Proof of Theorem fourierdlem65
Step | Hyp | Ref
| Expression |
1 | | fourierdlem65.l |
. . . . . 6
⊢ 𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦)) |
2 | 1 | a1i 11 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))) |
3 | | simpr 485 |
. . . . . . . 8
⊢ (((𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → 𝑦 = (𝐸‘(𝑆‘𝑗))) |
4 | | simpl 483 |
. . . . . . . 8
⊢ (((𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → (𝐸‘(𝑆‘𝑗)) = 𝐵) |
5 | 3, 4 | eqtrd 2778 |
. . . . . . 7
⊢ (((𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → 𝑦 = 𝐵) |
6 | 5 | iftrued 4467 |
. . . . . 6
⊢ (((𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → if(𝑦 = 𝐵, 𝐴, 𝑦) = 𝐴) |
7 | 6 | adantll 711 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → if(𝑦 = 𝐵, 𝐴, 𝑦) = 𝐴) |
8 | | fourierdlem65.p |
. . . . . . . . . . 11
⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
9 | | fourierdlem65.m |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℕ) |
10 | | fourierdlem65.q |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) |
11 | 8, 9, 10 | fourierdlem11 43659 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)) |
12 | 11 | simp1d 1141 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ ℝ) |
13 | 11 | simp2d 1142 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ ℝ) |
14 | 11 | simp3d 1143 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 < 𝐵) |
15 | | fourierdlem65.t |
. . . . . . . . 9
⊢ 𝑇 = (𝐵 − 𝐴) |
16 | | fourierdlem65.e |
. . . . . . . . 9
⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) |
17 | 12, 13, 14, 15, 16 | fourierdlem4 43652 |
. . . . . . . 8
⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵)) |
18 | 17 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐸:ℝ⟶(𝐴(,]𝐵)) |
19 | | fourierdlem65.c |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐶 ∈ ℝ) |
20 | | ioossre 13140 |
. . . . . . . . . . . . . . . 16
⊢ (𝐶(,)+∞) ⊆
ℝ |
21 | | fourierdlem65.d |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞)) |
22 | 20, 21 | sselid 3919 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐷 ∈ ℝ) |
23 | 19 | rexrd 11025 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐶 ∈
ℝ*) |
24 | | pnfxr 11029 |
. . . . . . . . . . . . . . . . 17
⊢ +∞
∈ ℝ* |
25 | 24 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → +∞ ∈
ℝ*) |
26 | | ioogtlb 43033 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐶 ∈ ℝ*
∧ +∞ ∈ ℝ* ∧ 𝐷 ∈ (𝐶(,)+∞)) → 𝐶 < 𝐷) |
27 | 23, 25, 21, 26 | syl3anc 1370 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐶 < 𝐷) |
28 | | fourierdlem65.o |
. . . . . . . . . . . . . . 15
⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m
(0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
29 | | id 22 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥) |
30 | 15 | eqcomi 2747 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐵 − 𝐴) = 𝑇 |
31 | 30 | oveq2i 7286 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 · (𝐵 − 𝐴)) = (𝑘 · 𝑇) |
32 | 31 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑥 → (𝑘 · (𝐵 − 𝐴)) = (𝑘 · 𝑇)) |
33 | 29, 32 | oveq12d 7293 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑥 → (𝑦 + (𝑘 · (𝐵 − 𝐴))) = (𝑥 + (𝑘 · 𝑇))) |
34 | 33 | eleq1d 2823 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = 𝑥 → ((𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄)) |
35 | 34 | rexbidv 3226 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = 𝑥 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄)) |
36 | 35 | cbvrabv 3426 |
. . . . . . . . . . . . . . . 16
⊢ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄} = {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} |
37 | 36 | uneq2i 4094 |
. . . . . . . . . . . . . . 15
⊢ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}) = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) |
38 | | fourierdlem65.n |
. . . . . . . . . . . . . . 15
⊢ 𝑁 = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1) |
39 | | fourierdlem65.s |
. . . . . . . . . . . . . . 15
⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))) |
40 | 15, 8, 9, 10, 19, 22, 27, 28, 37, 38, 39 | fourierdlem54 43701 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})))) |
41 | 40 | simpld 495 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁))) |
42 | 41 | simprd 496 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁)) |
43 | 41 | simpld 495 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℕ) |
44 | 28 | fourierdlem2 43650 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ → (𝑆 ∈ (𝑂‘𝑁) ↔ (𝑆 ∈ (ℝ ↑m
(0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))) |
45 | 43, 44 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑆 ∈ (𝑂‘𝑁) ↔ (𝑆 ∈ (ℝ ↑m
(0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))) |
46 | 42, 45 | mpbid 231 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑆 ∈ (ℝ ↑m
(0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))) |
47 | 46 | simpld 495 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆 ∈ (ℝ ↑m
(0...𝑁))) |
48 | | elmapi 8637 |
. . . . . . . . . 10
⊢ (𝑆 ∈ (ℝ
↑m (0...𝑁))
→ 𝑆:(0...𝑁)⟶ℝ) |
49 | 47, 48 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑆:(0...𝑁)⟶ℝ) |
50 | 49 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑆:(0...𝑁)⟶ℝ) |
51 | | elfzofz 13403 |
. . . . . . . . 9
⊢ (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ (0...𝑁)) |
52 | 51 | adantl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0...𝑁)) |
53 | 50, 52 | ffvelrnd 6962 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) ∈ ℝ) |
54 | 18, 53 | ffvelrnd 6962 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐸‘(𝑆‘𝑗)) ∈ (𝐴(,]𝐵)) |
55 | 54 | adantr 481 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘𝑗)) ∈ (𝐴(,]𝐵)) |
56 | 12 | ad2antrr 723 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐴 ∈ ℝ) |
57 | 2, 7, 55, 56 | fvmptd 6882 |
. . . 4
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐿‘(𝐸‘(𝑆‘𝑗))) = 𝐴) |
58 | 57 | oveq2d 7291 |
. . 3
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝐸‘(𝑆‘(𝑗 + 1))) − 𝐴)) |
59 | 13 | ad2antrr 723 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐵 ∈ ℝ) |
60 | 14 | ad2antrr 723 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐴 < 𝐵) |
61 | 53 | adantr 481 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘𝑗) ∈ ℝ) |
62 | | simpr 485 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘𝑗)) = 𝐵) |
63 | | fzofzp1 13484 |
. . . . . . . . 9
⊢ (𝑗 ∈ (0..^𝑁) → (𝑗 + 1) ∈ (0...𝑁)) |
64 | 63 | adantl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 + 1) ∈ (0...𝑁)) |
65 | 50, 64 | ffvelrnd 6962 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ∈ ℝ) |
66 | 65 | adantr 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘(𝑗 + 1)) ∈ ℝ) |
67 | | elfzoelz 13387 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ ℤ) |
68 | 67 | zred 12426 |
. . . . . . . . . 10
⊢ (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ ℝ) |
69 | 68 | adantl 482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ ℝ) |
70 | 69 | ltp1d 11905 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 < (𝑗 + 1)) |
71 | 40 | simprd 496 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))) |
72 | 71 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))) |
73 | | isorel 7197 |
. . . . . . . . 9
⊢ ((𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) ∧ (𝑗 ∈ (0...𝑁) ∧ (𝑗 + 1) ∈ (0...𝑁))) → (𝑗 < (𝑗 + 1) ↔ (𝑆‘𝑗) < (𝑆‘(𝑗 + 1)))) |
74 | 72, 52, 64, 73 | syl12anc 834 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 < (𝑗 + 1) ↔ (𝑆‘𝑗) < (𝑆‘(𝑗 + 1)))) |
75 | 70, 74 | mpbid 231 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) < (𝑆‘(𝑗 + 1))) |
76 | 75 | adantr 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘𝑗) < (𝑆‘(𝑗 + 1))) |
77 | | isof1o 7194 |
. . . . . . . . . . 11
⊢ (𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) → 𝑆:(0...𝑁)–1-1-onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) |
78 | | f1ofo 6723 |
. . . . . . . . . . 11
⊢ (𝑆:(0...𝑁)–1-1-onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}) → 𝑆:(0...𝑁)–onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) |
79 | 71, 77, 78 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆:(0...𝑁)–onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) |
80 | 79 | ad3antrrr 727 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → 𝑆:(0...𝑁)–onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) |
81 | 19 | ad2antrr 723 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → 𝐶 ∈ ℝ) |
82 | 22 | ad2antrr 723 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → 𝐷 ∈ ℝ) |
83 | 13, 12 | resubcld 11403 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
84 | 15, 83 | eqeltrid 2843 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑇 ∈ ℝ) |
85 | 84 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑇 ∈ ℝ) |
86 | 53, 85 | readdcld 11004 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + 𝑇) ∈ ℝ) |
87 | 86 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) ∈ ℝ) |
88 | 19 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 ∈ ℝ) |
89 | 28, 43, 42 | fourierdlem15 43663 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷)) |
90 | 89 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷)) |
91 | 90, 52 | ffvelrnd 6962 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) ∈ (𝐶[,]𝐷)) |
92 | 22 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐷 ∈ ℝ) |
93 | | elicc2 13144 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝑆‘𝑗) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘𝑗) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘𝑗) ∧ (𝑆‘𝑗) ≤ 𝐷))) |
94 | 88, 92, 93 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘𝑗) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘𝑗) ∧ (𝑆‘𝑗) ≤ 𝐷))) |
95 | 91, 94 | mpbid 231 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘𝑗) ∧ (𝑆‘𝑗) ≤ 𝐷)) |
96 | 95 | simp2d 1142 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 ≤ (𝑆‘𝑗)) |
97 | 12, 13 | posdifd 11562 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
98 | 14, 97 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 0 < (𝐵 − 𝐴)) |
99 | 98, 15 | breqtrrdi 5116 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 0 < 𝑇) |
100 | 84, 99 | elrpd 12769 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑇 ∈
ℝ+) |
101 | 100 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑇 ∈
ℝ+) |
102 | 53, 101 | ltaddrpd 12805 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) < ((𝑆‘𝑗) + 𝑇)) |
103 | 88, 53, 86, 96, 102 | lelttrd 11133 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 < ((𝑆‘𝑗) + 𝑇)) |
104 | 88, 86, 103 | ltled 11123 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 ≤ ((𝑆‘𝑗) + 𝑇)) |
105 | 104 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → 𝐶 ≤ ((𝑆‘𝑗) + 𝑇)) |
106 | 65 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → (𝑆‘(𝑗 + 1)) ∈ ℝ) |
107 | | simpr 485 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) |
108 | 87, 106 | ltnled 11122 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → (((𝑆‘𝑗) + 𝑇) < (𝑆‘(𝑗 + 1)) ↔ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇))) |
109 | 107, 108 | mpbird 256 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) < (𝑆‘(𝑗 + 1))) |
110 | 90, 64 | ffvelrnd 6962 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ∈ (𝐶[,]𝐷)) |
111 | | elicc2 13144 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝑆‘(𝑗 + 1)) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘(𝑗 + 1)) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘(𝑗 + 1)) ∧ (𝑆‘(𝑗 + 1)) ≤ 𝐷))) |
112 | 88, 92, 111 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘(𝑗 + 1)) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘(𝑗 + 1)) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘(𝑗 + 1)) ∧ (𝑆‘(𝑗 + 1)) ≤ 𝐷))) |
113 | 110, 112 | mpbid 231 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘(𝑗 + 1)) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘(𝑗 + 1)) ∧ (𝑆‘(𝑗 + 1)) ≤ 𝐷)) |
114 | 113 | simp3d 1143 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ≤ 𝐷) |
115 | 114 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → (𝑆‘(𝑗 + 1)) ≤ 𝐷) |
116 | 87, 106, 82, 109, 115 | ltletrd 11135 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) < 𝐷) |
117 | 87, 82, 116 | ltled 11123 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) ≤ 𝐷) |
118 | 81, 82, 87, 105, 117 | eliccd 43042 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) ∈ (𝐶[,]𝐷)) |
119 | 118 | adantlr 712 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) ∈ (𝐶[,]𝐷)) |
120 | 16 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))) |
121 | | id 22 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = (𝑆‘𝑗) → 𝑥 = (𝑆‘𝑗)) |
122 | | oveq2 7283 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = (𝑆‘𝑗) → (𝐵 − 𝑥) = (𝐵 − (𝑆‘𝑗))) |
123 | 122 | oveq1d 7290 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = (𝑆‘𝑗) → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − (𝑆‘𝑗)) / 𝑇)) |
124 | 123 | fveq2d 6778 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = (𝑆‘𝑗) → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇))) |
125 | 124 | oveq1d 7290 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = (𝑆‘𝑗) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) |
126 | 121, 125 | oveq12d 7293 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = (𝑆‘𝑗) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))) |
127 | 126 | adantl 482 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑥 = (𝑆‘𝑗)) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))) |
128 | 13 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐵 ∈ ℝ) |
129 | 128, 53 | resubcld 11403 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝑆‘𝑗)) ∈ ℝ) |
130 | 129, 101 | rerpdivcld 12803 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝑆‘𝑗)) / 𝑇) ∈ ℝ) |
131 | 130 | flcld 13518 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∈ ℤ) |
132 | 131 | zred 12426 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∈ ℝ) |
133 | 132, 85 | remulcld 11005 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇) ∈ ℝ) |
134 | 53, 133 | readdcld 11004 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) ∈ ℝ) |
135 | 120, 127,
53, 134 | fvmptd 6882 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐸‘(𝑆‘𝑗)) = ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))) |
136 | 135 | oveq1d 7290 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) = (((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) − (𝑆‘𝑗))) |
137 | 136 | oveq1d 7290 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) = ((((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) − (𝑆‘𝑗)) / 𝑇)) |
138 | 53 | recnd 11003 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) ∈ ℂ) |
139 | 133 | recnd 11003 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇) ∈ ℂ) |
140 | 138, 139 | pncan2d 11334 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) − (𝑆‘𝑗)) = ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) |
141 | 140 | oveq1d 7290 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) − (𝑆‘𝑗)) / 𝑇) = (((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇) / 𝑇)) |
142 | 132 | recnd 11003 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∈ ℂ) |
143 | 85 | recnd 11003 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑇 ∈ ℂ) |
144 | 101 | rpne0d 12777 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑇 ≠ 0) |
145 | 142, 143,
144 | divcan4d 11757 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇) / 𝑇) = (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇))) |
146 | 137, 141,
145 | 3eqtrd 2782 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) = (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇))) |
147 | 146, 131 | eqeltrd 2839 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) ∈ ℤ) |
148 | | peano2zm 12363 |
. . . . . . . . . . . . . 14
⊢ ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) ∈ ℤ → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) ∈
ℤ) |
149 | 147, 148 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) ∈
ℤ) |
150 | 149 | ad2antrr 723 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) ∈
ℤ) |
151 | 30 | oveq2i 7286 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴)) = (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇) |
152 | 151 | oveq2i 7286 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))) = (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇)) |
153 | 152 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))) = (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇))) |
154 | 135 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘𝑗)) = ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))) |
155 | | oveq1 7282 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐸‘(𝑆‘𝑗)) = 𝐵 → ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) = (𝐵 − (𝑆‘𝑗))) |
156 | 155 | eqcomd 2744 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐸‘(𝑆‘𝑗)) = 𝐵 → (𝐵 − (𝑆‘𝑗)) = ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))) |
157 | 156 | oveq1d 7290 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐸‘(𝑆‘𝑗)) = 𝐵 → ((𝐵 − (𝑆‘𝑗)) / 𝑇) = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) |
158 | 157 | fveq2d 6778 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐸‘(𝑆‘𝑗)) = 𝐵 → (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) = (⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇))) |
159 | 158 | oveq1d 7290 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐸‘(𝑆‘𝑗)) = 𝐵 → ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇) = ((⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) |
160 | 159 | oveq2d 7291 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐸‘(𝑆‘𝑗)) = 𝐵 → ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) = ((𝑆‘𝑗) + ((⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) · 𝑇))) |
161 | 160 | adantl 482 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) = ((𝑆‘𝑗) + ((⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) · 𝑇))) |
162 | 146, 142 | eqeltrd 2839 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) ∈ ℂ) |
163 | | 1cnd 10970 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 1 ∈ ℂ) |
164 | 162, 163,
143 | subdird 11432 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇) = (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − (1 · 𝑇))) |
165 | 84 | recnd 11003 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝑇 ∈ ℂ) |
166 | 165 | mulid2d 10993 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (1 · 𝑇) = 𝑇) |
167 | 166 | oveq2d 7291 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − (1 · 𝑇)) = (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − 𝑇)) |
168 | 167 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − (1 · 𝑇)) = (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − 𝑇)) |
169 | 164, 168 | eqtrd 2778 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇) = (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − 𝑇)) |
170 | 169 | oveq2d 7291 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇)) = (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − 𝑇))) |
171 | 162, 143 | mulcld 10995 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) ∈ ℂ) |
172 | 138, 143,
171 | ppncand 11372 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) − 𝑇)) = ((𝑆‘𝑗) + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇))) |
173 | | flid 13528 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) ∈ ℤ →
(⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) |
174 | 147, 173 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) |
175 | 174 | eqcomd 2744 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) = (⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇))) |
176 | 175 | oveq1d 7290 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) = ((⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) |
177 | 176 | oveq2d 7291 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇)) = ((𝑆‘𝑗) + ((⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) · 𝑇))) |
178 | 170, 172,
177 | 3eqtrrd 2783 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + ((⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) = (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇))) |
179 | 178 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝑆‘𝑗) + ((⌊‘(((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) = (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇))) |
180 | 154, 161,
179 | 3eqtrrd 2783 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · 𝑇)) = (𝐸‘(𝑆‘𝑗))) |
181 | 153, 180,
62 | 3eqtrd 2782 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))) = 𝐵) |
182 | 8 | fourierdlem2 43650 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))) |
183 | 9, 182 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))) |
184 | 10, 183 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑄 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))) |
185 | 184 | simprd 496 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))) |
186 | 185 | simpld 495 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵)) |
187 | 186 | simprd 496 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑄‘𝑀) = 𝐵) |
188 | 187 | eqcomd 2744 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐵 = (𝑄‘𝑀)) |
189 | 8, 9, 10 | fourierdlem15 43663 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) |
190 | | ffn 6600 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑄:(0...𝑀)⟶(𝐴[,]𝐵) → 𝑄 Fn (0...𝑀)) |
191 | 189, 190 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑄 Fn (0...𝑀)) |
192 | 9 | nnnn0d 12293 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
193 | | nn0uz 12620 |
. . . . . . . . . . . . . . . . . . 19
⊢
ℕ0 = (ℤ≥‘0) |
194 | 192, 193 | eleqtrdi 2849 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑀 ∈
(ℤ≥‘0)) |
195 | | eluzfz2 13264 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑀 ∈
(ℤ≥‘0) → 𝑀 ∈ (0...𝑀)) |
196 | 194, 195 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈ (0...𝑀)) |
197 | | fnfvelrn 6958 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑄 Fn (0...𝑀) ∧ 𝑀 ∈ (0...𝑀)) → (𝑄‘𝑀) ∈ ran 𝑄) |
198 | 191, 196,
197 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑄‘𝑀) ∈ ran 𝑄) |
199 | 188, 198 | eqeltrd 2839 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐵 ∈ ran 𝑄) |
200 | 199 | ad2antrr 723 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐵 ∈ ran 𝑄) |
201 | 181, 200 | eqeltrd 2839 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))) ∈ ran 𝑄) |
202 | 201 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))) ∈ ran 𝑄) |
203 | | oveq1 7282 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) → (𝑘 · (𝐵 − 𝐴)) = (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))) |
204 | 203 | oveq2d 7291 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) → (((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴))) = (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴)))) |
205 | 204 | eleq1d 2823 |
. . . . . . . . . . . . 13
⊢ (𝑘 = ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) → ((((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))) ∈ ran 𝑄)) |
206 | 205 | rspcev 3561 |
. . . . . . . . . . . 12
⊢
((((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) ∈ ℤ ∧ (((𝑆‘𝑗) + 𝑇) + (((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) − 1) · (𝐵 − 𝐴))) ∈ ran 𝑄) → ∃𝑘 ∈ ℤ (((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄) |
207 | 150, 202,
206 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ∃𝑘 ∈ ℤ (((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄) |
208 | | oveq1 7282 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = ((𝑆‘𝑗) + 𝑇) → (𝑦 + (𝑘 · (𝐵 − 𝐴))) = (((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴)))) |
209 | 208 | eleq1d 2823 |
. . . . . . . . . . . . 13
⊢ (𝑦 = ((𝑆‘𝑗) + 𝑇) → ((𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ (((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)) |
210 | 209 | rexbidv 3226 |
. . . . . . . . . . . 12
⊢ (𝑦 = ((𝑆‘𝑗) + 𝑇) → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)) |
211 | 210 | elrab 3624 |
. . . . . . . . . . 11
⊢ (((𝑆‘𝑗) + 𝑇) ∈ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄} ↔ (((𝑆‘𝑗) + 𝑇) ∈ (𝐶[,]𝐷) ∧ ∃𝑘 ∈ ℤ (((𝑆‘𝑗) + 𝑇) + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)) |
212 | 119, 207,
211 | sylanbrc 583 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) ∈ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}) |
213 | | elun2 4111 |
. . . . . . . . . 10
⊢ (((𝑆‘𝑗) + 𝑇) ∈ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄} → ((𝑆‘𝑗) + 𝑇) ∈ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) |
214 | 212, 213 | syl 17 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) ∈ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) |
215 | | foelrn 6982 |
. . . . . . . . 9
⊢ ((𝑆:(0...𝑁)–onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}) ∧ ((𝑆‘𝑗) + 𝑇) ∈ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) → ∃𝑖 ∈ (0...𝑁)((𝑆‘𝑗) + 𝑇) = (𝑆‘𝑖)) |
216 | 80, 214, 215 | syl2anc 584 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ∃𝑖 ∈ (0...𝑁)((𝑆‘𝑗) + 𝑇) = (𝑆‘𝑖)) |
217 | | eqcom 2745 |
. . . . . . . . 9
⊢ (((𝑆‘𝑗) + 𝑇) = (𝑆‘𝑖) ↔ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) |
218 | 217 | rexbii 3181 |
. . . . . . . 8
⊢
(∃𝑖 ∈
(0...𝑁)((𝑆‘𝑗) + 𝑇) = (𝑆‘𝑖) ↔ ∃𝑖 ∈ (0...𝑁)(𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) |
219 | 216, 218 | sylib 217 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ∃𝑖 ∈ (0...𝑁)(𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) |
220 | 102 | ad2antrr 723 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → (𝑆‘𝑗) < ((𝑆‘𝑗) + 𝑇)) |
221 | 217 | biimpri 227 |
. . . . . . . . . . . . . 14
⊢ ((𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇) → ((𝑆‘𝑗) + 𝑇) = (𝑆‘𝑖)) |
222 | 221 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) = (𝑆‘𝑖)) |
223 | 220, 222 | breqtrd 5100 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → (𝑆‘𝑗) < (𝑆‘𝑖)) |
224 | 109 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) + 𝑇) < (𝑆‘(𝑗 + 1))) |
225 | 222, 224 | eqbrtrrd 5098 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))) |
226 | 223, 225 | jca 512 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) |
227 | 226 | adantlr 712 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) |
228 | | simplll 772 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → (𝜑 ∧ 𝑗 ∈ (0..^𝑁))) |
229 | | simplr 766 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → 𝑖 ∈ (0...𝑁)) |
230 | | elfzelz 13256 |
. . . . . . . . . . . . 13
⊢ (𝑖 ∈ (0...𝑁) → 𝑖 ∈ ℤ) |
231 | 230 | ad2antlr 724 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) → 𝑖 ∈ ℤ) |
232 | 67 | ad3antlr 728 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) → 𝑗 ∈ ℤ) |
233 | | simpr 485 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘𝑖)) → (𝑆‘𝑗) < (𝑆‘𝑖)) |
234 | 72 | ad2antrr 723 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘𝑖)) → 𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))) |
235 | 52 | ad2antrr 723 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘𝑖)) → 𝑗 ∈ (0...𝑁)) |
236 | | simplr 766 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘𝑖)) → 𝑖 ∈ (0...𝑁)) |
237 | | isorel 7197 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑖 ∈ (0...𝑁))) → (𝑗 < 𝑖 ↔ (𝑆‘𝑗) < (𝑆‘𝑖))) |
238 | 234, 235,
236, 237 | syl12anc 834 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘𝑖)) → (𝑗 < 𝑖 ↔ (𝑆‘𝑗) < (𝑆‘𝑖))) |
239 | 233, 238 | mpbird 256 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘𝑖)) → 𝑗 < 𝑖) |
240 | 239 | adantrr 714 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) → 𝑗 < 𝑖) |
241 | | simpr 485 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))) → (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))) |
242 | 72 | ad2antrr 723 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))) → 𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))) |
243 | | simplr 766 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))) → 𝑖 ∈ (0...𝑁)) |
244 | 64 | ad2antrr 723 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))) → (𝑗 + 1) ∈ (0...𝑁)) |
245 | | isorel 7197 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑆 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) ∧ (𝑖 ∈ (0...𝑁) ∧ (𝑗 + 1) ∈ (0...𝑁))) → (𝑖 < (𝑗 + 1) ↔ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) |
246 | 242, 243,
244, 245 | syl12anc 834 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))) → (𝑖 < (𝑗 + 1) ↔ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) |
247 | 241, 246 | mpbird 256 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))) → 𝑖 < (𝑗 + 1)) |
248 | 247 | adantrl 713 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) → 𝑖 < (𝑗 + 1)) |
249 | | btwnnz 12396 |
. . . . . . . . . . . . 13
⊢ ((𝑗 ∈ ℤ ∧ 𝑗 < 𝑖 ∧ 𝑖 < (𝑗 + 1)) → ¬ 𝑖 ∈ ℤ) |
250 | 232, 240,
248, 249 | syl3anc 1370 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) ∧ ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) → ¬ 𝑖 ∈ ℤ) |
251 | 231, 250 | pm2.65da 814 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0...𝑁)) → ¬ ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) |
252 | 228, 229,
251 | syl2anc 584 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ 𝑖 ∈ (0...𝑁)) ∧ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) → ¬ ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) |
253 | 227, 252 | pm2.65da 814 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) ∧ 𝑖 ∈ (0...𝑁)) → ¬ (𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) |
254 | 253 | nrexdv 3198 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ¬ ∃𝑖 ∈ (0...𝑁)(𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) |
255 | 254 | adantlr 712 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) → ¬ ∃𝑖 ∈ (0...𝑁)(𝑆‘𝑖) = ((𝑆‘𝑗) + 𝑇)) |
256 | 219, 255 | condan 815 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)) |
257 | 61 | rexrd 11025 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘𝑗) ∈
ℝ*) |
258 | 84 | ad2antrr 723 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝑇 ∈ ℝ) |
259 | 61, 258 | readdcld 11004 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝑆‘𝑗) + 𝑇) ∈ ℝ) |
260 | | elioc2 13142 |
. . . . . . 7
⊢ (((𝑆‘𝑗) ∈ ℝ* ∧ ((𝑆‘𝑗) + 𝑇) ∈ ℝ) → ((𝑆‘(𝑗 + 1)) ∈ ((𝑆‘𝑗)(,]((𝑆‘𝑗) + 𝑇)) ↔ ((𝑆‘(𝑗 + 1)) ∈ ℝ ∧ (𝑆‘𝑗) < (𝑆‘(𝑗 + 1)) ∧ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)))) |
261 | 257, 259,
260 | syl2anc 584 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝑆‘(𝑗 + 1)) ∈ ((𝑆‘𝑗)(,]((𝑆‘𝑗) + 𝑇)) ↔ ((𝑆‘(𝑗 + 1)) ∈ ℝ ∧ (𝑆‘𝑗) < (𝑆‘(𝑗 + 1)) ∧ (𝑆‘(𝑗 + 1)) ≤ ((𝑆‘𝑗) + 𝑇)))) |
262 | 66, 76, 256, 261 | mpbir3and 1341 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘(𝑗 + 1)) ∈ ((𝑆‘𝑗)(,]((𝑆‘𝑗) + 𝑇))) |
263 | 56, 59, 60, 15, 16, 61, 62, 262 | fourierdlem26 43674 |
. . . 4
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘(𝑗 + 1))) = (𝐴 + ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)))) |
264 | 263 | oveq1d 7290 |
. . 3
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐸‘(𝑆‘(𝑗 + 1))) − 𝐴) = ((𝐴 + ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗))) − 𝐴)) |
265 | 56 | recnd 11003 |
. . . 4
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐴 ∈ ℂ) |
266 | 65 | recnd 11003 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ∈ ℂ) |
267 | 266, 138 | subcld 11332 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) ∈ ℂ) |
268 | 267 | adantr 481 |
. . . 4
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) ∈ ℂ) |
269 | 265, 268 | pncan2d 11334 |
. . 3
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐴 + ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗))) − 𝐴) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗))) |
270 | 58, 264, 269 | 3eqtrd 2782 |
. 2
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗))) |
271 | 1 | a1i 11 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))) |
272 | | eqcom 2745 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝐸‘(𝑆‘𝑗)) ↔ (𝐸‘(𝑆‘𝑗)) = 𝑦) |
273 | 272 | biimpi 215 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝐸‘(𝑆‘𝑗)) → (𝐸‘(𝑆‘𝑗)) = 𝑦) |
274 | 273 | adantl 482 |
. . . . . . . . . 10
⊢ ((¬
(𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → (𝐸‘(𝑆‘𝑗)) = 𝑦) |
275 | | neqne 2951 |
. . . . . . . . . . 11
⊢ (¬
(𝐸‘(𝑆‘𝑗)) = 𝐵 → (𝐸‘(𝑆‘𝑗)) ≠ 𝐵) |
276 | 275 | adantr 481 |
. . . . . . . . . 10
⊢ ((¬
(𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → (𝐸‘(𝑆‘𝑗)) ≠ 𝐵) |
277 | 274, 276 | eqnetrrd 3012 |
. . . . . . . . 9
⊢ ((¬
(𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → 𝑦 ≠ 𝐵) |
278 | 277 | neneqd 2948 |
. . . . . . . 8
⊢ ((¬
(𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → ¬ 𝑦 = 𝐵) |
279 | 278 | iffalsed 4470 |
. . . . . . 7
⊢ ((¬
(𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → if(𝑦 = 𝐵, 𝐴, 𝑦) = 𝑦) |
280 | | simpr 485 |
. . . . . . 7
⊢ ((¬
(𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → 𝑦 = (𝐸‘(𝑆‘𝑗))) |
281 | 279, 280 | eqtrd 2778 |
. . . . . 6
⊢ ((¬
(𝐸‘(𝑆‘𝑗)) = 𝐵 ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → if(𝑦 = 𝐵, 𝐴, 𝑦) = (𝐸‘(𝑆‘𝑗))) |
282 | 281 | adantll 711 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ 𝑦 = (𝐸‘(𝑆‘𝑗))) → if(𝑦 = 𝐵, 𝐴, 𝑦) = (𝐸‘(𝑆‘𝑗))) |
283 | 54 | adantr 481 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘𝑗)) ∈ (𝐴(,]𝐵)) |
284 | 271, 282,
283, 283 | fvmptd 6882 |
. . . 4
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐿‘(𝐸‘(𝑆‘𝑗))) = (𝐸‘(𝑆‘𝑗))) |
285 | 284 | oveq2d 7291 |
. . 3
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐸‘(𝑆‘𝑗)))) |
286 | | id 22 |
. . . . . . . 8
⊢ (𝑥 = (𝑆‘(𝑗 + 1)) → 𝑥 = (𝑆‘(𝑗 + 1))) |
287 | | oveq2 7283 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑆‘(𝑗 + 1)) → (𝐵 − 𝑥) = (𝐵 − (𝑆‘(𝑗 + 1)))) |
288 | 287 | oveq1d 7290 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑆‘(𝑗 + 1)) → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) |
289 | 288 | fveq2d 6778 |
. . . . . . . . 9
⊢ (𝑥 = (𝑆‘(𝑗 + 1)) → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇))) |
290 | 289 | oveq1d 7290 |
. . . . . . . 8
⊢ (𝑥 = (𝑆‘(𝑗 + 1)) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)) |
291 | 286, 290 | oveq12d 7293 |
. . . . . . 7
⊢ (𝑥 = (𝑆‘(𝑗 + 1)) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇))) |
292 | 291 | adantl 482 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑥 = (𝑆‘(𝑗 + 1))) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇))) |
293 | 128, 65 | resubcld 11403 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝑆‘(𝑗 + 1))) ∈ ℝ) |
294 | 293, 101 | rerpdivcld 12803 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) ∈ ℝ) |
295 | 294 | flcld 13518 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ∈ ℤ) |
296 | 295 | zred 12426 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ∈ ℝ) |
297 | 296, 85 | remulcld 11005 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇) ∈ ℝ) |
298 | 65, 297 | readdcld 11004 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)) ∈ ℝ) |
299 | 120, 292,
65, 298 | fvmptd 6882 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐸‘(𝑆‘(𝑗 + 1))) = ((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇))) |
300 | 299, 135 | oveq12d 7293 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐸‘(𝑆‘𝑗))) = (((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)) − ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)))) |
301 | 300 | adantr 481 |
. . 3
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐸‘(𝑆‘𝑗))) = (((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)) − ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)))) |
302 | | flle 13519 |
. . . . . . . . . . . . 13
⊢ (((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) ∈ ℝ →
(⌊‘((𝐵 −
(𝑆‘(𝑗 + 1))) / 𝑇)) ≤ ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) |
303 | 294, 302 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ≤ ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) |
304 | 53, 65, 75 | ltled 11123 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) ≤ (𝑆‘(𝑗 + 1))) |
305 | 53, 65, 128, 304 | lesub2dd 11592 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝑆‘(𝑗 + 1))) ≤ (𝐵 − (𝑆‘𝑗))) |
306 | 293, 129,
101, 305 | lediv1dd 12830 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) |
307 | 296, 294,
130, 303, 306 | letrd 11132 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) |
308 | 307 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) |
309 | 296 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ∈ ℝ) |
310 | | 1red 10976 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → 1 ∈
ℝ) |
311 | 309, 310 | readdcld 11004 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ∈ ℝ) |
312 | 130 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → ((𝐵 − (𝑆‘𝑗)) / 𝑇) ∈ ℝ) |
313 | | simpr 485 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) |
314 | 311, 312,
313 | nltled 11125 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) |
315 | 314 | adantlr 712 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) |
316 | 79 | ad3antrrr 727 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑆:(0...𝑁)–onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) |
317 | 88 | ad2antrr 723 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝐶 ∈ ℝ) |
318 | 92 | ad2antrr 723 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝐷 ∈ ℝ) |
319 | | fourierdlem65.z |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑍 = ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) |
320 | 135, 134 | eqeltrd 2839 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐸‘(𝑆‘𝑗)) ∈ ℝ) |
321 | 128, 320 | resubcld 11403 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝐸‘(𝑆‘𝑗))) ∈ ℝ) |
322 | 53, 321 | readdcld 11004 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) ∈ ℝ) |
323 | 319, 322 | eqeltrid 2843 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑍 ∈ ℝ) |
324 | 323 | ad2antrr 723 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑍 ∈ ℝ) |
325 | 12 | rexrd 11025 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
326 | 325 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 ∈
ℝ*) |
327 | | elioc2 13142 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈ ℝ)
→ ((𝐸‘(𝑆‘𝑗)) ∈ (𝐴(,]𝐵) ↔ ((𝐸‘(𝑆‘𝑗)) ∈ ℝ ∧ 𝐴 < (𝐸‘(𝑆‘𝑗)) ∧ (𝐸‘(𝑆‘𝑗)) ≤ 𝐵))) |
328 | 326, 128,
327 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘𝑗)) ∈ (𝐴(,]𝐵) ↔ ((𝐸‘(𝑆‘𝑗)) ∈ ℝ ∧ 𝐴 < (𝐸‘(𝑆‘𝑗)) ∧ (𝐸‘(𝑆‘𝑗)) ≤ 𝐵))) |
329 | 54, 328 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘𝑗)) ∈ ℝ ∧ 𝐴 < (𝐸‘(𝑆‘𝑗)) ∧ (𝐸‘(𝑆‘𝑗)) ≤ 𝐵)) |
330 | 329 | simp3d 1143 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐸‘(𝑆‘𝑗)) ≤ 𝐵) |
331 | 128, 320 | subge0d 11565 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (0 ≤ (𝐵 − (𝐸‘(𝑆‘𝑗))) ↔ (𝐸‘(𝑆‘𝑗)) ≤ 𝐵)) |
332 | 330, 331 | mpbird 256 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 0 ≤ (𝐵 − (𝐸‘(𝑆‘𝑗)))) |
333 | 53, 321 | addge01d 11563 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (0 ≤ (𝐵 − (𝐸‘(𝑆‘𝑗))) ↔ (𝑆‘𝑗) ≤ ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))))) |
334 | 332, 333 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) ≤ ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗))))) |
335 | 88, 53, 322, 96, 334 | letrd 11132 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 ≤ ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗))))) |
336 | 335, 319 | breqtrrdi 5116 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 ≤ 𝑍) |
337 | 336 | ad2antrr 723 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝐶 ≤ 𝑍) |
338 | 65 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (𝑆‘(𝑗 + 1)) ∈ ℝ) |
339 | 294 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) ∈ ℝ) |
340 | | reflcl 13516 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) ∈ ℝ →
(⌊‘((𝐵 −
(𝑆‘(𝑗 + 1))) / 𝑇)) ∈ ℝ) |
341 | | peano2re 11148 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((⌊‘((𝐵
− (𝑆‘(𝑗 + 1))) / 𝑇)) ∈ ℝ →
((⌊‘((𝐵 −
(𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ∈ ℝ) |
342 | 339, 340,
341 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ∈ ℝ) |
343 | 128 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝐵 ∈ ℝ) |
344 | 343, 324 | resubcld 11403 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (𝐵 − 𝑍) ∈ ℝ) |
345 | 101 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑇 ∈
ℝ+) |
346 | 344, 345 | rerpdivcld 12803 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − 𝑍) / 𝑇) ∈ ℝ) |
347 | | flltp1 13520 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) ∈ ℝ → ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) |
348 | 294, 347 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) |
349 | 348 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) |
350 | 295 | peano2zd 12429 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ∈ ℤ) |
351 | 350 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ∈ ℤ) |
352 | 130 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − (𝑆‘𝑗)) / 𝑇) ∈ ℝ) |
353 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) |
354 | 321, 101 | rerpdivcld 12803 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇) ∈ ℝ) |
355 | 354 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇) ∈ ℝ) |
356 | 12 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 ∈ ℝ) |
357 | 329 | simp2d 1142 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 < (𝐸‘(𝑆‘𝑗))) |
358 | 356, 320,
128, 357 | ltsub2dd 11588 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝐸‘(𝑆‘𝑗))) < (𝐵 − 𝐴)) |
359 | 358, 15 | breqtrrdi 5116 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝐸‘(𝑆‘𝑗))) < 𝑇) |
360 | 321, 85, 101, 359 | ltdiv1dd 12829 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇) < (𝑇 / 𝑇)) |
361 | 143, 144 | dividd 11749 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑇 / 𝑇) = 1) |
362 | 360, 361 | breqtrd 5100 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇) < 1) |
363 | 362 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇) < 1) |
364 | 129 | recnd 11003 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝑆‘𝑗)) ∈ ℂ) |
365 | 321 | recnd 11003 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝐸‘(𝑆‘𝑗))) ∈ ℂ) |
366 | 364, 365,
143, 144 | divsubdird 11790 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝑆‘𝑗)) − (𝐵 − (𝐸‘(𝑆‘𝑗)))) / 𝑇) = (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇))) |
367 | 366 | eqcomd 2744 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇)) = (((𝐵 − (𝑆‘𝑗)) − (𝐵 − (𝐸‘(𝑆‘𝑗)))) / 𝑇)) |
368 | 128 | recnd 11003 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐵 ∈ ℂ) |
369 | 320 | recnd 11003 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐸‘(𝑆‘𝑗)) ∈ ℂ) |
370 | 368, 138,
369 | nnncan1d 11366 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝑆‘𝑗)) − (𝐵 − (𝐸‘(𝑆‘𝑗)))) = ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))) |
371 | 370 | oveq1d 7290 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝑆‘𝑗)) − (𝐵 − (𝐸‘(𝑆‘𝑗)))) / 𝑇) = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) |
372 | 367, 371 | eqtrd 2778 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇)) = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) |
373 | 372, 147 | eqeltrd 2839 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇)) ∈ ℤ) |
374 | 373 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇)) ∈ ℤ) |
375 | 351, 352,
353, 355, 363, 374 | zltlesub 42824 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇))) |
376 | 319 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑍 = ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗))))) |
377 | 376 | oveq2d 7291 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − 𝑍) = (𝐵 − ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))))) |
378 | 138, 368,
369 | addsub12d 11355 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) = (𝐵 + ((𝑆‘𝑗) − (𝐸‘(𝑆‘𝑗))))) |
379 | 368, 369,
138 | subsub2d 11361 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))) = (𝐵 + ((𝑆‘𝑗) − (𝐸‘(𝑆‘𝑗))))) |
380 | 378, 379 | eqtr4d 2781 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) = (𝐵 − ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)))) |
381 | 380 | oveq2d 7291 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗))))) = (𝐵 − (𝐵 − ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))))) |
382 | 369, 138 | subcld 11332 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) ∈ ℂ) |
383 | 368, 382 | nncand 11337 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − (𝐵 − ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)))) = ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))) |
384 | 377, 381,
383 | 3eqtrd 2782 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − 𝑍) = ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))) |
385 | 384 | oveq1d 7290 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − 𝑍) / 𝑇) = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇)) |
386 | 371, 367,
385 | 3eqtr4d 2788 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇)) = ((𝐵 − 𝑍) / 𝑇)) |
387 | 386 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (((𝐵 − (𝑆‘𝑗)) / 𝑇) − ((𝐵 − (𝐸‘(𝑆‘𝑗))) / 𝑇)) = ((𝐵 − 𝑍) / 𝑇)) |
388 | 375, 387 | breqtrd 5100 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − 𝑍) / 𝑇)) |
389 | 339, 342,
346, 349, 388 | ltletrd 11135 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) < ((𝐵 − 𝑍) / 𝑇)) |
390 | 293 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (𝐵 − (𝑆‘(𝑗 + 1))) ∈ ℝ) |
391 | 390, 344,
345 | ltdiv1d 12817 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ((𝐵 − (𝑆‘(𝑗 + 1))) < (𝐵 − 𝑍) ↔ ((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇) < ((𝐵 − 𝑍) / 𝑇))) |
392 | 389, 391 | mpbird 256 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (𝐵 − (𝑆‘(𝑗 + 1))) < (𝐵 − 𝑍)) |
393 | 324, 338,
343 | ltsub2d 11585 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (𝑍 < (𝑆‘(𝑗 + 1)) ↔ (𝐵 − (𝑆‘(𝑗 + 1))) < (𝐵 − 𝑍))) |
394 | 392, 393 | mpbird 256 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑍 < (𝑆‘(𝑗 + 1))) |
395 | 114 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (𝑆‘(𝑗 + 1)) ≤ 𝐷) |
396 | 324, 338,
318, 394, 395 | ltletrd 11135 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑍 < 𝐷) |
397 | 324, 318,
396 | ltled 11123 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑍 ≤ 𝐷) |
398 | 317, 318,
324, 337, 397 | eliccd 43042 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑍 ∈ (𝐶[,]𝐷)) |
399 | 30 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − 𝐴) = 𝑇) |
400 | 399 | oveq2d 7291 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴)) = ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇)) |
401 | 382, 143,
144 | divcan1d 11752 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · 𝑇) = ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))) |
402 | 400, 401 | eqtrd 2778 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴)) = ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))) |
403 | 376, 402 | oveq12d 7293 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑍 + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴))) = (((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) + ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)))) |
404 | 138, 365 | addcomd 11177 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) = ((𝐵 − (𝐸‘(𝑆‘𝑗))) + (𝑆‘𝑗))) |
405 | 404 | oveq1d 7290 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) + ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))) = (((𝐵 − (𝐸‘(𝑆‘𝑗))) + (𝑆‘𝑗)) + ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)))) |
406 | 365, 138,
369 | ppncand 11372 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝐸‘(𝑆‘𝑗))) + (𝑆‘𝑗)) + ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))) = ((𝐵 − (𝐸‘(𝑆‘𝑗))) + (𝐸‘(𝑆‘𝑗)))) |
407 | 368, 369 | npcand 11336 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐵 − (𝐸‘(𝑆‘𝑗))) + (𝐸‘(𝑆‘𝑗))) = 𝐵) |
408 | 406, 407 | eqtrd 2778 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝐵 − (𝐸‘(𝑆‘𝑗))) + (𝑆‘𝑗)) + ((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗))) = 𝐵) |
409 | 403, 405,
408 | 3eqtrd 2782 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑍 + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴))) = 𝐵) |
410 | 199 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐵 ∈ ran 𝑄) |
411 | 409, 410 | eqeltrd 2839 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑍 + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴))) ∈ ran 𝑄) |
412 | | oveq1 7282 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) → (𝑘 · (𝐵 − 𝐴)) = ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴))) |
413 | 412 | oveq2d 7291 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) → (𝑍 + (𝑘 · (𝐵 − 𝐴))) = (𝑍 + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴)))) |
414 | 413 | eleq1d 2823 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = (((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) → ((𝑍 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ (𝑍 + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴))) ∈ ran 𝑄)) |
415 | 414 | rspcev 3561 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) ∈ ℤ ∧ (𝑍 + ((((𝐸‘(𝑆‘𝑗)) − (𝑆‘𝑗)) / 𝑇) · (𝐵 − 𝐴))) ∈ ran 𝑄) → ∃𝑘 ∈ ℤ (𝑍 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄) |
416 | 147, 411,
415 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ∃𝑘 ∈ ℤ (𝑍 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄) |
417 | 416 | ad2antrr 723 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ∃𝑘 ∈ ℤ (𝑍 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄) |
418 | | oveq1 7282 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑍 → (𝑦 + (𝑘 · (𝐵 − 𝐴))) = (𝑍 + (𝑘 · (𝐵 − 𝐴)))) |
419 | 418 | eleq1d 2823 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = 𝑍 → ((𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ (𝑍 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)) |
420 | 419 | rexbidv 3226 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = 𝑍 → (∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑍 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)) |
421 | 420 | elrab 3624 |
. . . . . . . . . . . . . . . 16
⊢ (𝑍 ∈ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄} ↔ (𝑍 ∈ (𝐶[,]𝐷) ∧ ∃𝑘 ∈ ℤ (𝑍 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄)) |
422 | 398, 417,
421 | sylanbrc 583 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑍 ∈ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}) |
423 | | elun2 4111 |
. . . . . . . . . . . . . . 15
⊢ (𝑍 ∈ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄} → 𝑍 ∈ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) |
424 | 422, 423 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → 𝑍 ∈ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) |
425 | | foelrn 6982 |
. . . . . . . . . . . . . 14
⊢ ((𝑆:(0...𝑁)–onto→({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}) ∧ 𝑍 ∈ ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) → ∃𝑖 ∈ (0...𝑁)𝑍 = (𝑆‘𝑖)) |
426 | 316, 424,
425 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ∃𝑖 ∈ (0...𝑁)𝑍 = (𝑆‘𝑖)) |
427 | 53 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘𝑗) ∈ ℝ) |
428 | 321 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐵 − (𝐸‘(𝑆‘𝑗))) ∈ ℝ) |
429 | 320 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘𝑗)) ∈ ℝ) |
430 | 13 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐵 ∈ ℝ) |
431 | 330 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘𝑗)) ≤ 𝐵) |
432 | 275 | necomd 2999 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (¬
(𝐸‘(𝑆‘𝑗)) = 𝐵 → 𝐵 ≠ (𝐸‘(𝑆‘𝑗))) |
433 | 432 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 𝐵 ≠ (𝐸‘(𝑆‘𝑗))) |
434 | 429, 430,
431, 433 | leneltd 11129 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐸‘(𝑆‘𝑗)) < 𝐵) |
435 | 429, 430 | posdifd 11562 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐸‘(𝑆‘𝑗)) < 𝐵 ↔ 0 < (𝐵 − (𝐸‘(𝑆‘𝑗))))) |
436 | 434, 435 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → 0 < (𝐵 − (𝐸‘(𝑆‘𝑗)))) |
437 | 428, 436 | elrpd 12769 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝐵 − (𝐸‘(𝑆‘𝑗))) ∈
ℝ+) |
438 | 427, 437 | ltaddrpd 12805 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘𝑗) < ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗))))) |
439 | 438, 319 | breqtrrdi 5116 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘𝑗) < 𝑍) |
440 | 439 | ad2antrr 723 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∧ 𝑍 = (𝑆‘𝑖)) → (𝑆‘𝑗) < 𝑍) |
441 | | simpr 485 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∧ 𝑍 = (𝑆‘𝑖)) → 𝑍 = (𝑆‘𝑖)) |
442 | 440, 441 | breqtrd 5100 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∧ 𝑍 = (𝑆‘𝑖)) → (𝑆‘𝑗) < (𝑆‘𝑖)) |
443 | 394 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∧ 𝑍 = (𝑆‘𝑖)) → 𝑍 < (𝑆‘(𝑗 + 1))) |
444 | 441, 443 | eqbrtrrd 5098 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∧ 𝑍 = (𝑆‘𝑖)) → (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))) |
445 | 442, 444 | jca 512 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) ∧ 𝑍 = (𝑆‘𝑖)) → ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) |
446 | 445 | ex 413 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (𝑍 = (𝑆‘𝑖) → ((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))))) |
447 | 446 | reximdv 3202 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → (∃𝑖 ∈ (0...𝑁)𝑍 = (𝑆‘𝑖) → ∃𝑖 ∈ (0...𝑁)((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1))))) |
448 | 426, 447 | mpd 15 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇)) → ∃𝑖 ∈ (0...𝑁)((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) |
449 | 315, 448 | syldan 591 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → ∃𝑖 ∈ (0...𝑁)((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) |
450 | 251 | nrexdv 3198 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ¬ ∃𝑖 ∈ (0...𝑁)((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) |
451 | 450 | ad2antrr 723 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) ∧ ¬ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) → ¬ ∃𝑖 ∈ (0...𝑁)((𝑆‘𝑗) < (𝑆‘𝑖) ∧ (𝑆‘𝑖) < (𝑆‘(𝑗 + 1)))) |
452 | 449, 451 | condan 815 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)) |
453 | 308, 452 | jca 512 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇) ∧ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1))) |
454 | 130 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐵 − (𝑆‘𝑗)) / 𝑇) ∈ ℝ) |
455 | 295 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ∈ ℤ) |
456 | | flbi 13536 |
. . . . . . . . . 10
⊢ ((((𝐵 − (𝑆‘𝑗)) / 𝑇) ∈ ℝ ∧
(⌊‘((𝐵 −
(𝑆‘(𝑗 + 1))) / 𝑇)) ∈ ℤ) →
((⌊‘((𝐵 −
(𝑆‘𝑗)) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ↔ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇) ∧ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)))) |
457 | 454, 455,
456 | syl2anc 584 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ↔ ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) ≤ ((𝐵 − (𝑆‘𝑗)) / 𝑇) ∧ ((𝐵 − (𝑆‘𝑗)) / 𝑇) < ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) + 1)))) |
458 | 453, 457 | mpbird 256 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇))) |
459 | 458 | eqcomd 2744 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇))) |
460 | 459 | oveq1d 7290 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) |
461 | 460 | oveq2d 7291 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)) = ((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))) |
462 | 461 | oveq1d 7290 |
. . . 4
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)) − ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))) = (((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) − ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)))) |
463 | 266 | adantr 481 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘(𝑗 + 1)) ∈ ℂ) |
464 | 138 | adantr 481 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (𝑆‘𝑗) ∈ ℂ) |
465 | 139 | adantr 481 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇) ∈ ℂ) |
466 | 463, 464,
465 | pnpcan2d 11370 |
. . . 4
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇)) − ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗))) |
467 | 462, 466 | eqtrd 2778 |
. . 3
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → (((𝑆‘(𝑗 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝑗 + 1))) / 𝑇)) · 𝑇)) − ((𝑆‘𝑗) + ((⌊‘((𝐵 − (𝑆‘𝑗)) / 𝑇)) · 𝑇))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗))) |
468 | 285, 301,
467 | 3eqtrd 2782 |
. 2
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ ¬ (𝐸‘(𝑆‘𝑗)) = 𝐵) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗))) |
469 | 270, 468 | pm2.61dan 810 |
1
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗))) |