Proof of Theorem fourierdlem109
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fourierdlem109.a | . . . 4
⊢ (𝜑 → 𝐴 ∈ ℝ) | 
| 2 | 1 | adantr 480 | . . 3
⊢ ((𝜑 ∧ 0 < 𝑋) → 𝐴 ∈ ℝ) | 
| 3 |  | fourierdlem109.b | . . . 4
⊢ (𝜑 → 𝐵 ∈ ℝ) | 
| 4 | 3 | adantr 480 | . . 3
⊢ ((𝜑 ∧ 0 < 𝑋) → 𝐵 ∈ ℝ) | 
| 5 |  | fourierdlem109.t | . . 3
⊢ 𝑇 = (𝐵 − 𝐴) | 
| 6 |  | fourierdlem109.x | . . . . 5
⊢ (𝜑 → 𝑋 ∈ ℝ) | 
| 7 | 6 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ 0 < 𝑋) → 𝑋 ∈ ℝ) | 
| 8 |  | simpr 484 | . . . 4
⊢ ((𝜑 ∧ 0 < 𝑋) → 0 < 𝑋) | 
| 9 | 7, 8 | elrpd 13075 | . . 3
⊢ ((𝜑 ∧ 0 < 𝑋) → 𝑋 ∈
ℝ+) | 
| 10 |  | fourierdlem109.p | . . 3
⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) | 
| 11 |  | fourierdlem109.m | . . . 4
⊢ (𝜑 → 𝑀 ∈ ℕ) | 
| 12 | 11 | adantr 480 | . . 3
⊢ ((𝜑 ∧ 0 < 𝑋) → 𝑀 ∈ ℕ) | 
| 13 |  | fourierdlem109.q | . . . 4
⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) | 
| 14 | 13 | adantr 480 | . . 3
⊢ ((𝜑 ∧ 0 < 𝑋) → 𝑄 ∈ (𝑃‘𝑀)) | 
| 15 |  | fourierdlem109.f | . . . 4
⊢ (𝜑 → 𝐹:ℝ⟶ℂ) | 
| 16 | 15 | adantr 480 | . . 3
⊢ ((𝜑 ∧ 0 < 𝑋) → 𝐹:ℝ⟶ℂ) | 
| 17 |  | fourierdlem109.fper | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) | 
| 18 | 17 | adantlr 715 | . . 3
⊢ (((𝜑 ∧ 0 < 𝑋) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) | 
| 19 |  | fourierdlem109.fcn | . . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) | 
| 20 | 19 | adantlr 715 | . . 3
⊢ (((𝜑 ∧ 0 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) | 
| 21 |  | fourierdlem109.r | . . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) | 
| 22 | 21 | adantlr 715 | . . 3
⊢ (((𝜑 ∧ 0 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) | 
| 23 |  | fourierdlem109.l | . . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) | 
| 24 | 23 | adantlr 715 | . . 3
⊢ (((𝜑 ∧ 0 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) | 
| 25 | 2, 4, 5, 9, 10, 12, 14, 16, 18, 20, 22, 24 | fourierdlem108 46234 | . 2
⊢ ((𝜑 ∧ 0 < 𝑋) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) | 
| 26 |  | oveq2 7440 | . . . . . . 7
⊢ (𝑋 = 0 → (𝐴 − 𝑋) = (𝐴 − 0)) | 
| 27 | 1 | recnd 11290 | . . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ ℂ) | 
| 28 | 27 | subid1d 11610 | . . . . . . 7
⊢ (𝜑 → (𝐴 − 0) = 𝐴) | 
| 29 | 26, 28 | sylan9eqr 2798 | . . . . . 6
⊢ ((𝜑 ∧ 𝑋 = 0) → (𝐴 − 𝑋) = 𝐴) | 
| 30 |  | oveq2 7440 | . . . . . . 7
⊢ (𝑋 = 0 → (𝐵 − 𝑋) = (𝐵 − 0)) | 
| 31 | 3 | recnd 11290 | . . . . . . . 8
⊢ (𝜑 → 𝐵 ∈ ℂ) | 
| 32 | 31 | subid1d 11610 | . . . . . . 7
⊢ (𝜑 → (𝐵 − 0) = 𝐵) | 
| 33 | 30, 32 | sylan9eqr 2798 | . . . . . 6
⊢ ((𝜑 ∧ 𝑋 = 0) → (𝐵 − 𝑋) = 𝐵) | 
| 34 | 29, 33 | oveq12d 7450 | . . . . 5
⊢ ((𝜑 ∧ 𝑋 = 0) → ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) = (𝐴[,]𝐵)) | 
| 35 | 34 | itgeq1d 45977 | . . . 4
⊢ ((𝜑 ∧ 𝑋 = 0) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) | 
| 36 | 35 | adantlr 715 | . . 3
⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ 𝑋 = 0) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) | 
| 37 |  | simpll 766 | . . . 4
⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → 𝜑) | 
| 38 | 37, 6 | syl 17 | . . . . 5
⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → 𝑋 ∈ ℝ) | 
| 39 |  | 0red 11265 | . . . . 5
⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → 0 ∈
ℝ) | 
| 40 |  | simpr 484 | . . . . . 6
⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → ¬ 𝑋 = 0) | 
| 41 | 40 | neqned 2946 | . . . . 5
⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → 𝑋 ≠ 0) | 
| 42 |  | simplr 768 | . . . . 5
⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → ¬ 0 < 𝑋) | 
| 43 | 38, 39, 41, 42 | lttri5d 45316 | . . . 4
⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → 𝑋 < 0) | 
| 44 | 6 | recnd 11290 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑋 ∈ ℂ) | 
| 45 | 27, 44 | subcld 11621 | . . . . . . . . . . 11
⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℂ) | 
| 46 | 45, 44 | subnegd 11628 | . . . . . . . . . 10
⊢ (𝜑 → ((𝐴 − 𝑋) − -𝑋) = ((𝐴 − 𝑋) + 𝑋)) | 
| 47 | 27, 44 | npcand 11625 | . . . . . . . . . 10
⊢ (𝜑 → ((𝐴 − 𝑋) + 𝑋) = 𝐴) | 
| 48 | 46, 47 | eqtrd 2776 | . . . . . . . . 9
⊢ (𝜑 → ((𝐴 − 𝑋) − -𝑋) = 𝐴) | 
| 49 | 31, 44 | subcld 11621 | . . . . . . . . . . 11
⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℂ) | 
| 50 | 49, 44 | subnegd 11628 | . . . . . . . . . 10
⊢ (𝜑 → ((𝐵 − 𝑋) − -𝑋) = ((𝐵 − 𝑋) + 𝑋)) | 
| 51 | 31, 44 | npcand 11625 | . . . . . . . . . 10
⊢ (𝜑 → ((𝐵 − 𝑋) + 𝑋) = 𝐵) | 
| 52 | 50, 51 | eqtrd 2776 | . . . . . . . . 9
⊢ (𝜑 → ((𝐵 − 𝑋) − -𝑋) = 𝐵) | 
| 53 | 48, 52 | oveq12d 7450 | . . . . . . . 8
⊢ (𝜑 → (((𝐴 − 𝑋) − -𝑋)[,]((𝐵 − 𝑋) − -𝑋)) = (𝐴[,]𝐵)) | 
| 54 | 53 | eqcomd 2742 | . . . . . . 7
⊢ (𝜑 → (𝐴[,]𝐵) = (((𝐴 − 𝑋) − -𝑋)[,]((𝐵 − 𝑋) − -𝑋))) | 
| 55 | 54 | itgeq1d 45977 | . . . . . 6
⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫(((𝐴 − 𝑋) − -𝑋)[,]((𝐵 − 𝑋) − -𝑋))(𝐹‘𝑥) d𝑥) | 
| 56 | 55 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 𝑋 < 0) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫(((𝐴 − 𝑋) − -𝑋)[,]((𝐵 − 𝑋) − -𝑋))(𝐹‘𝑥) d𝑥) | 
| 57 | 1, 6 | resubcld 11692 | . . . . . . 7
⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ) | 
| 58 | 57 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑋 < 0) → (𝐴 − 𝑋) ∈ ℝ) | 
| 59 | 3, 6 | resubcld 11692 | . . . . . . 7
⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ) | 
| 60 | 59 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑋 < 0) → (𝐵 − 𝑋) ∈ ℝ) | 
| 61 |  | eqid 2736 | . . . . . 6
⊢ ((𝐵 − 𝑋) − (𝐴 − 𝑋)) = ((𝐵 − 𝑋) − (𝐴 − 𝑋)) | 
| 62 | 6 | renegcld 11691 | . . . . . . . 8
⊢ (𝜑 → -𝑋 ∈ ℝ) | 
| 63 | 62 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑋 < 0) → -𝑋 ∈ ℝ) | 
| 64 | 6 | lt0neg1d 11833 | . . . . . . . 8
⊢ (𝜑 → (𝑋 < 0 ↔ 0 < -𝑋)) | 
| 65 | 64 | biimpa 476 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑋 < 0) → 0 < -𝑋) | 
| 66 | 63, 65 | elrpd 13075 | . . . . . 6
⊢ ((𝜑 ∧ 𝑋 < 0) → -𝑋 ∈
ℝ+) | 
| 67 |  | fourierdlem109.o | . . . . . . 7
⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) | 
| 68 |  | fveq2 6905 | . . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → (𝑝‘𝑖) = (𝑝‘𝑗)) | 
| 69 |  | oveq1 7439 | . . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1)) | 
| 70 | 69 | fveq2d 6909 | . . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑗 + 1))) | 
| 71 | 68, 70 | breq12d 5155 | . . . . . . . . . . . 12
⊢ (𝑖 = 𝑗 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))) | 
| 72 | 71 | cbvralvw 3236 | . . . . . . . . . . 11
⊢
(∀𝑖 ∈
(0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1))) | 
| 73 | 72 | anbi2i 623 | . . . . . . . . . 10
⊢ ((((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))) | 
| 74 | 73 | a1i 11 | . . . . . . . . 9
⊢ (𝑝 ∈ (ℝ
↑m (0...𝑚))
→ ((((𝑝‘0) =
(𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1))))) | 
| 75 | 74 | rabbiia 3439 | . . . . . . . 8
⊢ {𝑝 ∈ (ℝ
↑m (0...𝑚))
∣ (((𝑝‘0) =
(𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} = {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))} | 
| 76 | 75 | mpteq2i 5246 | . . . . . . 7
⊢ (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ
↑m (0...𝑚))
∣ (((𝑝‘0) =
(𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))}) | 
| 77 | 67, 76 | eqtri 2764 | . . . . . 6
⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))}) | 
| 78 | 10, 11, 13 | fourierdlem11 46138 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)) | 
| 79 | 78 | simp3d 1144 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐴 < 𝐵) | 
| 80 | 1, 3, 6, 79 | ltsub1dd 11876 | . . . . . . . . . 10
⊢ (𝜑 → (𝐴 − 𝑋) < (𝐵 − 𝑋)) | 
| 81 |  | fourierdlem109.h | . . . . . . . . . 10
⊢ 𝐻 = ({(𝐴 − 𝑋), (𝐵 − 𝑋)} ∪ {𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) | 
| 82 |  | fourierdlem109.n | . . . . . . . . . 10
⊢ 𝑁 = ((♯‘𝐻) − 1) | 
| 83 |  | fourierdlem109.16 | . . . . . . . . . 10
⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻)) | 
| 84 | 5, 10, 11, 13, 57, 59, 80, 67, 81, 82, 83 | fourierdlem54 46180 | . . . . . . . . 9
⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻))) | 
| 85 | 84 | simpld 494 | . . . . . . . 8
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁))) | 
| 86 | 85 | simpld 494 | . . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℕ) | 
| 87 | 86 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑋 < 0) → 𝑁 ∈ ℕ) | 
| 88 | 85 | simprd 495 | . . . . . . 7
⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁)) | 
| 89 | 88 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑋 < 0) → 𝑆 ∈ (𝑂‘𝑁)) | 
| 90 | 15 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑋 < 0) → 𝐹:ℝ⟶ℂ) | 
| 91 | 31, 27, 44 | nnncan2d 11656 | . . . . . . . . . . . 12
⊢ (𝜑 → ((𝐵 − 𝑋) − (𝐴 − 𝑋)) = (𝐵 − 𝐴)) | 
| 92 | 91, 5 | eqtr4di 2794 | . . . . . . . . . . 11
⊢ (𝜑 → ((𝐵 − 𝑋) − (𝐴 − 𝑋)) = 𝑇) | 
| 93 | 92 | oveq2d 7448 | . . . . . . . . . 10
⊢ (𝜑 → (𝑥 + ((𝐵 − 𝑋) − (𝐴 − 𝑋))) = (𝑥 + 𝑇)) | 
| 94 | 93 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 + ((𝐵 − 𝑋) − (𝐴 − 𝑋))) = (𝑥 + 𝑇)) | 
| 95 | 94 | fveq2d 6909 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + ((𝐵 − 𝑋) − (𝐴 − 𝑋)))) = (𝐹‘(𝑥 + 𝑇))) | 
| 96 | 95, 17 | eqtrd 2776 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + ((𝐵 − 𝑋) − (𝐴 − 𝑋)))) = (𝐹‘𝑥)) | 
| 97 | 96 | adantlr 715 | . . . . . 6
⊢ (((𝜑 ∧ 𝑋 < 0) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + ((𝐵 − 𝑋) − (𝐴 − 𝑋)))) = (𝐹‘𝑥)) | 
| 98 | 11 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑀 ∈ ℕ) | 
| 99 | 13 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑄 ∈ (𝑃‘𝑀)) | 
| 100 | 15 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℂ) | 
| 101 | 17 | adantlr 715 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) | 
| 102 | 19 | adantlr 715 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) | 
| 103 | 57 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴 − 𝑋) ∈ ℝ) | 
| 104 | 57 | rexrd 11312 | . . . . . . . . . 10
⊢ (𝜑 → (𝐴 − 𝑋) ∈
ℝ*) | 
| 105 |  | pnfxr 11316 | . . . . . . . . . . 11
⊢ +∞
∈ ℝ* | 
| 106 | 105 | a1i 11 | . . . . . . . . . 10
⊢ (𝜑 → +∞ ∈
ℝ*) | 
| 107 | 59 | ltpnfd 13164 | . . . . . . . . . 10
⊢ (𝜑 → (𝐵 − 𝑋) < +∞) | 
| 108 | 104, 106,
59, 80, 107 | eliood 45516 | . . . . . . . . 9
⊢ (𝜑 → (𝐵 − 𝑋) ∈ ((𝐴 − 𝑋)(,)+∞)) | 
| 109 | 108 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − 𝑋) ∈ ((𝐴 − 𝑋)(,)+∞)) | 
| 110 |  | oveq1 7439 | . . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → (𝑥 + (𝑘 · 𝑇)) = (𝑦 + (𝑘 · 𝑇))) | 
| 111 | 110 | eleq1d 2825 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → ((𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄)) | 
| 112 | 111 | rexbidv 3178 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄)) | 
| 113 | 112 | cbvrabv 3446 | . . . . . . . . . 10
⊢ {𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄} | 
| 114 | 113 | uneq2i 4164 | . . . . . . . . 9
⊢ ({(𝐴 − 𝑋), (𝐵 − 𝑋)} ∪ {𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({(𝐴 − 𝑋), (𝐵 − 𝑋)} ∪ {𝑦 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) | 
| 115 | 81, 114 | eqtri 2764 | . . . . . . . 8
⊢ 𝐻 = ({(𝐴 − 𝑋), (𝐵 − 𝑋)} ∪ {𝑦 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) | 
| 116 |  | fourierdlem109.17 | . . . . . . . 8
⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) | 
| 117 |  | fourierdlem109.18 | . . . . . . . 8
⊢ 𝐽 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦)) | 
| 118 |  | simpr 484 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0..^𝑁)) | 
| 119 |  | eqid 2736 | . . . . . . . 8
⊢ ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))) = ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))) | 
| 120 |  | eqid 2736 | . . . . . . . 8
⊢ (𝐹 ↾ ((𝐽‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1))))) = (𝐹 ↾ ((𝐽‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1))))) | 
| 121 |  | eqid 2736 | . . . . . . . 8
⊢ (𝑦 ∈ (((𝐽‘(𝐸‘(𝑆‘𝑗))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))(,)((𝐸‘(𝑆‘(𝑗 + 1))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))) ↦ ((𝐹 ↾ ((𝐽‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))‘(𝑦 − ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1))))))) = (𝑦 ∈ (((𝐽‘(𝐸‘(𝑆‘𝑗))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))(,)((𝐸‘(𝑆‘(𝑗 + 1))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))) ↦ ((𝐹 ↾ ((𝐽‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))‘(𝑦 − ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1))))))) | 
| 122 |  | fourierdlem109.19 | . . . . . . . . 9
⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < )) | 
| 123 |  | fveq2 6905 | . . . . . . . . . . . . 13
⊢ (𝑗 = 𝑖 → (𝑄‘𝑗) = (𝑄‘𝑖)) | 
| 124 | 123 | breq1d 5152 | . . . . . . . . . . . 12
⊢ (𝑗 = 𝑖 → ((𝑄‘𝑗) ≤ (𝐽‘(𝐸‘𝑥)) ↔ (𝑄‘𝑖) ≤ (𝐽‘(𝐸‘𝑥)))) | 
| 125 | 124 | cbvrabv 3446 | . . . . . . . . . . 11
⊢ {𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) ≤ (𝐽‘(𝐸‘𝑥))} = {𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐽‘(𝐸‘𝑥))} | 
| 126 | 125 | supeq1i 9488 | . . . . . . . . . 10
⊢
sup({𝑗 ∈
(0..^𝑀) ∣ (𝑄‘𝑗) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < ) = sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < ) | 
| 127 | 126 | mpteq2i 5246 | . . . . . . . . 9
⊢ (𝑥 ∈ ℝ ↦
sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < )) = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < )) | 
| 128 | 122, 127 | eqtri 2764 | . . . . . . . 8
⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < )) | 
| 129 | 10, 5, 98, 99, 100, 101, 102, 103, 109, 67, 115, 82, 83, 116, 117, 118, 119, 120, 121, 128 | fourierdlem90 46216 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)) | 
| 130 | 129 | adantlr 715 | . . . . . 6
⊢ (((𝜑 ∧ 𝑋 < 0) ∧ 𝑗 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)) | 
| 131 | 21 | adantlr 715 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖))) | 
| 132 |  | eqid 2736 | . . . . . . . 8
⊢ (𝑖 ∈ (0..^𝑀) ↦ 𝑅) = (𝑖 ∈ (0..^𝑀) ↦ 𝑅) | 
| 133 | 10, 5, 98, 99, 100, 101, 102, 131, 103, 109, 67, 115, 82, 83, 116, 117, 118, 119, 128, 132 | fourierdlem89 46215 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → if((𝐽‘(𝐸‘(𝑆‘𝑗))) = (𝑄‘(𝐼‘(𝑆‘𝑗))), ((𝑖 ∈ (0..^𝑀) ↦ 𝑅)‘(𝐼‘(𝑆‘𝑗))), (𝐹‘(𝐽‘(𝐸‘(𝑆‘𝑗))))) ∈ ((𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) limℂ (𝑆‘𝑗))) | 
| 134 | 133 | adantlr 715 | . . . . . 6
⊢ (((𝜑 ∧ 𝑋 < 0) ∧ 𝑗 ∈ (0..^𝑁)) → if((𝐽‘(𝐸‘(𝑆‘𝑗))) = (𝑄‘(𝐼‘(𝑆‘𝑗))), ((𝑖 ∈ (0..^𝑀) ↦ 𝑅)‘(𝐼‘(𝑆‘𝑗))), (𝐹‘(𝐽‘(𝐸‘(𝑆‘𝑗))))) ∈ ((𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) limℂ (𝑆‘𝑗))) | 
| 135 | 23 | adantlr 715 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) | 
| 136 |  | eqid 2736 | . . . . . . . 8
⊢ (𝑖 ∈ (0..^𝑀) ↦ 𝐿) = (𝑖 ∈ (0..^𝑀) ↦ 𝐿) | 
| 137 | 10, 5, 98, 99, 100, 101, 102, 135, 103, 109, 67, 115, 82, 83, 116, 117, 118, 119, 128, 136 | fourierdlem91 46217 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → if((𝐸‘(𝑆‘(𝑗 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)), ((𝑖 ∈ (0..^𝑀) ↦ 𝐿)‘(𝐼‘(𝑆‘𝑗))), (𝐹‘(𝐸‘(𝑆‘(𝑗 + 1))))) ∈ ((𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) limℂ (𝑆‘(𝑗 + 1)))) | 
| 138 | 137 | adantlr 715 | . . . . . 6
⊢ (((𝜑 ∧ 𝑋 < 0) ∧ 𝑗 ∈ (0..^𝑁)) → if((𝐸‘(𝑆‘(𝑗 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)), ((𝑖 ∈ (0..^𝑀) ↦ 𝐿)‘(𝐼‘(𝑆‘𝑗))), (𝐹‘(𝐸‘(𝑆‘(𝑗 + 1))))) ∈ ((𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) limℂ (𝑆‘(𝑗 + 1)))) | 
| 139 | 58, 60, 61, 66, 77, 87, 89, 90, 97, 130, 134, 138 | fourierdlem108 46234 | . . . . 5
⊢ ((𝜑 ∧ 𝑋 < 0) → ∫(((𝐴 − 𝑋) − -𝑋)[,]((𝐵 − 𝑋) − -𝑋))(𝐹‘𝑥) d𝑥 = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) | 
| 140 | 56, 139 | eqtr2d 2777 | . . . 4
⊢ ((𝜑 ∧ 𝑋 < 0) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) | 
| 141 | 37, 43, 140 | syl2anc 584 | . . 3
⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) | 
| 142 | 36, 141 | pm2.61dan 812 | . 2
⊢ ((𝜑 ∧ ¬ 0 < 𝑋) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) | 
| 143 | 25, 142 | pm2.61dan 812 | 1
⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) |