 1)) + (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) − (𝐸‘(𝑆‘(𝐽 + 1)))))) |
| 305 | 190, 297,
190 | sub32d 11652 |
. . . . . . . . 9
⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))) |
| 306 | 190 | subidd 11608 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = 0) |
| 307 | 306 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − (𝐸‘(𝑆‘(𝐽 + 1)))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)))) |
| 308 | | df-neg 11495 |
. . . . . . . . . 10
⊢ -((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) = (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) |
| 309 | 191, 296 | negsubdi2d 11636 |
. . . . . . . . . 10
⊢ (𝜑 → -((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽)) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1)))) |
| 310 | 308, 309 | eqtr3id 2791 |
. . . . . . . . 9
⊢ (𝜑 → (0 − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1)))) |
| 311 | 305, 307,
310 | 3eqtrd 2781 |
. . . . . . . 8
⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) − (𝐸‘(𝑆‘(𝐽 + 1)))) = ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1)))) |
| 312 | 311 | oveq2d 7447 |
. . . . . . 7
⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) − (𝐸‘(𝑆‘(𝐽 + 1))))) = ((𝑆‘(𝐽 + 1)) + ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1))))) |
| 313 | 191, 296 | pncan3d 11623 |
. . . . . . 7
⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((𝑆‘𝐽) − (𝑆‘(𝐽 + 1)))) = (𝑆‘𝐽)) |
| 314 | 304, 312,
313 | 3eqtrd 2781 |
. . . . . 6
⊢ (𝜑 → (((𝐸‘(𝑆‘(𝐽 + 1))) − ((𝑆‘(𝐽 + 1)) − (𝑆‘𝐽))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = (𝑆‘𝐽)) |
| 315 | 290, 302,
314 | 3eqtrd 2781 |
. . . . 5
⊢ (𝜑 → ((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈) = (𝑆‘𝐽)) |
| 316 | 315 | oveq2d 7447 |
. . . 4
⊢ (𝜑 → ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) limℂ ((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)) = ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) limℂ (𝑆‘𝐽))) |
| 317 | 288, 316 | eleqtrd 2843 |
. . 3
⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) ∈ ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) limℂ (𝑆‘𝐽))) |
| 318 | 166 | oveq2i 7442 |
. . . . . . . 8
⊢ ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = ((𝐸‘(𝑆‘(𝐽 + 1))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) |
| 319 | 190, 191 | pncan3d 11623 |
. . . . . . . 8
⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) + ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))) = (𝑆‘(𝐽 + 1))) |
| 320 | 318, 319 | eqtrid 2789 |
. . . . . . 7
⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈) = (𝑆‘(𝐽 + 1))) |
| 321 | 315, 320 | oveq12d 7449 |
. . . . . 6
⊢ (𝜑 → (((𝑍‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) = ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) |
| 322 | 171, 321 | eqtr3d 2779 |
. . . . 5
⊢ (𝜑 → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)} = ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) |
| 323 | 322 | reseq2d 5997 |
. . . 4
⊢ (𝜑 → (𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) = (𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))))) |
| 324 | 323 | oveq1d 7446 |
. . 3
⊢ (𝜑 → ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑍‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))𝑥 = (𝑦 + 𝑈)}) limℂ (𝑆‘𝐽)) = ((𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) limℂ (𝑆‘𝐽))) |
| 325 | 317, 324 | eleqtrd 2843 |
. 2
⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), ((𝐹 ↾ ((𝑄‘(𝐼‘(𝑆‘𝐽)))(,)(𝑄‘((𝐼‘(𝑆‘𝐽)) + 1))))‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) ∈ ((𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) limℂ (𝑆‘𝐽))) |
| 326 | 156, 325 | eqeltrd 2841 |
1
⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), (𝐹‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) ∈ ((𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) limℂ (𝑆‘𝐽))) |