| Step | Hyp | Ref
| Expression |
| 1 | | ioossre 13448 |
. . . . . . . 8
⊢ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ ℝ |
| 2 | 1 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ ℝ) |
| 3 | 2 | sselda 3983 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → 𝑠 ∈ ℝ) |
| 4 | | iftrue 4531 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ dom 𝐺 → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) = (𝐺‘𝑠)) |
| 5 | 4 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ dom 𝐺) → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) = (𝐺‘𝑠)) |
| 6 | | fourierdlem97.f |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
| 7 | | ssid 4006 |
. . . . . . . . . . . . . 14
⊢ ℝ
⊆ ℝ |
| 8 | | dvfre 25989 |
. . . . . . . . . . . . . 14
⊢ ((𝐹:ℝ⟶ℝ ∧
ℝ ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
| 9 | 6, 7, 8 | sylancl 586 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
| 10 | | fourierdlem97.g |
. . . . . . . . . . . . . 14
⊢ 𝐺 = (ℝ D 𝐹) |
| 11 | 10 | feq1i 6727 |
. . . . . . . . . . . . 13
⊢ (𝐺:dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ
D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
| 12 | 9, 11 | sylibr 234 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐺:dom (ℝ D 𝐹)⟶ℝ) |
| 13 | 12 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ dom 𝐺) → 𝐺:dom (ℝ D 𝐹)⟶ℝ) |
| 14 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑠 ∈ dom 𝐺 → 𝑠 ∈ dom 𝐺) |
| 15 | 10 | dmeqi 5915 |
. . . . . . . . . . . . 13
⊢ dom 𝐺 = dom (ℝ D 𝐹) |
| 16 | 14, 15 | eleqtrdi 2851 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ dom 𝐺 → 𝑠 ∈ dom (ℝ D 𝐹)) |
| 17 | 16 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ dom 𝐺) → 𝑠 ∈ dom (ℝ D 𝐹)) |
| 18 | 13, 17 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ dom 𝐺) → (𝐺‘𝑠) ∈ ℝ) |
| 19 | 5, 18 | eqeltrd 2841 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ dom 𝐺) → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) ∈ ℝ) |
| 20 | 19 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑠 ∈ dom 𝐺) → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) ∈ ℝ) |
| 21 | | iffalse 4534 |
. . . . . . . . . 10
⊢ (¬
𝑠 ∈ dom 𝐺 → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) = 0) |
| 22 | | 0red 11264 |
. . . . . . . . . 10
⊢ (¬
𝑠 ∈ dom 𝐺 → 0 ∈
ℝ) |
| 23 | 21, 22 | eqeltrd 2841 |
. . . . . . . . 9
⊢ (¬
𝑠 ∈ dom 𝐺 → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) ∈ ℝ) |
| 24 | 23 | adantl 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ ¬ 𝑠 ∈ dom 𝐺) → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) ∈ ℝ) |
| 25 | 20, 24 | pm2.61dan 813 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) ∈ ℝ) |
| 26 | 3, 25 | syldan 591 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) ∈ ℝ) |
| 27 | | fourierdlem97.h |
. . . . . . 7
⊢ 𝐻 = (𝑠 ∈ ℝ ↦ if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0)) |
| 28 | 27 | fvmpt2 7027 |
. . . . . 6
⊢ ((𝑠 ∈ ℝ ∧ if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) ∈ ℝ) → (𝐻‘𝑠) = if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0)) |
| 29 | 3, 26, 28 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → (𝐻‘𝑠) = if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0)) |
| 30 | | fourierdlem97.t |
. . . . . . . . . 10
⊢ 𝑇 = (𝐵 − 𝐴) |
| 31 | | fourierdlem97.p |
. . . . . . . . . 10
⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
| 32 | | fourierdlem97.m |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ ℕ) |
| 33 | | fourierdlem97.q |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) |
| 34 | | fourierdlem97.c |
. . . . . . . . . 10
⊢ (𝜑 → 𝐶 ∈ ℝ) |
| 35 | | fourierdlem97.d |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞)) |
| 36 | | elioore 13417 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ (𝐶(,)+∞) → 𝐷 ∈ ℝ) |
| 37 | 35, 36 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐷 ∈ ℝ) |
| 38 | 34 | rexrd 11311 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐶 ∈
ℝ*) |
| 39 | | pnfxr 11315 |
. . . . . . . . . . . 12
⊢ +∞
∈ ℝ* |
| 40 | 39 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → +∞ ∈
ℝ*) |
| 41 | | ioogtlb 45508 |
. . . . . . . . . . 11
⊢ ((𝐶 ∈ ℝ*
∧ +∞ ∈ ℝ* ∧ 𝐷 ∈ (𝐶(,)+∞)) → 𝐶 < 𝐷) |
| 42 | 38, 40, 35, 41 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐶 < 𝐷) |
| 43 | | oveq1 7438 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑥 → (𝑦 + (ℎ · 𝑇)) = (𝑥 + (ℎ · 𝑇))) |
| 44 | 43 | eleq1d 2826 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑥 → ((𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄 ↔ (𝑥 + (ℎ · 𝑇)) ∈ ran 𝑄)) |
| 45 | 44 | rexbidv 3179 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑥 → (∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄 ↔ ∃ℎ ∈ ℤ (𝑥 + (ℎ · 𝑇)) ∈ ran 𝑄)) |
| 46 | 45 | cbvrabv 3447 |
. . . . . . . . . . 11
⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄} = {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃ℎ ∈ ℤ (𝑥 + (ℎ · 𝑇)) ∈ ran 𝑄} |
| 47 | 46 | uneq2i 4165 |
. . . . . . . . . 10
⊢ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}) = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃ℎ ∈ ℤ (𝑥 + (ℎ · 𝑇)) ∈ ran 𝑄}) |
| 48 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑙 → (𝑘 · 𝑇) = (𝑙 · 𝑇)) |
| 49 | 48 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑙 → (𝑦 + (𝑘 · 𝑇)) = (𝑦 + (𝑙 · 𝑇))) |
| 50 | 49 | eleq1d 2826 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑙 → ((𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄)) |
| 51 | 50 | cbvrexvw 3238 |
. . . . . . . . . . . . . . . 16
⊢
(∃𝑘 ∈
ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄) |
| 52 | 51 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (𝐶[,]𝐷) → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄)) |
| 53 | 52 | rabbiia 3440 |
. . . . . . . . . . . . . 14
⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄} |
| 54 | 53 | uneq2i 4165 |
. . . . . . . . . . . . 13
⊢ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄}) |
| 55 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑙 = ℎ → (𝑙 · 𝑇) = (ℎ · 𝑇)) |
| 56 | 55 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑙 = ℎ → (𝑦 + (𝑙 · 𝑇)) = (𝑦 + (ℎ · 𝑇))) |
| 57 | 56 | eleq1d 2826 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑙 = ℎ → ((𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄)) |
| 58 | 57 | cbvrexvw 3238 |
. . . . . . . . . . . . . . . 16
⊢
(∃𝑙 ∈
ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄 ↔ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄) |
| 59 | 58 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (𝐶[,]𝐷) → (∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄 ↔ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄)) |
| 60 | 59 | rabbiia 3440 |
. . . . . . . . . . . . . 14
⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄} |
| 61 | 60 | uneq2i 4165 |
. . . . . . . . . . . . 13
⊢ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄}) = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}) |
| 62 | 54, 61 | eqtri 2765 |
. . . . . . . . . . . 12
⊢ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}) |
| 63 | 62 | fveq2i 6909 |
. . . . . . . . . . 11
⊢
(♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) = (♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄})) |
| 64 | 63 | oveq1i 7441 |
. . . . . . . . . 10
⊢
((♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1) = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄})) − 1) |
| 65 | | fourierdlem97.v |
. . . . . . . . . 10
⊢ 𝑉 = (℩𝑔𝑔 Isom < , <
((0...((♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}))) |
| 66 | | fourierdlem97.j |
. . . . . . . . . 10
⊢ (𝜑 → 𝐽 ∈ (0..^((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1))) |
| 67 | | oveq1 7438 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = ℎ → (𝑘 · 𝑇) = (ℎ · 𝑇)) |
| 68 | 67 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ (𝑘 = ℎ → ((𝑄‘0) + (𝑘 · 𝑇)) = ((𝑄‘0) + (ℎ · 𝑇))) |
| 69 | 68 | breq1d 5153 |
. . . . . . . . . . . 12
⊢ (𝑘 = ℎ → (((𝑄‘0) + (𝑘 · 𝑇)) ≤ (𝑉‘𝐽) ↔ ((𝑄‘0) + (ℎ · 𝑇)) ≤ (𝑉‘𝐽))) |
| 70 | 69 | cbvrabv 3447 |
. . . . . . . . . . 11
⊢ {𝑘 ∈ ℤ ∣ ((𝑄‘0) + (𝑘 · 𝑇)) ≤ (𝑉‘𝐽)} = {ℎ ∈ ℤ ∣ ((𝑄‘0) + (ℎ · 𝑇)) ≤ (𝑉‘𝐽)} |
| 71 | 70 | supeq1i 9487 |
. . . . . . . . . 10
⊢
sup({𝑘 ∈
ℤ ∣ ((𝑄‘0) + (𝑘 · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) = sup({ℎ ∈ ℤ ∣ ((𝑄‘0) + (ℎ · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) |
| 72 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑒 → (𝑄‘𝑗) = (𝑄‘𝑒)) |
| 73 | 72 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑒 → ((𝑄‘𝑗) + (sup({𝑘 ∈ ℤ ∣ ((𝑄‘0) + (𝑘 · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) · 𝑇)) = ((𝑄‘𝑒) + (sup({𝑘 ∈ ℤ ∣ ((𝑄‘0) + (𝑘 · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) · 𝑇))) |
| 74 | 73 | breq1d 5153 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑒 → (((𝑄‘𝑗) + (sup({𝑘 ∈ ℤ ∣ ((𝑄‘0) + (𝑘 · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) · 𝑇)) ≤ (𝑉‘𝐽) ↔ ((𝑄‘𝑒) + (sup({𝑘 ∈ ℤ ∣ ((𝑄‘0) + (𝑘 · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) · 𝑇)) ≤ (𝑉‘𝐽))) |
| 75 | 74 | cbvrabv 3447 |
. . . . . . . . . . 11
⊢ {𝑗 ∈ (0..^𝑀) ∣ ((𝑄‘𝑗) + (sup({𝑘 ∈ ℤ ∣ ((𝑄‘0) + (𝑘 · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) · 𝑇)) ≤ (𝑉‘𝐽)} = {𝑒 ∈ (0..^𝑀) ∣ ((𝑄‘𝑒) + (sup({𝑘 ∈ ℤ ∣ ((𝑄‘0) + (𝑘 · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) · 𝑇)) ≤ (𝑉‘𝐽)} |
| 76 | 75 | supeq1i 9487 |
. . . . . . . . . 10
⊢
sup({𝑗 ∈
(0..^𝑀) ∣ ((𝑄‘𝑗) + (sup({𝑘 ∈ ℤ ∣ ((𝑄‘0) + (𝑘 · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) = sup({𝑒 ∈ (0..^𝑀) ∣ ((𝑄‘𝑒) + (sup({𝑘 ∈ ℤ ∣ ((𝑄‘0) + (𝑘 · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) |
| 77 | 30, 31, 32, 33, 34, 37, 42, 47, 64, 65, 66, 71, 76 | fourierdlem64 46185 |
. . . . . . . . 9
⊢ (𝜑 → ((sup({𝑗 ∈ (0..^𝑀) ∣ ((𝑄‘𝑗) + (sup({𝑘 ∈ ℤ ∣ ((𝑄‘0) + (𝑘 · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) ∈ (0..^𝑀) ∧ sup({𝑘 ∈ ℤ ∣ ((𝑄‘0) + (𝑘 · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) ∈ ℤ) ∧
∃𝑖 ∈ (0..^𝑀)∃𝑙 ∈ ℤ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇))))) |
| 78 | 77 | simprd 495 |
. . . . . . . 8
⊢ (𝜑 → ∃𝑖 ∈ (0..^𝑀)∃𝑙 ∈ ℤ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) |
| 79 | | simpl1 1192 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) ∧ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) ∧ 𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → 𝜑) |
| 80 | | simpl2l 1227 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) ∧ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) ∧ 𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → 𝑖 ∈ (0..^𝑀)) |
| 81 | | fourierdlem97.qcn |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
| 82 | | cncff 24919 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ) |
| 83 | 81, 82 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ) |
| 84 | | ffun 6739 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐺:dom (ℝ D 𝐹)⟶ℝ → Fun
𝐺) |
| 85 | 12, 84 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → Fun 𝐺) |
| 86 | 85 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → Fun 𝐺) |
| 87 | | ffvresb 7145 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (Fun
𝐺 → ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ ↔ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(𝑠 ∈ dom 𝐺 ∧ (𝐺‘𝑠) ∈ ℂ))) |
| 88 | 86, 87 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ ↔ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(𝑠 ∈ dom 𝐺 ∧ (𝐺‘𝑠) ∈ ℂ))) |
| 89 | 83, 88 | mpbid 232 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(𝑠 ∈ dom 𝐺 ∧ (𝐺‘𝑠) ∈ ℂ)) |
| 90 | 89 | r19.21bi 3251 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝑠 ∈ dom 𝐺 ∧ (𝐺‘𝑠) ∈ ℂ)) |
| 91 | 90 | simpld 494 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ dom 𝐺) |
| 92 | 91 | ralrimiva 3146 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑠 ∈ dom 𝐺) |
| 93 | | dfss3 3972 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐺 ↔ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑠 ∈ dom 𝐺) |
| 94 | 92, 93 | sylibr 234 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐺) |
| 95 | 79, 80, 94 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) ∧ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) ∧ 𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐺) |
| 96 | | simpl2 1193 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) ∧ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) ∧ 𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) |
| 97 | 79, 96 | jca 511 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) ∧ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) ∧ 𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ))) |
| 98 | | simpl3 1194 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) ∧ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) ∧ 𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) |
| 99 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) ∧ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) ∧ 𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → 𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) |
| 100 | 98, 99 | sseldd 3984 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) ∧ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) ∧ 𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) |
| 101 | 31 | fourierdlem2 46124 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))) |
| 102 | 32, 101 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))) |
| 103 | 33, 102 | mpbid 232 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝑄 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))) |
| 104 | 103 | simpld 494 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝑄 ∈ (ℝ ↑m
(0...𝑀))) |
| 105 | | elmapi 8889 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑄 ∈ (ℝ
↑m (0...𝑀))
→ 𝑄:(0...𝑀)⟶ℝ) |
| 106 | 104, 105 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ) |
| 107 | 106 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ) |
| 108 | | elfzofz 13715 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀)) |
| 109 | 108 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀)) |
| 110 | 107, 109 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ) |
| 111 | 110 | rexrd 11311 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈
ℝ*) |
| 112 | 111 | adantrr 717 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) → (𝑄‘𝑖) ∈
ℝ*) |
| 113 | 112 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → (𝑄‘𝑖) ∈
ℝ*) |
| 114 | | fzofzp1 13803 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀)) |
| 115 | 114 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀)) |
| 116 | 107, 115 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
| 117 | 116 | adantrr 717 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
| 118 | 117 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
| 119 | 118 | rexrd 11311 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → (𝑄‘(𝑖 + 1)) ∈
ℝ*) |
| 120 | | elioore 13417 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇))) → 𝑡 ∈ ℝ) |
| 121 | 120 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → 𝑡 ∈ ℝ) |
| 122 | | zre 12617 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑙 ∈ ℤ → 𝑙 ∈
ℝ) |
| 123 | 122 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) → 𝑙 ∈ ℝ) |
| 124 | 123 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → 𝑙 ∈ ℝ) |
| 125 | | fourierdlem97.a |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 126 | | fourierdlem97.b |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 127 | 125, 126 | resubcld 11691 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
| 128 | 30, 127 | eqeltrid 2845 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑇 ∈ ℝ) |
| 129 | 128 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → 𝑇 ∈ ℝ) |
| 130 | 124, 129 | remulcld 11291 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → (𝑙 · 𝑇) ∈ ℝ) |
| 131 | 121, 130 | resubcld 11691 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → (𝑡 − (𝑙 · 𝑇)) ∈ ℝ) |
| 132 | 110 | adantrr 717 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) → (𝑄‘𝑖) ∈ ℝ) |
| 133 | 122 | ad2antll 729 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) → 𝑙 ∈ ℝ) |
| 134 | 128 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) → 𝑇 ∈ ℝ) |
| 135 | 133, 134 | remulcld 11291 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) → (𝑙 · 𝑇) ∈ ℝ) |
| 136 | 132, 135 | readdcld 11290 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) → ((𝑄‘𝑖) + (𝑙 · 𝑇)) ∈ ℝ) |
| 137 | 136 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) → ((𝑄‘𝑖) + (𝑙 · 𝑇)) ∈
ℝ*) |
| 138 | 137 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → ((𝑄‘𝑖) + (𝑙 · 𝑇)) ∈
ℝ*) |
| 139 | 117, 135 | readdcld 11290 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) → ((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)) ∈ ℝ) |
| 140 | 139 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) → ((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)) ∈
ℝ*) |
| 141 | 140 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → ((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)) ∈
ℝ*) |
| 142 | | simpr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) |
| 143 | | ioogtlb 45508 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑄‘𝑖) + (𝑙 · 𝑇)) ∈ ℝ* ∧ ((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)) ∈ ℝ* ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → ((𝑄‘𝑖) + (𝑙 · 𝑇)) < 𝑡) |
| 144 | 138, 141,
142, 143 | syl3anc 1373 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → ((𝑄‘𝑖) + (𝑙 · 𝑇)) < 𝑡) |
| 145 | 132 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → (𝑄‘𝑖) ∈ ℝ) |
| 146 | 145, 130,
121 | ltaddsubd 11863 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → (((𝑄‘𝑖) + (𝑙 · 𝑇)) < 𝑡 ↔ (𝑄‘𝑖) < (𝑡 − (𝑙 · 𝑇)))) |
| 147 | 144, 146 | mpbid 232 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → (𝑄‘𝑖) < (𝑡 − (𝑙 · 𝑇))) |
| 148 | | iooltub 45523 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑄‘𝑖) + (𝑙 · 𝑇)) ∈ ℝ* ∧ ((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)) ∈ ℝ* ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → 𝑡 < ((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇))) |
| 149 | 138, 141,
142, 148 | syl3anc 1373 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → 𝑡 < ((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇))) |
| 150 | 121, 130,
118 | ltsubaddd 11859 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → ((𝑡 − (𝑙 · 𝑇)) < (𝑄‘(𝑖 + 1)) ↔ 𝑡 < ((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) |
| 151 | 149, 150 | mpbird 257 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → (𝑡 − (𝑙 · 𝑇)) < (𝑄‘(𝑖 + 1))) |
| 152 | 113, 119,
131, 147, 151 | eliood 45511 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → (𝑡 − (𝑙 · 𝑇)) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
| 153 | 97, 100, 152 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) ∧ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) ∧ 𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → (𝑡 − (𝑙 · 𝑇)) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
| 154 | 95, 153 | sseldd 3984 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) ∧ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) ∧ 𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → (𝑡 − (𝑙 · 𝑇)) ∈ dom 𝐺) |
| 155 | | elioore 13417 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) → 𝑡 ∈ ℝ) |
| 156 | | recn 11245 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 ∈ ℝ → 𝑡 ∈
ℂ) |
| 157 | 156 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℂ) |
| 158 | | zcn 12618 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑙 ∈ ℤ → 𝑙 ∈
ℂ) |
| 159 | 158 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝑙 ∈ ℂ) |
| 160 | 128 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝑇 ∈ ℂ) |
| 161 | 160 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝑇 ∈ ℂ) |
| 162 | 159, 161 | mulcld 11281 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → (𝑙 · 𝑇) ∈ ℂ) |
| 163 | 157, 162 | npcand 11624 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → ((𝑡 − (𝑙 · 𝑇)) + (𝑙 · 𝑇)) = 𝑡) |
| 164 | 163 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → 𝑡 = ((𝑡 − (𝑙 · 𝑇)) + (𝑙 · 𝑇))) |
| 165 | 164 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑡 ∈ ℝ) ∧ (𝑡 − (𝑙 · 𝑇)) ∈ dom 𝐺) → 𝑡 = ((𝑡 − (𝑙 · 𝑇)) + (𝑙 · 𝑇))) |
| 166 | | ovex 7464 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 − (𝑙 · 𝑇)) ∈ V |
| 167 | | eleq1 2829 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑠 = (𝑡 − (𝑙 · 𝑇)) → (𝑠 ∈ dom 𝐺 ↔ (𝑡 − (𝑙 · 𝑇)) ∈ dom 𝐺)) |
| 168 | 167 | anbi2d 630 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑠 = (𝑡 − (𝑙 · 𝑇)) → (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑠 ∈ dom 𝐺) ↔ ((𝜑 ∧ 𝑙 ∈ ℤ) ∧ (𝑡 − (𝑙 · 𝑇)) ∈ dom 𝐺))) |
| 169 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 = (𝑡 − (𝑙 · 𝑇)) → (𝑠 + (𝑙 · 𝑇)) = ((𝑡 − (𝑙 · 𝑇)) + (𝑙 · 𝑇))) |
| 170 | 169 | eleq1d 2826 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑠 = (𝑡 − (𝑙 · 𝑇)) → ((𝑠 + (𝑙 · 𝑇)) ∈ dom 𝐺 ↔ ((𝑡 − (𝑙 · 𝑇)) + (𝑙 · 𝑇)) ∈ dom 𝐺)) |
| 171 | 169 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 = (𝑡 − (𝑙 · 𝑇)) → (𝐺‘(𝑠 + (𝑙 · 𝑇))) = (𝐺‘((𝑡 − (𝑙 · 𝑇)) + (𝑙 · 𝑇)))) |
| 172 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 = (𝑡 − (𝑙 · 𝑇)) → (𝐺‘𝑠) = (𝐺‘(𝑡 − (𝑙 · 𝑇)))) |
| 173 | 171, 172 | eqeq12d 2753 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑠 = (𝑡 − (𝑙 · 𝑇)) → ((𝐺‘(𝑠 + (𝑙 · 𝑇))) = (𝐺‘𝑠) ↔ (𝐺‘((𝑡 − (𝑙 · 𝑇)) + (𝑙 · 𝑇))) = (𝐺‘(𝑡 − (𝑙 · 𝑇))))) |
| 174 | 170, 173 | anbi12d 632 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑠 = (𝑡 − (𝑙 · 𝑇)) → (((𝑠 + (𝑙 · 𝑇)) ∈ dom 𝐺 ∧ (𝐺‘(𝑠 + (𝑙 · 𝑇))) = (𝐺‘𝑠)) ↔ (((𝑡 − (𝑙 · 𝑇)) + (𝑙 · 𝑇)) ∈ dom 𝐺 ∧ (𝐺‘((𝑡 − (𝑙 · 𝑇)) + (𝑙 · 𝑇))) = (𝐺‘(𝑡 − (𝑙 · 𝑇)))))) |
| 175 | 168, 174 | imbi12d 344 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 = (𝑡 − (𝑙 · 𝑇)) → ((((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑠 ∈ dom 𝐺) → ((𝑠 + (𝑙 · 𝑇)) ∈ dom 𝐺 ∧ (𝐺‘(𝑠 + (𝑙 · 𝑇))) = (𝐺‘𝑠))) ↔ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ (𝑡 − (𝑙 · 𝑇)) ∈ dom 𝐺) → (((𝑡 − (𝑙 · 𝑇)) + (𝑙 · 𝑇)) ∈ dom 𝐺 ∧ (𝐺‘((𝑡 − (𝑙 · 𝑇)) + (𝑙 · 𝑇))) = (𝐺‘(𝑡 − (𝑙 · 𝑇))))))) |
| 176 | | ax-resscn 11212 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ℝ
⊆ ℂ |
| 177 | 176 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ℝ ⊆
ℂ) |
| 178 | 6, 177 | fssd 6753 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝐹:ℝ⟶ℂ) |
| 179 | 178 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑙 ∈ ℤ) → 𝐹:ℝ⟶ℂ) |
| 180 | 122 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑙 ∈ ℤ) → 𝑙 ∈ ℝ) |
| 181 | 128 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑙 ∈ ℤ) → 𝑇 ∈ ℝ) |
| 182 | 180, 181 | remulcld 11291 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑙 ∈ ℤ) → (𝑙 · 𝑇) ∈ ℝ) |
| 183 | 178 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑠 ∈ ℝ) → 𝐹:ℝ⟶ℂ) |
| 184 | 128 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑠 ∈ ℝ) → 𝑇 ∈ ℝ) |
| 185 | | simplr 769 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑠 ∈ ℝ) → 𝑙 ∈ ℤ) |
| 186 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℝ) |
| 187 | | fourierdlem97.fper |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
| 188 | 187 | ad4ant14 752 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
| 189 | 183, 184,
185, 186, 188 | fperiodmul 45316 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑠 ∈ ℝ) → (𝐹‘(𝑠 + (𝑙 · 𝑇))) = (𝐹‘𝑠)) |
| 190 | 179, 182,
189, 10 | fperdvper 45934 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑠 ∈ dom 𝐺) → ((𝑠 + (𝑙 · 𝑇)) ∈ dom 𝐺 ∧ (𝐺‘(𝑠 + (𝑙 · 𝑇))) = (𝐺‘𝑠))) |
| 191 | 166, 175,
190 | vtocl 3558 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ (𝑡 − (𝑙 · 𝑇)) ∈ dom 𝐺) → (((𝑡 − (𝑙 · 𝑇)) + (𝑙 · 𝑇)) ∈ dom 𝐺 ∧ (𝐺‘((𝑡 − (𝑙 · 𝑇)) + (𝑙 · 𝑇))) = (𝐺‘(𝑡 − (𝑙 · 𝑇))))) |
| 192 | 191 | simpld 494 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ (𝑡 − (𝑙 · 𝑇)) ∈ dom 𝐺) → ((𝑡 − (𝑙 · 𝑇)) + (𝑙 · 𝑇)) ∈ dom 𝐺) |
| 193 | 192 | adantlr 715 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑡 ∈ ℝ) ∧ (𝑡 − (𝑙 · 𝑇)) ∈ dom 𝐺) → ((𝑡 − (𝑙 · 𝑇)) + (𝑙 · 𝑇)) ∈ dom 𝐺) |
| 194 | 165, 193 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑡 ∈ ℝ) ∧ (𝑡 − (𝑙 · 𝑇)) ∈ dom 𝐺) → 𝑡 ∈ dom 𝐺) |
| 195 | 194 | ex 412 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑡 ∈ ℝ) → ((𝑡 − (𝑙 · 𝑇)) ∈ dom 𝐺 → 𝑡 ∈ dom 𝐺)) |
| 196 | 155, 195 | sylan2 593 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑙 ∈ ℤ) ∧ 𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → ((𝑡 − (𝑙 · 𝑇)) ∈ dom 𝐺 → 𝑡 ∈ dom 𝐺)) |
| 197 | 196 | adantlrl 720 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ)) ∧ 𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → ((𝑡 − (𝑙 · 𝑇)) ∈ dom 𝐺 → 𝑡 ∈ dom 𝐺)) |
| 198 | 197 | 3adantl3 1169 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) ∧ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) ∧ 𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → ((𝑡 − (𝑙 · 𝑇)) ∈ dom 𝐺 → 𝑡 ∈ dom 𝐺)) |
| 199 | 154, 198 | mpd 15 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) ∧ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) ∧ 𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → 𝑡 ∈ dom 𝐺) |
| 200 | 199 | ralrimiva 3146 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) ∧ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → ∀𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))𝑡 ∈ dom 𝐺) |
| 201 | | dfss3 3972 |
. . . . . . . . . . 11
⊢ (((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ dom 𝐺 ↔ ∀𝑡 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))𝑡 ∈ dom 𝐺) |
| 202 | 200, 201 | sylibr 234 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) ∧ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))) → ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ dom 𝐺) |
| 203 | 202 | 3exp 1120 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑖 ∈ (0..^𝑀) ∧ 𝑙 ∈ ℤ) → (((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇))) → ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ dom 𝐺))) |
| 204 | 203 | rexlimdvv 3212 |
. . . . . . . 8
⊢ (𝜑 → (∃𝑖 ∈ (0..^𝑀)∃𝑙 ∈ ℤ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇))) → ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ dom 𝐺)) |
| 205 | 78, 204 | mpd 15 |
. . . . . . 7
⊢ (𝜑 → ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ dom 𝐺) |
| 206 | 205 | sselda 3983 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → 𝑠 ∈ dom 𝐺) |
| 207 | 206 | iftrued 4533 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) = (𝐺‘𝑠)) |
| 208 | 29, 207 | eqtr2d 2778 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) → (𝐺‘𝑠) = (𝐻‘𝑠)) |
| 209 | 208 | mpteq2dva 5242 |
. . 3
⊢ (𝜑 → (𝑠 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ↦ (𝐺‘𝑠)) = (𝑠 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ↦ (𝐻‘𝑠))) |
| 210 | 15 | a1i 11 |
. . . . . 6
⊢ (𝜑 → dom 𝐺 = dom (ℝ D 𝐹)) |
| 211 | 210 | feq2d 6722 |
. . . . 5
⊢ (𝜑 → (𝐺:dom 𝐺⟶ℝ ↔ 𝐺:dom (ℝ D 𝐹)⟶ℝ)) |
| 212 | 12, 211 | mpbird 257 |
. . . 4
⊢ (𝜑 → 𝐺:dom 𝐺⟶ℝ) |
| 213 | 212, 205 | feqresmpt 6978 |
. . 3
⊢ (𝜑 → (𝐺 ↾ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) = (𝑠 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ↦ (𝐺‘𝑠))) |
| 214 | 25, 27 | fmptd 7134 |
. . . 4
⊢ (𝜑 → 𝐻:ℝ⟶ℝ) |
| 215 | 214, 2 | feqresmpt 6978 |
. . 3
⊢ (𝜑 → (𝐻 ↾ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) = (𝑠 ∈ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ↦ (𝐻‘𝑠))) |
| 216 | 209, 213,
215 | 3eqtr4d 2787 |
. 2
⊢ (𝜑 → (𝐺 ↾ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) = (𝐻 ↾ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))))) |
| 217 | 214, 177 | fssd 6753 |
. . 3
⊢ (𝜑 → 𝐻:ℝ⟶ℂ) |
| 218 | 27 | a1i 11 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) → 𝐻 = (𝑠 ∈ ℝ ↦ if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0))) |
| 219 | | eleq1 2829 |
. . . . . . . . 9
⊢ (𝑠 = (𝑥 + 𝑇) → (𝑠 ∈ dom 𝐺 ↔ (𝑥 + 𝑇) ∈ dom 𝐺)) |
| 220 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑠 = (𝑥 + 𝑇) → (𝐺‘𝑠) = (𝐺‘(𝑥 + 𝑇))) |
| 221 | 219, 220 | ifbieq1d 4550 |
. . . . . . . 8
⊢ (𝑠 = (𝑥 + 𝑇) → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) = if((𝑥 + 𝑇) ∈ dom 𝐺, (𝐺‘(𝑥 + 𝑇)), 0)) |
| 222 | 178, 128,
187, 10 | fperdvper 45934 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐺) → ((𝑥 + 𝑇) ∈ dom 𝐺 ∧ (𝐺‘(𝑥 + 𝑇)) = (𝐺‘𝑥))) |
| 223 | 222 | simpld 494 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐺) → (𝑥 + 𝑇) ∈ dom 𝐺) |
| 224 | 223 | iftrued 4533 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐺) → if((𝑥 + 𝑇) ∈ dom 𝐺, (𝐺‘(𝑥 + 𝑇)), 0) = (𝐺‘(𝑥 + 𝑇))) |
| 225 | 221, 224 | sylan9eqr 2799 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐺) ∧ 𝑠 = (𝑥 + 𝑇)) → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) = (𝐺‘(𝑥 + 𝑇))) |
| 226 | 225 | adantllr 719 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) ∧ 𝑠 = (𝑥 + 𝑇)) → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) = (𝐺‘(𝑥 + 𝑇))) |
| 227 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) |
| 228 | 128 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℝ) |
| 229 | 227, 228 | readdcld 11290 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 + 𝑇) ∈ ℝ) |
| 230 | 229 | adantr 480 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) → (𝑥 + 𝑇) ∈ ℝ) |
| 231 | 212 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) → 𝐺:dom 𝐺⟶ℝ) |
| 232 | 223 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) → (𝑥 + 𝑇) ∈ dom 𝐺) |
| 233 | 231, 232 | ffvelcdmd 7105 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) → (𝐺‘(𝑥 + 𝑇)) ∈ ℝ) |
| 234 | 218, 226,
230, 233 | fvmptd 7023 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) → (𝐻‘(𝑥 + 𝑇)) = (𝐺‘(𝑥 + 𝑇))) |
| 235 | 222 | simprd 495 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐺) → (𝐺‘(𝑥 + 𝑇)) = (𝐺‘𝑥)) |
| 236 | 235 | adantlr 715 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) → (𝐺‘(𝑥 + 𝑇)) = (𝐺‘𝑥)) |
| 237 | | eleq1 2829 |
. . . . . . . . 9
⊢ (𝑠 = 𝑥 → (𝑠 ∈ dom 𝐺 ↔ 𝑥 ∈ dom 𝐺)) |
| 238 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑠 = 𝑥 → (𝐺‘𝑠) = (𝐺‘𝑥)) |
| 239 | 237, 238 | ifbieq1d 4550 |
. . . . . . . 8
⊢ (𝑠 = 𝑥 → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) = if(𝑥 ∈ dom 𝐺, (𝐺‘𝑥), 0)) |
| 240 | 239 | adantl 481 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) ∧ 𝑠 = 𝑥) → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) = if(𝑥 ∈ dom 𝐺, (𝐺‘𝑥), 0)) |
| 241 | | simplr 769 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) → 𝑥 ∈ ℝ) |
| 242 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐺) → 𝑥 ∈ dom 𝐺) |
| 243 | 242 | iftrued 4533 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐺) → if(𝑥 ∈ dom 𝐺, (𝐺‘𝑥), 0) = (𝐺‘𝑥)) |
| 244 | 212 | ffvelcdmda 7104 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐺) → (𝐺‘𝑥) ∈ ℝ) |
| 245 | 243, 244 | eqeltrd 2841 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐺) → if(𝑥 ∈ dom 𝐺, (𝐺‘𝑥), 0) ∈ ℝ) |
| 246 | 245 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) → if(𝑥 ∈ dom 𝐺, (𝐺‘𝑥), 0) ∈ ℝ) |
| 247 | 218, 240,
241, 246 | fvmptd 7023 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) → (𝐻‘𝑥) = if(𝑥 ∈ dom 𝐺, (𝐺‘𝑥), 0)) |
| 248 | | simpr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) → 𝑥 ∈ dom 𝐺) |
| 249 | 248 | iftrued 4533 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) → if(𝑥 ∈ dom 𝐺, (𝐺‘𝑥), 0) = (𝐺‘𝑥)) |
| 250 | 247, 249 | eqtr2d 2778 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) → (𝐺‘𝑥) = (𝐻‘𝑥)) |
| 251 | 234, 236,
250 | 3eqtrd 2781 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ dom 𝐺) → (𝐻‘(𝑥 + 𝑇)) = (𝐻‘𝑥)) |
| 252 | 229 | recnd 11289 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 + 𝑇) ∈ ℂ) |
| 253 | 228 | recnd 11289 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℂ) |
| 254 | 252, 253 | negsubd 11626 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑥 + 𝑇) + -𝑇) = ((𝑥 + 𝑇) − 𝑇)) |
| 255 | 227 | recnd 11289 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ) |
| 256 | 255, 253 | pncand 11621 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑥 + 𝑇) − 𝑇) = 𝑥) |
| 257 | 254, 256 | eqtr2d 2778 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 = ((𝑥 + 𝑇) + -𝑇)) |
| 258 | 257 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 + 𝑇) ∈ dom 𝐺) → 𝑥 = ((𝑥 + 𝑇) + -𝑇)) |
| 259 | | simpr 484 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 + 𝑇) ∈ dom 𝐺) → (𝑥 + 𝑇) ∈ dom 𝐺) |
| 260 | | simpll 767 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 + 𝑇) ∈ dom 𝐺) → 𝜑) |
| 261 | 260, 259 | jca 511 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 + 𝑇) ∈ dom 𝐺) → (𝜑 ∧ (𝑥 + 𝑇) ∈ dom 𝐺)) |
| 262 | | eleq1 2829 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝑥 + 𝑇) → (𝑦 ∈ dom 𝐺 ↔ (𝑥 + 𝑇) ∈ dom 𝐺)) |
| 263 | 262 | anbi2d 630 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝑥 + 𝑇) → ((𝜑 ∧ 𝑦 ∈ dom 𝐺) ↔ (𝜑 ∧ (𝑥 + 𝑇) ∈ dom 𝐺))) |
| 264 | | oveq1 7438 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑥 + 𝑇) → (𝑦 + -𝑇) = ((𝑥 + 𝑇) + -𝑇)) |
| 265 | 264 | eleq1d 2826 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝑥 + 𝑇) → ((𝑦 + -𝑇) ∈ dom 𝐺 ↔ ((𝑥 + 𝑇) + -𝑇) ∈ dom 𝐺)) |
| 266 | 264 | fveq2d 6910 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑥 + 𝑇) → (𝐺‘(𝑦 + -𝑇)) = (𝐺‘((𝑥 + 𝑇) + -𝑇))) |
| 267 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑥 + 𝑇) → (𝐺‘𝑦) = (𝐺‘(𝑥 + 𝑇))) |
| 268 | 266, 267 | eqeq12d 2753 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝑥 + 𝑇) → ((𝐺‘(𝑦 + -𝑇)) = (𝐺‘𝑦) ↔ (𝐺‘((𝑥 + 𝑇) + -𝑇)) = (𝐺‘(𝑥 + 𝑇)))) |
| 269 | 265, 268 | anbi12d 632 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝑥 + 𝑇) → (((𝑦 + -𝑇) ∈ dom 𝐺 ∧ (𝐺‘(𝑦 + -𝑇)) = (𝐺‘𝑦)) ↔ (((𝑥 + 𝑇) + -𝑇) ∈ dom 𝐺 ∧ (𝐺‘((𝑥 + 𝑇) + -𝑇)) = (𝐺‘(𝑥 + 𝑇))))) |
| 270 | 263, 269 | imbi12d 344 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝑥 + 𝑇) → (((𝜑 ∧ 𝑦 ∈ dom 𝐺) → ((𝑦 + -𝑇) ∈ dom 𝐺 ∧ (𝐺‘(𝑦 + -𝑇)) = (𝐺‘𝑦))) ↔ ((𝜑 ∧ (𝑥 + 𝑇) ∈ dom 𝐺) → (((𝑥 + 𝑇) + -𝑇) ∈ dom 𝐺 ∧ (𝐺‘((𝑥 + 𝑇) + -𝑇)) = (𝐺‘(𝑥 + 𝑇)))))) |
| 271 | 128 | renegcld 11690 |
. . . . . . . . . . . 12
⊢ (𝜑 → -𝑇 ∈ ℝ) |
| 272 | 160 | mulm1d 11715 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (-1 · 𝑇) = -𝑇) |
| 273 | 272 | eqcomd 2743 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → -𝑇 = (-1 · 𝑇)) |
| 274 | 273 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → -𝑇 = (-1 · 𝑇)) |
| 275 | 274 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + -𝑇) = (𝑦 + (-1 · 𝑇))) |
| 276 | 275 | fveq2d 6910 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + -𝑇)) = (𝐹‘(𝑦 + (-1 · 𝑇)))) |
| 277 | 178 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐹:ℝ⟶ℂ) |
| 278 | 128 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑇 ∈ ℝ) |
| 279 | | 1zzd 12648 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ∈
ℤ) |
| 280 | 279 | znegcld 12724 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → -1 ∈
ℤ) |
| 281 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) |
| 282 | 187 | adantlr 715 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) |
| 283 | 277, 278,
280, 281, 282 | fperiodmul 45316 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + (-1 · 𝑇))) = (𝐹‘𝑦)) |
| 284 | 276, 283 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘(𝑦 + -𝑇)) = (𝐹‘𝑦)) |
| 285 | 178, 271,
284, 10 | fperdvper 45934 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐺) → ((𝑦 + -𝑇) ∈ dom 𝐺 ∧ (𝐺‘(𝑦 + -𝑇)) = (𝐺‘𝑦))) |
| 286 | 270, 285 | vtoclg 3554 |
. . . . . . . . . 10
⊢ ((𝑥 + 𝑇) ∈ dom 𝐺 → ((𝜑 ∧ (𝑥 + 𝑇) ∈ dom 𝐺) → (((𝑥 + 𝑇) + -𝑇) ∈ dom 𝐺 ∧ (𝐺‘((𝑥 + 𝑇) + -𝑇)) = (𝐺‘(𝑥 + 𝑇))))) |
| 287 | 259, 261,
286 | sylc 65 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 + 𝑇) ∈ dom 𝐺) → (((𝑥 + 𝑇) + -𝑇) ∈ dom 𝐺 ∧ (𝐺‘((𝑥 + 𝑇) + -𝑇)) = (𝐺‘(𝑥 + 𝑇)))) |
| 288 | 287 | simpld 494 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 + 𝑇) ∈ dom 𝐺) → ((𝑥 + 𝑇) + -𝑇) ∈ dom 𝐺) |
| 289 | 258, 288 | eqeltrd 2841 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 + 𝑇) ∈ dom 𝐺) → 𝑥 ∈ dom 𝐺) |
| 290 | 289 | stoic1a 1772 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ dom 𝐺) → ¬ (𝑥 + 𝑇) ∈ dom 𝐺) |
| 291 | 290 | iffalsed 4536 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ dom 𝐺) → if((𝑥 + 𝑇) ∈ dom 𝐺, (𝐺‘(𝑥 + 𝑇)), 0) = 0) |
| 292 | 27 | a1i 11 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ dom 𝐺) → 𝐻 = (𝑠 ∈ ℝ ↦ if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0))) |
| 293 | 221 | adantl 481 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ dom 𝐺) ∧ 𝑠 = (𝑥 + 𝑇)) → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) = if((𝑥 + 𝑇) ∈ dom 𝐺, (𝐺‘(𝑥 + 𝑇)), 0)) |
| 294 | 229 | adantr 480 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ dom 𝐺) → (𝑥 + 𝑇) ∈ ℝ) |
| 295 | | 0red 11264 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ dom 𝐺) → 0 ∈ ℝ) |
| 296 | 291, 295 | eqeltrd 2841 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ dom 𝐺) → if((𝑥 + 𝑇) ∈ dom 𝐺, (𝐺‘(𝑥 + 𝑇)), 0) ∈ ℝ) |
| 297 | 292, 293,
294, 296 | fvmptd 7023 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ dom 𝐺) → (𝐻‘(𝑥 + 𝑇)) = if((𝑥 + 𝑇) ∈ dom 𝐺, (𝐺‘(𝑥 + 𝑇)), 0)) |
| 298 | | simpr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ dom 𝐺) → ¬ 𝑥 ∈ dom 𝐺) |
| 299 | 298 | iffalsed 4536 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ dom 𝐺) → if(𝑥 ∈ dom 𝐺, (𝐺‘𝑥), 0) = 0) |
| 300 | 239, 299 | sylan9eqr 2799 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ dom 𝐺) ∧ 𝑠 = 𝑥) → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) = 0) |
| 301 | | simplr 769 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ dom 𝐺) → 𝑥 ∈ ℝ) |
| 302 | 292, 300,
301, 295 | fvmptd 7023 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ dom 𝐺) → (𝐻‘𝑥) = 0) |
| 303 | 291, 297,
302 | 3eqtr4d 2787 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ dom 𝐺) → (𝐻‘(𝑥 + 𝑇)) = (𝐻‘𝑥)) |
| 304 | 251, 303 | pm2.61dan 813 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐻‘(𝑥 + 𝑇)) = (𝐻‘𝑥)) |
| 305 | | elioore 13417 |
. . . . . . . . . 10
⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑠 ∈ ℝ) |
| 306 | 305 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑠 ∈ ℝ) |
| 307 | 305, 25 | sylan2 593 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) ∈ ℝ) |
| 308 | 306, 307,
28 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻‘𝑠) = if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0)) |
| 309 | 308 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻‘𝑠) = if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0)) |
| 310 | 91 | iftrued 4533 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0) = (𝐺‘𝑠)) |
| 311 | 309, 310 | eqtrd 2777 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐻‘𝑠) = (𝐺‘𝑠)) |
| 312 | 311 | mpteq2dva 5242 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠)) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐺‘𝑠))) |
| 313 | 214 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐻:ℝ⟶ℝ) |
| 314 | | ioossre 13448 |
. . . . . . 7
⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ |
| 315 | 314 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ) |
| 316 | 313, 315 | feqresmpt 6978 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐻‘𝑠))) |
| 317 | 212 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐺:dom 𝐺⟶ℝ) |
| 318 | 317, 94 | feqresmpt 6978 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↦ (𝐺‘𝑠))) |
| 319 | 312, 316,
318 | 3eqtr4d 2787 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) |
| 320 | 319, 81 | eqeltrd 2841 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
| 321 | | eqid 2737 |
. . 3
⊢ (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ
↑m (0...𝑚))
∣ (((𝑝‘0) =
𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
| 322 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑧 = 𝑦 → (𝑧 + (𝑙 · 𝑇)) = (𝑦 + (𝑙 · 𝑇))) |
| 323 | 322 | eleq1d 2826 |
. . . . . . 7
⊢ (𝑧 = 𝑦 → ((𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄)) |
| 324 | 323 | rexbidv 3179 |
. . . . . 6
⊢ (𝑧 = 𝑦 → (∃𝑙 ∈ ℤ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄)) |
| 325 | 324 | cbvrabv 3447 |
. . . . 5
⊢ {𝑧 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄} |
| 326 | 325 | uneq2i 4165 |
. . . 4
⊢ ({𝐶, 𝐷} ∪ {𝑧 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄}) = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄}) |
| 327 | 326 | eqcomi 2746 |
. . 3
⊢ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄}) = ({𝐶, 𝐷} ∪ {𝑧 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄}) |
| 328 | 54 | fveq2i 6909 |
. . . 4
⊢
(♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) = (♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄})) |
| 329 | 328 | oveq1i 7441 |
. . 3
⊢
((♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1) = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄})) − 1) |
| 330 | | isoeq5 7341 |
. . . . . 6
⊢ (({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄}) = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}) → (𝑔 Isom < , <
((0...((♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑔 Isom < , <
((0...((♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄})))) |
| 331 | 61, 330 | ax-mp 5 |
. . . . 5
⊢ (𝑔 Isom < , <
((0...((♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑔 Isom < , <
((0...((♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}))) |
| 332 | 331 | iotabii 6546 |
. . . 4
⊢
(℩𝑔𝑔 Isom < , <
((0...((♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄}))) = (℩𝑔𝑔 Isom < , <
((0...((♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}))) |
| 333 | | isoeq1 7337 |
. . . . 5
⊢ (𝑓 = 𝑔 → (𝑓 Isom < , <
((0...((♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄})) ↔ 𝑔 Isom < , <
((0...((♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄})))) |
| 334 | 333 | cbviotavw 6522 |
. . . 4
⊢
(℩𝑓𝑓 Isom < , <
((0...((♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄}))) = (℩𝑔𝑔 Isom < , <
((0...((♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄}))) |
| 335 | 332, 334,
65 | 3eqtr4ri 2776 |
. . 3
⊢ 𝑉 = (℩𝑓𝑓 Isom < , <
((0...((♯‘({𝐶,
𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑙 ∈ ℤ (𝑦 + (𝑙 · 𝑇)) ∈ ran 𝑄}))) |
| 336 | | id 22 |
. . . . 5
⊢ (𝑣 = 𝑥 → 𝑣 = 𝑥) |
| 337 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑣 = 𝑥 → (𝐵 − 𝑣) = (𝐵 − 𝑥)) |
| 338 | 337 | oveq1d 7446 |
. . . . . . 7
⊢ (𝑣 = 𝑥 → ((𝐵 − 𝑣) / 𝑇) = ((𝐵 − 𝑥) / 𝑇)) |
| 339 | 338 | fveq2d 6910 |
. . . . . 6
⊢ (𝑣 = 𝑥 → (⌊‘((𝐵 − 𝑣) / 𝑇)) = (⌊‘((𝐵 − 𝑥) / 𝑇))) |
| 340 | 339 | oveq1d 7446 |
. . . . 5
⊢ (𝑣 = 𝑥 → ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) |
| 341 | 336, 340 | oveq12d 7449 |
. . . 4
⊢ (𝑣 = 𝑥 → (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) |
| 342 | 341 | cbvmptv 5255 |
. . 3
⊢ (𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇))) = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) |
| 343 | | eqeq1 2741 |
. . . . 5
⊢ (𝑢 = 𝑧 → (𝑢 = 𝐵 ↔ 𝑧 = 𝐵)) |
| 344 | | id 22 |
. . . . 5
⊢ (𝑢 = 𝑧 → 𝑢 = 𝑧) |
| 345 | 343, 344 | ifbieq2d 4552 |
. . . 4
⊢ (𝑢 = 𝑧 → if(𝑢 = 𝐵, 𝐴, 𝑢) = if(𝑧 = 𝐵, 𝐴, 𝑧)) |
| 346 | 345 | cbvmptv 5255 |
. . 3
⊢ (𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢)) = (𝑧 ∈ (𝐴(,]𝐵) ↦ if(𝑧 = 𝐵, 𝐴, 𝑧)) |
| 347 | | eqid 2737 |
. . 3
⊢ ((𝑉‘(𝐽 + 1)) − ((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1)))) = ((𝑉‘(𝐽 + 1)) − ((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1)))) |
| 348 | | eqid 2737 |
. . 3
⊢ (𝐻 ↾ (((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘𝐽)))(,)((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1))))) = (𝐻 ↾ (((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘𝐽)))(,)((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1))))) |
| 349 | | eqid 2737 |
. . 3
⊢ (𝑧 ∈ ((((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘𝐽))) + ((𝑉‘(𝐽 + 1)) − ((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1)))))(,)(((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1))) + ((𝑉‘(𝐽 + 1)) − ((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1)))))) ↦ ((𝐻 ↾ (((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘𝐽)))(,)((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1)))))‘(𝑧 − ((𝑉‘(𝐽 + 1)) − ((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1))))))) = (𝑧 ∈ ((((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘𝐽))) + ((𝑉‘(𝐽 + 1)) − ((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1)))))(,)(((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1))) + ((𝑉‘(𝐽 + 1)) − ((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1)))))) ↦ ((𝐻 ↾ (((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘𝐽)))(,)((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1)))))‘(𝑧 − ((𝑉‘(𝐽 + 1)) − ((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1))))))) |
| 350 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑖 = 𝑡 → (𝑄‘𝑖) = (𝑄‘𝑡)) |
| 351 | 350 | breq1d 5153 |
. . . . . . 7
⊢ (𝑖 = 𝑡 → ((𝑄‘𝑖) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑥)) ↔ (𝑄‘𝑡) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑥)))) |
| 352 | 351 | cbvrabv 3447 |
. . . . . 6
⊢ {𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑥))} = {𝑡 ∈ (0..^𝑀) ∣ (𝑄‘𝑡) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑥))} |
| 353 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑥 → ((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑤) = ((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑥)) |
| 354 | 353 | fveq2d 6910 |
. . . . . . . . 9
⊢ (𝑤 = 𝑥 → ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑤)) = ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑥))) |
| 355 | 354 | eqcomd 2743 |
. . . . . . . 8
⊢ (𝑤 = 𝑥 → ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑥)) = ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑤))) |
| 356 | 355 | breq2d 5155 |
. . . . . . 7
⊢ (𝑤 = 𝑥 → ((𝑄‘𝑡) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑥)) ↔ (𝑄‘𝑡) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑤)))) |
| 357 | 356 | rabbidv 3444 |
. . . . . 6
⊢ (𝑤 = 𝑥 → {𝑡 ∈ (0..^𝑀) ∣ (𝑄‘𝑡) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑥))} = {𝑡 ∈ (0..^𝑀) ∣ (𝑄‘𝑡) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑤))}) |
| 358 | 352, 357 | eqtr2id 2790 |
. . . . 5
⊢ (𝑤 = 𝑥 → {𝑡 ∈ (0..^𝑀) ∣ (𝑄‘𝑡) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑤))} = {𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑥))}) |
| 359 | 358 | supeq1d 9486 |
. . . 4
⊢ (𝑤 = 𝑥 → sup({𝑡 ∈ (0..^𝑀) ∣ (𝑄‘𝑡) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑤))}, ℝ, < ) = sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑥))}, ℝ, < )) |
| 360 | 359 | cbvmptv 5255 |
. . 3
⊢ (𝑤 ∈ ℝ ↦
sup({𝑡 ∈ (0..^𝑀) ∣ (𝑄‘𝑡) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑤))}, ℝ, < )) = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑥))}, ℝ, < )) |
| 361 | 31, 30, 32, 33, 217, 304, 320, 34, 35, 321, 327, 329, 335, 342, 346, 66, 347, 348, 349, 360 | fourierdlem90 46211 |
. 2
⊢ (𝜑 → (𝐻 ↾ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) ∈ (((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))–cn→ℂ)) |
| 362 | 216, 361 | eqeltrd 2841 |
1
⊢ (𝜑 → (𝐺 ↾ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) ∈ (((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))–cn→ℂ)) |