Step | Hyp | Ref
| Expression |
1 | | fourierdlem49.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℝ) |
2 | | fourierdlem49.b |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ ℝ) |
3 | | fourierdlem49.altb |
. . . . . 6
⊢ (𝜑 → 𝐴 < 𝐵) |
4 | | fourierdlem49.t |
. . . . . 6
⊢ 𝑇 = (𝐵 − 𝐴) |
5 | | fourierdlem49.e |
. . . . . . 7
⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍‘𝑥))) |
6 | | ovex 6937 |
. . . . . . . . . 10
⊢
((⌊‘((𝐵
− 𝑥) / 𝑇)) · 𝑇) ∈ V |
7 | | fourierdlem49.z |
. . . . . . . . . . 11
⊢ 𝑍 = (𝑥 ∈ ℝ ↦
((⌊‘((𝐵 −
𝑥) / 𝑇)) · 𝑇)) |
8 | 7 | fvmpt2 6538 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧
((⌊‘((𝐵 −
𝑥) / 𝑇)) · 𝑇) ∈ V) → (𝑍‘𝑥) = ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) |
9 | 6, 8 | mpan2 684 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ → (𝑍‘𝑥) = ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) |
10 | 9 | oveq2d 6921 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ → (𝑥 + (𝑍‘𝑥)) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) |
11 | 10 | mpteq2ia 4963 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍‘𝑥))) = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) |
12 | 5, 11 | eqtri 2849 |
. . . . . 6
⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) |
13 | 1, 2, 3, 4, 12 | fourierdlem4 41122 |
. . . . 5
⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵)) |
14 | | fourierdlem49.x |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ ℝ) |
15 | 13, 14 | ffvelrnd 6609 |
. . . 4
⊢ (𝜑 → (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) |
16 | | simpr 479 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → (𝐸‘𝑋) ∈ ran 𝑄) |
17 | | fourierdlem49.q |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) |
18 | | fourierdlem49.m |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ∈ ℕ) |
19 | | fourierdlem49.p |
. . . . . . . . . . . . . . 15
⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚
(0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
20 | 19 | fourierdlem2 41120 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑𝑚
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))) |
21 | 18, 20 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑𝑚
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))) |
22 | 17, 21 | mpbid 224 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑄 ∈ (ℝ ↑𝑚
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))) |
23 | 22 | simpld 490 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑄 ∈ (ℝ ↑𝑚
(0...𝑀))) |
24 | | elmapi 8144 |
. . . . . . . . . . 11
⊢ (𝑄 ∈ (ℝ
↑𝑚 (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ) |
25 | 23, 24 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ) |
26 | | ffn 6278 |
. . . . . . . . . 10
⊢ (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀)) |
27 | 25, 26 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑄 Fn (0...𝑀)) |
28 | 27 | ad2antrr 719 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → 𝑄 Fn (0...𝑀)) |
29 | | fvelrnb 6490 |
. . . . . . . 8
⊢ (𝑄 Fn (0...𝑀) → ((𝐸‘𝑋) ∈ ran 𝑄 ↔ ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = (𝐸‘𝑋))) |
30 | 28, 29 | syl 17 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → ((𝐸‘𝑋) ∈ ran 𝑄 ↔ ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = (𝐸‘𝑋))) |
31 | 16, 30 | mpbid 224 |
. . . . . 6
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = (𝐸‘𝑋)) |
32 | | 1zzd 11736 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 1 ∈ ℤ) |
33 | | elfzelz 12635 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ) |
34 | 33 | ad2antlr 720 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ∈ ℤ) |
35 | | 1e0p1 11864 |
. . . . . . . . . . . . . . . . 17
⊢ 1 = (0 +
1) |
36 | 35 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 1 = (0 + 1)) |
37 | 34 | zred 11810 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ∈ ℝ) |
38 | | elfzle1 12637 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ (0...𝑀) → 0 ≤ 𝑗) |
39 | 38 | ad2antlr 720 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 0 ≤ 𝑗) |
40 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑄‘𝑗) = (𝐸‘𝑋) → (𝑄‘𝑗) = (𝐸‘𝑋)) |
41 | 40 | eqcomd 2831 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑄‘𝑗) = (𝐸‘𝑋) → (𝐸‘𝑋) = (𝑄‘𝑗)) |
42 | 41 | ad2antlr 720 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝐸‘𝑋) = (𝑄‘𝑗)) |
43 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 = 0 → (𝑄‘𝑗) = (𝑄‘0)) |
44 | 43 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝑄‘𝑗) = (𝑄‘0)) |
45 | 22 | simprld 790 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵)) |
46 | 45 | simpld 490 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑄‘0) = 𝐴) |
47 | 46 | ad2antrr 719 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝑄‘0) = 𝐴) |
48 | 42, 44, 47 | 3eqtrd 2865 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝐸‘𝑋) = 𝐴) |
49 | 48 | adantllr 712 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝐸‘𝑋) = 𝐴) |
50 | 49 | adantllr 712 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝐸‘𝑋) = 𝐴) |
51 | 1 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → 𝐴 ∈ ℝ) |
52 | 1 | rexrd 10406 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
53 | 52 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → 𝐴 ∈
ℝ*) |
54 | 2 | rexrd 10406 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
55 | 54 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → 𝐵 ∈
ℝ*) |
56 | | simpr 479 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) |
57 | | iocgtlb 40523 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → 𝐴 < (𝐸‘𝑋)) |
58 | 53, 55, 56, 57 | syl3anc 1496 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → 𝐴 < (𝐸‘𝑋)) |
59 | 51, 58 | gtned 10491 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → (𝐸‘𝑋) ≠ 𝐴) |
60 | 59 | neneqd 3004 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → ¬ (𝐸‘𝑋) = 𝐴) |
61 | 60 | ad3antrrr 723 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → ¬ (𝐸‘𝑋) = 𝐴) |
62 | 50, 61 | pm2.65da 853 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → ¬ 𝑗 = 0) |
63 | 62 | neqned 3006 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ≠ 0) |
64 | 37, 39, 63 | ne0gt0d 10493 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 0 < 𝑗) |
65 | | 0zd 11716 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 0 ∈ ℤ) |
66 | | zltp1le 11755 |
. . . . . . . . . . . . . . . . . 18
⊢ ((0
∈ ℤ ∧ 𝑗
∈ ℤ) → (0 < 𝑗 ↔ (0 + 1) ≤ 𝑗)) |
67 | 65, 34, 66 | syl2anc 581 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (0 < 𝑗 ↔ (0 + 1) ≤ 𝑗)) |
68 | 64, 67 | mpbid 224 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (0 + 1) ≤ 𝑗) |
69 | 36, 68 | eqbrtrd 4895 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 1 ≤ 𝑗) |
70 | | eluz2 11974 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈
(ℤ≥‘1) ↔ (1 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 1 ≤
𝑗)) |
71 | 32, 34, 69, 70 | syl3anbrc 1449 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ∈
(ℤ≥‘1)) |
72 | | nnuz 12005 |
. . . . . . . . . . . . . 14
⊢ ℕ =
(ℤ≥‘1) |
73 | 71, 72 | syl6eleqr 2917 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ∈ ℕ) |
74 | | nnm1nn0 11661 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ → (𝑗 − 1) ∈
ℕ0) |
75 | 73, 74 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈
ℕ0) |
76 | | nn0uz 12004 |
. . . . . . . . . . . . 13
⊢
ℕ0 = (ℤ≥‘0) |
77 | 76 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → ℕ0 =
(ℤ≥‘0)) |
78 | 75, 77 | eleqtrd 2908 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈
(ℤ≥‘0)) |
79 | 18 | nnzd 11809 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ ℤ) |
80 | 79 | ad3antrrr 723 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑀 ∈ ℤ) |
81 | | peano2zm 11748 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℤ → (𝑗 − 1) ∈
ℤ) |
82 | 33, 81 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ∈ ℤ) |
83 | 82 | zred 11810 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ∈ ℝ) |
84 | 33 | zred 11810 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ) |
85 | | elfzel2 12633 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℤ) |
86 | 85 | zred 11810 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℝ) |
87 | 84 | ltm1d 11286 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) < 𝑗) |
88 | | elfzle2 12638 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ≤ 𝑀) |
89 | 83, 84, 86, 87, 88 | ltletrd 10516 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) < 𝑀) |
90 | 89 | ad2antlr 720 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) < 𝑀) |
91 | | elfzo2 12768 |
. . . . . . . . . . 11
⊢ ((𝑗 − 1) ∈ (0..^𝑀) ↔ ((𝑗 − 1) ∈
(ℤ≥‘0) ∧ 𝑀 ∈ ℤ ∧ (𝑗 − 1) < 𝑀)) |
92 | 78, 80, 90, 91 | syl3anbrc 1449 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈ (0..^𝑀)) |
93 | 25 | ad3antrrr 723 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑄:(0...𝑀)⟶ℝ) |
94 | 34, 81 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈ ℤ) |
95 | 65, 80, 94 | 3jca 1164 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑗 − 1) ∈
ℤ)) |
96 | 75 | nn0ge0d 11681 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 0 ≤ (𝑗 − 1)) |
97 | 83, 86, 89 | ltled 10504 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ≤ 𝑀) |
98 | 97 | ad2antlr 720 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ≤ 𝑀) |
99 | 95, 96, 98 | jca32 513 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑗 − 1) ∈ ℤ)
∧ (0 ≤ (𝑗 − 1)
∧ (𝑗 − 1) ≤
𝑀))) |
100 | | elfz2 12626 |
. . . . . . . . . . . . . . 15
⊢ ((𝑗 − 1) ∈ (0...𝑀) ↔ ((0 ∈ ℤ
∧ 𝑀 ∈ ℤ
∧ (𝑗 − 1) ∈
ℤ) ∧ (0 ≤ (𝑗
− 1) ∧ (𝑗 −
1) ≤ 𝑀))) |
101 | 99, 100 | sylibr 226 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈ (0...𝑀)) |
102 | 93, 101 | ffvelrnd 6609 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) ∈ ℝ) |
103 | 102 | rexrd 10406 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) ∈
ℝ*) |
104 | 25 | ffvelrnda 6608 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈ ℝ) |
105 | 104 | rexrd 10406 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈
ℝ*) |
106 | 105 | adantlr 708 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈
ℝ*) |
107 | 106 | adantr 474 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘𝑗) ∈
ℝ*) |
108 | | iocssre 12541 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈ ℝ)
→ (𝐴(,]𝐵) ⊆
ℝ) |
109 | 52, 2, 108 | syl2anc 581 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ) |
110 | 109 | sselda 3827 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → (𝐸‘𝑋) ∈ ℝ) |
111 | 110 | rexrd 10406 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → (𝐸‘𝑋) ∈
ℝ*) |
112 | 111 | ad2antrr 719 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈
ℝ*) |
113 | | simplll 793 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝜑) |
114 | | ovex 6937 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 − 1) ∈
V |
115 | | eleq1 2894 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = (𝑗 − 1) → (𝑖 ∈ (0..^𝑀) ↔ (𝑗 − 1) ∈ (0..^𝑀))) |
116 | 115 | anbi2d 624 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = (𝑗 − 1) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ (𝑗 − 1) ∈ (0..^𝑀)))) |
117 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = (𝑗 − 1) → (𝑄‘𝑖) = (𝑄‘(𝑗 − 1))) |
118 | | oveq1 6912 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = (𝑗 − 1) → (𝑖 + 1) = ((𝑗 − 1) + 1)) |
119 | 118 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = (𝑗 − 1) → (𝑄‘(𝑖 + 1)) = (𝑄‘((𝑗 − 1) + 1))) |
120 | 117, 119 | breq12d 4886 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = (𝑗 − 1) → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘(𝑗 − 1)) < (𝑄‘((𝑗 − 1) + 1)))) |
121 | 116, 120 | imbi12d 336 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = (𝑗 − 1) → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ (𝑗 − 1) ∈ (0..^𝑀)) → (𝑄‘(𝑗 − 1)) < (𝑄‘((𝑗 − 1) + 1))))) |
122 | 22 | simprrd 792 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) |
123 | 122 | r19.21bi 3141 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) |
124 | 114, 121,
123 | vtocl 3475 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑗 − 1) ∈ (0..^𝑀)) → (𝑄‘(𝑗 − 1)) < (𝑄‘((𝑗 − 1) + 1))) |
125 | 113, 92, 124 | syl2anc 581 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) < (𝑄‘((𝑗 − 1) + 1))) |
126 | 33 | zcnd 11811 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℂ) |
127 | | 1cnd 10351 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ (0...𝑀) → 1 ∈ ℂ) |
128 | 126, 127 | npcand 10717 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 − 1) + 1) = 𝑗) |
129 | 128 | eqcomd 2831 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 = ((𝑗 − 1) + 1)) |
130 | 129 | fveq2d 6437 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ (0...𝑀) → (𝑄‘𝑗) = (𝑄‘((𝑗 − 1) + 1))) |
131 | 130 | eqcomd 2831 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ (0...𝑀) → (𝑄‘((𝑗 − 1) + 1)) = (𝑄‘𝑗)) |
132 | 131 | ad2antlr 720 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘((𝑗 − 1) + 1)) = (𝑄‘𝑗)) |
133 | 125, 132 | breqtrd 4899 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) < (𝑄‘𝑗)) |
134 | | simpr 479 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘𝑗) = (𝐸‘𝑋)) |
135 | 133, 134 | breqtrd 4899 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) < (𝐸‘𝑋)) |
136 | 109, 15 | sseldd 3828 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐸‘𝑋) ∈ ℝ) |
137 | 136 | leidd 10918 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐸‘𝑋) ≤ (𝐸‘𝑋)) |
138 | 137 | ad2antrr 719 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ≤ (𝐸‘𝑋)) |
139 | 41 | adantl 475 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) = (𝑄‘𝑗)) |
140 | 138, 139 | breqtrd 4899 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ≤ (𝑄‘𝑗)) |
141 | 140 | adantllr 712 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ≤ (𝑄‘𝑗)) |
142 | 103, 107,
112, 135, 141 | eliocd 40529 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ((𝑄‘(𝑗 − 1))(,](𝑄‘𝑗))) |
143 | 130 | oveq2d 6921 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ (0...𝑀) → ((𝑄‘(𝑗 − 1))(,](𝑄‘𝑗)) = ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1)))) |
144 | 143 | ad2antlr 720 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → ((𝑄‘(𝑗 − 1))(,](𝑄‘𝑗)) = ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1)))) |
145 | 142, 144 | eleqtrd 2908 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1)))) |
146 | 117, 119 | oveq12d 6923 |
. . . . . . . . . . . 12
⊢ (𝑖 = (𝑗 − 1) → ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) = ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1)))) |
147 | 146 | eleq2d 2892 |
. . . . . . . . . . 11
⊢ (𝑖 = (𝑗 − 1) → ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) ↔ (𝐸‘𝑋) ∈ ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1))))) |
148 | 147 | rspcev 3526 |
. . . . . . . . . 10
⊢ (((𝑗 − 1) ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1)))) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
149 | 92, 145, 148 | syl2anc 581 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
150 | 149 | ex 403 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑄‘𝑗) = (𝐸‘𝑋) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))) |
151 | 150 | adantlr 708 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑄‘𝑗) = (𝐸‘𝑋) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))) |
152 | 151 | rexlimdva 3240 |
. . . . . 6
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → (∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = (𝐸‘𝑋) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))) |
153 | 31, 152 | mpd 15 |
. . . . 5
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
154 | 18 | ad2antrr 719 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → 𝑀 ∈ ℕ) |
155 | 25 | ad2antrr 719 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ) |
156 | | iocssicc 12550 |
. . . . . . . . . 10
⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵) |
157 | 46 | eqcomd 2831 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 = (𝑄‘0)) |
158 | 45 | simprd 491 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑄‘𝑀) = 𝐵) |
159 | 158 | eqcomd 2831 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 = (𝑄‘𝑀)) |
160 | 157, 159 | oveq12d 6923 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀))) |
161 | 156, 160 | syl5sseq 3878 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ((𝑄‘0)[,](𝑄‘𝑀))) |
162 | 161 | sselda 3827 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → (𝐸‘𝑋) ∈ ((𝑄‘0)[,](𝑄‘𝑀))) |
163 | 162 | adantr 474 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → (𝐸‘𝑋) ∈ ((𝑄‘0)[,](𝑄‘𝑀))) |
164 | | simpr 479 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → ¬ (𝐸‘𝑋) ∈ ran 𝑄) |
165 | | fveq2 6433 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑗 → (𝑄‘𝑘) = (𝑄‘𝑗)) |
166 | 165 | breq1d 4883 |
. . . . . . . . 9
⊢ (𝑘 = 𝑗 → ((𝑄‘𝑘) < (𝐸‘𝑋) ↔ (𝑄‘𝑗) < (𝐸‘𝑋))) |
167 | 166 | cbvrabv 3412 |
. . . . . . . 8
⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < (𝐸‘𝑋)} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < (𝐸‘𝑋)} |
168 | 167 | supeq1i 8622 |
. . . . . . 7
⊢
sup({𝑘 ∈
(0..^𝑀) ∣ (𝑄‘𝑘) < (𝐸‘𝑋)}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < (𝐸‘𝑋)}, ℝ, < ) |
169 | 154, 155,
163, 164, 168 | fourierdlem25 41143 |
. . . . . 6
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
170 | | ioossioc 40512 |
. . . . . . . . 9
⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) |
171 | 170 | sseli 3823 |
. . . . . . . 8
⊢ ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
172 | 171 | a1i 11 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))) |
173 | 172 | reximdva 3225 |
. . . . . 6
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → (∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))) |
174 | 169, 173 | mpd 15 |
. . . . 5
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
175 | 153, 174 | pm2.61dan 849 |
. . . 4
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
176 | 15, 175 | mpdan 680 |
. . 3
⊢ (𝜑 → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
177 | | fourierdlem49.f |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
178 | | frel 6283 |
. . . . . . . . . . 11
⊢ (𝐹:𝐷⟶ℝ → Rel 𝐹) |
179 | 177, 178 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → Rel 𝐹) |
180 | | resindm 5681 |
. . . . . . . . . . 11
⊢ (Rel
𝐹 → (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ dom 𝐹)) = (𝐹 ↾ (-∞(,)(𝐸‘𝑋)))) |
181 | 180 | eqcomd 2831 |
. . . . . . . . . 10
⊢ (Rel
𝐹 → (𝐹 ↾ (-∞(,)(𝐸‘𝑋))) = (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ dom 𝐹))) |
182 | 179, 181 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 ↾ (-∞(,)(𝐸‘𝑋))) = (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ dom 𝐹))) |
183 | | fdm 6286 |
. . . . . . . . . . . 12
⊢ (𝐹:𝐷⟶ℝ → dom 𝐹 = 𝐷) |
184 | 177, 183 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → dom 𝐹 = 𝐷) |
185 | 184 | ineq2d 4041 |
. . . . . . . . . 10
⊢ (𝜑 → ((-∞(,)(𝐸‘𝑋)) ∩ dom 𝐹) = ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) |
186 | 185 | reseq2d 5629 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ dom 𝐹)) = (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷))) |
187 | 182, 186 | eqtrd 2861 |
. . . . . . . 8
⊢ (𝜑 → (𝐹 ↾ (-∞(,)(𝐸‘𝑋))) = (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷))) |
188 | 187 | 3ad2ant1 1169 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐹 ↾ (-∞(,)(𝐸‘𝑋))) = (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷))) |
189 | 188 | oveq1d 6920 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) = ((𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) limℂ (𝐸‘𝑋))) |
190 | | ax-resscn 10309 |
. . . . . . . . . . 11
⊢ ℝ
⊆ ℂ |
191 | 190 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℝ ⊆
ℂ) |
192 | 177, 191 | fssd 6292 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:𝐷⟶ℂ) |
193 | 192 | 3ad2ant1 1169 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝐹:𝐷⟶ℂ) |
194 | | inss2 4058 |
. . . . . . . . 9
⊢
((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ⊆ 𝐷 |
195 | 194 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ⊆ 𝐷) |
196 | 193, 195 | fssresd 6308 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)):((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)⟶ℂ) |
197 | | mnfxr 10414 |
. . . . . . . . . 10
⊢ -∞
∈ ℝ* |
198 | 197 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -∞ ∈
ℝ*) |
199 | 25 | adantr 474 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ) |
200 | | elfzofz 12780 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀)) |
201 | 200 | adantl 475 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀)) |
202 | 199, 201 | ffvelrnd 6609 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ) |
203 | 202 | rexrd 10406 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈
ℝ*) |
204 | 202 | mnfltd 12244 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -∞ < (𝑄‘𝑖)) |
205 | 198, 203,
204 | xrltled 12269 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -∞ ≤ (𝑄‘𝑖)) |
206 | | iooss1 12498 |
. . . . . . . . . 10
⊢
((-∞ ∈ ℝ* ∧ -∞ ≤ (𝑄‘𝑖)) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ (-∞(,)(𝐸‘𝑋))) |
207 | 197, 205,
206 | sylancr 583 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ (-∞(,)(𝐸‘𝑋))) |
208 | 207 | 3adant3 1168 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ (-∞(,)(𝐸‘𝑋))) |
209 | | fzofzp1 12860 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀)) |
210 | 209 | adantl 475 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀)) |
211 | 199, 210 | ffvelrnd 6609 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
212 | 211 | 3adant3 1168 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
213 | 212 | rexrd 10406 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈
ℝ*) |
214 | 202 | 3adant3 1168 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ) |
215 | 214 | rexrd 10406 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈
ℝ*) |
216 | | simp3 1174 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
217 | | iocleub 40524 |
. . . . . . . . . . 11
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ* ∧
(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ≤ (𝑄‘(𝑖 + 1))) |
218 | 215, 213,
216, 217 | syl3anc 1496 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ≤ (𝑄‘(𝑖 + 1))) |
219 | | iooss2 12499 |
. . . . . . . . . 10
⊢ (((𝑄‘(𝑖 + 1)) ∈ ℝ* ∧
(𝐸‘𝑋) ≤ (𝑄‘(𝑖 + 1))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
220 | 213, 218,
219 | syl2anc 581 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
221 | | fourierdlem49.cn |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
222 | | cncff 23066 |
. . . . . . . . . . . . 13
⊢ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ) |
223 | | fdm 6286 |
. . . . . . . . . . . . 13
⊢ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ → dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
224 | 221, 222,
223 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
225 | | ssdmres 5656 |
. . . . . . . . . . . 12
⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
226 | 224, 225 | sylibr 226 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹) |
227 | 184 | adantr 474 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom 𝐹 = 𝐷) |
228 | 226, 227 | sseqtrd 3866 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷) |
229 | 228 | 3adant3 1168 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷) |
230 | 220, 229 | sstrd 3837 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ 𝐷) |
231 | 208, 230 | ssind 4061 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) |
232 | | fourierdlem49.d |
. . . . . . . . . 10
⊢ (𝜑 → 𝐷 ⊆ ℝ) |
233 | 232, 191 | sstrd 3837 |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 ⊆ ℂ) |
234 | 233 | 3ad2ant1 1169 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝐷 ⊆ ℂ) |
235 | 194, 234 | syl5ss 3838 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ⊆ ℂ) |
236 | | eqid 2825 |
. . . . . . 7
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
237 | | eqid 2825 |
. . . . . . 7
⊢
((TopOpen‘ℂfld) ↾t
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) =
((TopOpen‘ℂfld) ↾t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) |
238 | 136 | 3ad2ant1 1169 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ℝ) |
239 | 238 | rexrd 10406 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈
ℝ*) |
240 | | iocgtlb 40523 |
. . . . . . . . . 10
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ* ∧
(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝐸‘𝑋)) |
241 | 215, 213,
216, 240 | syl3anc 1496 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝐸‘𝑋)) |
242 | 238 | leidd 10918 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ≤ (𝐸‘𝑋)) |
243 | 215, 239,
239, 241, 242 | eliocd 40529 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) |
244 | | ioounsn 12589 |
. . . . . . . . . . 11
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝐸‘𝑋) ∈ ℝ* ∧ (𝑄‘𝑖) < (𝐸‘𝑋)) → (((𝑄‘𝑖)(,)(𝐸‘𝑋)) ∪ {(𝐸‘𝑋)}) = ((𝑄‘𝑖)(,](𝐸‘𝑋))) |
245 | 215, 239,
241, 244 | syl3anc 1496 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖)(,)(𝐸‘𝑋)) ∪ {(𝐸‘𝑋)}) = ((𝑄‘𝑖)(,](𝐸‘𝑋))) |
246 | 245 | fveq2d 6437 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) →
((int‘((TopOpen‘ℂfld) ↾t
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))‘(((𝑄‘𝑖)(,)(𝐸‘𝑋)) ∪ {(𝐸‘𝑋)})) =
((int‘((TopOpen‘ℂfld) ↾t
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))‘((𝑄‘𝑖)(,](𝐸‘𝑋)))) |
247 | 236 | cnfldtop 22957 |
. . . . . . . . . . 11
⊢
(TopOpen‘ℂfld) ∈ Top |
248 | | ovex 6937 |
. . . . . . . . . . . . 13
⊢
(-∞(,)(𝐸‘𝑋)) ∈ V |
249 | 248 | inex1 5024 |
. . . . . . . . . . . 12
⊢
((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∈ V |
250 | | snex 5129 |
. . . . . . . . . . . 12
⊢ {(𝐸‘𝑋)} ∈ V |
251 | 249, 250 | unex 7216 |
. . . . . . . . . . 11
⊢
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ∈ V |
252 | | resttop 21335 |
. . . . . . . . . . 11
⊢
(((TopOpen‘ℂfld) ∈ Top ∧
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ∈ V) →
((TopOpen‘ℂfld) ↾t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∈ Top) |
253 | 247, 251,
252 | mp2an 685 |
. . . . . . . . . 10
⊢
((TopOpen‘ℂfld) ↾t
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∈ Top |
254 | | retop 22935 |
. . . . . . . . . . . . 13
⊢
(topGen‘ran (,)) ∈ Top |
255 | 254 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (topGen‘ran (,)) ∈
Top) |
256 | 251 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ∈ V) |
257 | | iooretop 22939 |
. . . . . . . . . . . . 13
⊢ ((𝑄‘𝑖)(,)+∞) ∈ (topGen‘ran
(,)) |
258 | 257 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)+∞) ∈ (topGen‘ran
(,))) |
259 | | elrestr 16442 |
. . . . . . . . . . . 12
⊢
(((topGen‘ran (,)) ∈ Top ∧ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ∈ V ∧ ((𝑄‘𝑖)(,)+∞) ∈ (topGen‘ran (,)))
→ (((𝑄‘𝑖)(,)+∞) ∩
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∈ ((topGen‘ran (,))
↾t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
260 | 255, 256,
258, 259 | syl3anc 1496 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∈ ((topGen‘ran (,))
↾t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
261 | | simpr 479 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 = (𝐸‘𝑋)) → 𝑥 = (𝐸‘𝑋)) |
262 | | pnfxr 10410 |
. . . . . . . . . . . . . . . . . . . 20
⊢ +∞
∈ ℝ* |
263 | 262 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → +∞ ∈
ℝ*) |
264 | 238 | ltpnfd 12241 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) < +∞) |
265 | 215, 263,
238, 241, 264 | eliood 40519 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)+∞)) |
266 | | snidg 4427 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐸‘𝑋) ∈ ℝ → (𝐸‘𝑋) ∈ {(𝐸‘𝑋)}) |
267 | | elun2 4008 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐸‘𝑋) ∈ {(𝐸‘𝑋)} → (𝐸‘𝑋) ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) |
268 | 266, 267 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐸‘𝑋) ∈ ℝ → (𝐸‘𝑋) ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) |
269 | 136, 268 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐸‘𝑋) ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) |
270 | 269 | 3ad2ant1 1169 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) |
271 | 265, 270 | elind 4025 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
272 | 271 | adantr 474 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
273 | 261, 272 | eqeltrd 2906 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
274 | 273 | adantlr 708 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
275 | 215 | adantr 474 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝑄‘𝑖) ∈
ℝ*) |
276 | 262 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → +∞ ∈
ℝ*) |
277 | 203 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝑄‘𝑖) ∈
ℝ*) |
278 | 136 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐸‘𝑋) ∈ ℝ) |
279 | 278 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝐸‘𝑋) ∈ ℝ) |
280 | | iocssre 12541 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝐸‘𝑋) ∈ ℝ) → ((𝑄‘𝑖)(,](𝐸‘𝑋)) ⊆ ℝ) |
281 | 277, 279,
280 | syl2anc 581 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → ((𝑄‘𝑖)(,](𝐸‘𝑋)) ⊆ ℝ) |
282 | | simpr 479 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) |
283 | 281, 282 | sseldd 3828 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ∈ ℝ) |
284 | 283 | 3adantl3 1215 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ∈ ℝ) |
285 | 279 | rexrd 10406 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝐸‘𝑋) ∈
ℝ*) |
286 | | iocgtlb 40523 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝐸‘𝑋) ∈ ℝ* ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝑄‘𝑖) < 𝑥) |
287 | 277, 285,
282, 286 | syl3anc 1496 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝑄‘𝑖) < 𝑥) |
288 | 287 | 3adantl3 1215 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝑄‘𝑖) < 𝑥) |
289 | 284 | ltpnfd 12241 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 < +∞) |
290 | 275, 276,
284, 288, 289 | eliood 40519 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ∈ ((𝑄‘𝑖)(,)+∞)) |
291 | 290 | adantr 474 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ ((𝑄‘𝑖)(,)+∞)) |
292 | 197 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → -∞ ∈
ℝ*) |
293 | 285 | adantr 474 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈
ℝ*) |
294 | 283 | adantr 474 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ ℝ) |
295 | 294 | mnfltd 12244 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → -∞ < 𝑥) |
296 | 136 | ad3antrrr 723 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ℝ) |
297 | | iocleub 40524 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝐸‘𝑋) ∈ ℝ* ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ≤ (𝐸‘𝑋)) |
298 | 277, 285,
282, 297 | syl3anc 1496 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ≤ (𝐸‘𝑋)) |
299 | 298 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ≤ (𝐸‘𝑋)) |
300 | | neqne 3007 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (¬
𝑥 = (𝐸‘𝑋) → 𝑥 ≠ (𝐸‘𝑋)) |
301 | 300 | necomd 3054 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (¬
𝑥 = (𝐸‘𝑋) → (𝐸‘𝑋) ≠ 𝑥) |
302 | 301 | adantl 475 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ≠ 𝑥) |
303 | 294, 296,
299, 302 | leneltd 10510 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 < (𝐸‘𝑋)) |
304 | 292, 293,
294, 295, 303 | eliood 40519 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) |
305 | 304 | 3adantll3 40021 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) |
306 | 229 | ad2antrr 719 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷) |
307 | 275 | adantr 474 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝑄‘𝑖) ∈
ℝ*) |
308 | 213 | ad2antrr 719 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝑄‘(𝑖 + 1)) ∈
ℝ*) |
309 | 284 | adantr 474 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ ℝ) |
310 | 288 | adantr 474 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝑄‘𝑖) < 𝑥) |
311 | 238 | ad2antrr 719 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ℝ) |
312 | 212 | ad2antrr 719 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
313 | 303 | 3adantll3 40021 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 < (𝐸‘𝑋)) |
314 | 218 | ad2antrr 719 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ≤ (𝑄‘(𝑖 + 1))) |
315 | 309, 311,
312, 313, 314 | ltletrd 10516 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 < (𝑄‘(𝑖 + 1))) |
316 | 307, 308,
309, 310, 315 | eliood 40519 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
317 | 306, 316 | sseldd 3828 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ 𝐷) |
318 | 305, 317 | elind 4025 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) |
319 | | elun1 4007 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) → 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) |
320 | 318, 319 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) |
321 | 291, 320 | elind 4025 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
322 | 274, 321 | pm2.61dan 849 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
323 | 215 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → (𝑄‘𝑖) ∈
ℝ*) |
324 | 239 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → (𝐸‘𝑋) ∈
ℝ*) |
325 | | elinel1 4026 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) → 𝑥 ∈ ((𝑄‘𝑖)(,)+∞)) |
326 | | elioore 12493 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)+∞) → 𝑥 ∈ ℝ) |
327 | 326 | rexrd 10406 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)+∞) → 𝑥 ∈ ℝ*) |
328 | 325, 327 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) → 𝑥 ∈ ℝ*) |
329 | 328 | adantl 475 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → 𝑥 ∈ ℝ*) |
330 | 203 | adantr 474 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → (𝑄‘𝑖) ∈
ℝ*) |
331 | 262 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → +∞ ∈
ℝ*) |
332 | 325 | adantl 475 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → 𝑥 ∈ ((𝑄‘𝑖)(,)+∞)) |
333 | | ioogtlb 40516 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ +∞
∈ ℝ* ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)+∞)) → (𝑄‘𝑖) < 𝑥) |
334 | 330, 331,
332, 333 | syl3anc 1496 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → (𝑄‘𝑖) < 𝑥) |
335 | 334 | 3adantl3 1215 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → (𝑄‘𝑖) < 𝑥) |
336 | | elinel2 4027 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) → 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) |
337 | | elsni 4414 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ {(𝐸‘𝑋)} → 𝑥 = (𝐸‘𝑋)) |
338 | 337 | adantl 475 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 = (𝐸‘𝑋)) |
339 | 137 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ {(𝐸‘𝑋)}) → (𝐸‘𝑋) ≤ (𝐸‘𝑋)) |
340 | 338, 339 | eqbrtrd 4895 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 ≤ (𝐸‘𝑋)) |
341 | 340 | adantlr 708 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∧ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 ≤ (𝐸‘𝑋)) |
342 | | simpll 785 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∧ ¬ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝜑) |
343 | | elunnel2 40016 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ∧ ¬ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) |
344 | 343 | adantll 707 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∧ ¬ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) |
345 | | elinel1 4026 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) → 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) |
346 | | elioore 12493 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ (-∞(,)(𝐸‘𝑋)) → 𝑥 ∈ ℝ) |
347 | 346 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → 𝑥 ∈ ℝ) |
348 | 136 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → (𝐸‘𝑋) ∈ ℝ) |
349 | 197 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → -∞ ∈
ℝ*) |
350 | 348 | rexrd 10406 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → (𝐸‘𝑋) ∈
ℝ*) |
351 | | simpr 479 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) |
352 | | iooltub 40532 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((-∞ ∈ ℝ* ∧ (𝐸‘𝑋) ∈ ℝ* ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → 𝑥 < (𝐸‘𝑋)) |
353 | 349, 350,
351, 352 | syl3anc 1496 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → 𝑥 < (𝐸‘𝑋)) |
354 | 347, 348,
353 | ltled 10504 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → 𝑥 ≤ (𝐸‘𝑋)) |
355 | 345, 354 | sylan2 588 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) → 𝑥 ≤ (𝐸‘𝑋)) |
356 | 342, 344,
355 | syl2anc 581 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∧ ¬ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 ≤ (𝐸‘𝑋)) |
357 | 341, 356 | pm2.61dan 849 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) → 𝑥 ≤ (𝐸‘𝑋)) |
358 | 357 | adantlr 708 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) → 𝑥 ≤ (𝐸‘𝑋)) |
359 | 336, 358 | sylan2 588 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → 𝑥 ≤ (𝐸‘𝑋)) |
360 | 359 | 3adantl3 1215 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → 𝑥 ≤ (𝐸‘𝑋)) |
361 | 323, 324,
329, 335, 360 | eliocd 40529 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) |
362 | 322, 361 | impbida 837 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋)) ↔ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))) |
363 | 362 | eqrdv 2823 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,](𝐸‘𝑋)) = (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
364 | | ioossre 12523 |
. . . . . . . . . . . . . 14
⊢
(-∞(,)(𝐸‘𝑋)) ⊆ ℝ |
365 | | ssinss1 4066 |
. . . . . . . . . . . . . 14
⊢
((-∞(,)(𝐸‘𝑋)) ⊆ ℝ →
((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ⊆ ℝ) |
366 | 364, 365 | mp1i 13 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ⊆ ℝ) |
367 | 238 | snssd 4558 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → {(𝐸‘𝑋)} ⊆ ℝ) |
368 | 366, 367 | unssd 4016 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ⊆ ℝ) |
369 | | eqid 2825 |
. . . . . . . . . . . . 13
⊢
(topGen‘ran (,)) = (topGen‘ran (,)) |
370 | 236, 369 | rerest 22977 |
. . . . . . . . . . . 12
⊢
((((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ⊆ ℝ →
((TopOpen‘ℂfld) ↾t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) = ((topGen‘ran (,))
↾t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
371 | 368, 370 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) →
((TopOpen‘ℂfld) ↾t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) = ((topGen‘ran (,))
↾t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
372 | 260, 363,
371 | 3eltr4d 2921 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,](𝐸‘𝑋)) ∈
((TopOpen‘ℂfld) ↾t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
373 | | isopn3i 21257 |
. . . . . . . . . 10
⊢
((((TopOpen‘ℂfld) ↾t
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∈ Top ∧ ((𝑄‘𝑖)(,](𝐸‘𝑋)) ∈
((TopOpen‘ℂfld) ↾t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) →
((int‘((TopOpen‘ℂfld) ↾t
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))‘((𝑄‘𝑖)(,](𝐸‘𝑋))) = ((𝑄‘𝑖)(,](𝐸‘𝑋))) |
374 | 253, 372,
373 | sylancr 583 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) →
((int‘((TopOpen‘ℂfld) ↾t
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))‘((𝑄‘𝑖)(,](𝐸‘𝑋))) = ((𝑄‘𝑖)(,](𝐸‘𝑋))) |
375 | 246, 374 | eqtr2d 2862 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,](𝐸‘𝑋)) =
((int‘((TopOpen‘ℂfld) ↾t
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))‘(((𝑄‘𝑖)(,)(𝐸‘𝑋)) ∪ {(𝐸‘𝑋)}))) |
376 | 243, 375 | eleqtrd 2908 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈
((int‘((TopOpen‘ℂfld) ↾t
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})))‘(((𝑄‘𝑖)(,)(𝐸‘𝑋)) ∪ {(𝐸‘𝑋)}))) |
377 | 196, 231,
235, 236, 237, 376 | limcres 24049 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) = ((𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) limℂ (𝐸‘𝑋))) |
378 | 231 | resabs1d 5664 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) = (𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋)))) |
379 | 378 | oveq1d 6920 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) |
380 | 189, 377,
379 | 3eqtr2d 2867 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) |
381 | 184 | feq2d 6264 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ↔ 𝐹:𝐷⟶ℂ)) |
382 | 192, 381 | mpbird 249 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ) |
383 | 382 | adantr 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → 𝐹:dom 𝐹⟶ℂ) |
384 | 383 | 3ad2antl1 1242 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → 𝐹:dom 𝐹⟶ℂ) |
385 | | ioosscn 40515 |
. . . . . . . . . 10
⊢ ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ ℂ |
386 | 385 | a1i 11 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ ℂ) |
387 | 184 | eqcomd 2831 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐷 = dom 𝐹) |
388 | 387 | 3ad2ant1 1169 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝐷 = dom 𝐹) |
389 | 230, 388 | sseqtrd 3866 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ dom 𝐹) |
390 | 389 | adantr 474 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ dom 𝐹) |
391 | 7 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑍 = (𝑥 ∈ ℝ ↦
((⌊‘((𝐵 −
𝑥) / 𝑇)) · 𝑇))) |
392 | | oveq2 6913 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑋 → (𝐵 − 𝑥) = (𝐵 − 𝑋)) |
393 | 392 | oveq1d 6920 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑋 → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − 𝑋) / 𝑇)) |
394 | 393 | fveq2d 6437 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑋 → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − 𝑋) / 𝑇))) |
395 | 394 | oveq1d 6920 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑋 → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
396 | 395 | adantl 475 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
397 | 2, 14 | resubcld 10782 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ) |
398 | 2, 1 | resubcld 10782 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
399 | 4, 398 | syl5eqel 2910 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑇 ∈ ℝ) |
400 | 1, 2 | posdifd 10939 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
401 | 3, 400 | mpbid 224 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 0 < (𝐵 − 𝐴)) |
402 | 4 | eqcomi 2834 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐵 − 𝐴) = 𝑇 |
403 | 402 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐵 − 𝐴) = 𝑇) |
404 | 401, 403 | breqtrd 4899 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 0 < 𝑇) |
405 | 404 | gt0ne0d 10916 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑇 ≠ 0) |
406 | 397, 399,
405 | redivcld 11179 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝐵 − 𝑋) / 𝑇) ∈ ℝ) |
407 | 406 | flcld 12894 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
408 | 407 | zred 11810 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℝ) |
409 | 408, 399 | remulcld 10387 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) ∈ ℝ) |
410 | 391, 396,
14, 409 | fvmptd 6535 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑍‘𝑋) = ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
411 | 410, 409 | eqeltrd 2906 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑍‘𝑋) ∈ ℝ) |
412 | 411 | recnd 10385 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑍‘𝑋) ∈ ℂ) |
413 | 412 | adantr 474 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → (𝑍‘𝑋) ∈ ℂ) |
414 | 413 | 3ad2antl1 1242 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → (𝑍‘𝑋) ∈ ℂ) |
415 | 414 | negcld 10700 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → -(𝑍‘𝑋) ∈ ℂ) |
416 | | eqid 2825 |
. . . . . . . . 9
⊢ {𝑧 ∈ ℂ ∣
∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))} = {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))} |
417 | | ioosscn 40515 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ ℂ |
418 | 417 | sseli 3823 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) → 𝑦 ∈ ℂ) |
419 | 418 | adantl 475 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ ℂ) |
420 | 412 | adantr 474 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑍‘𝑋) ∈ ℂ) |
421 | 419, 420 | pncand 10714 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = 𝑦) |
422 | 421 | eqcomd 2831 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 = ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋))) |
423 | 422 | 3ad2antl1 1242 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 = ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋))) |
424 | 410 | oveq2d 6921 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = ((𝑦 + (𝑍‘𝑋)) − ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
425 | 424 | adantr 474 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = ((𝑦 + (𝑍‘𝑋)) − ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
426 | 419, 420 | addcld 10376 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ℂ) |
427 | 409 | recnd 10385 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) ∈ ℂ) |
428 | 427 | adantr 474 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) ∈ ℂ) |
429 | 426, 428 | negsubd 10719 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) + -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) − ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
430 | 407 | zcnd 11811 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℂ) |
431 | 399 | recnd 10385 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑇 ∈ ℂ) |
432 | 430, 431 | mulneg1d 10807 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) = -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
433 | 432 | eqcomd 2831 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
434 | 433 | oveq2d 6921 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑦 + (𝑍‘𝑋)) + -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
435 | 434 | adantr 474 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) + -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
436 | 425, 429,
435 | 3eqtr2d 2867 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
437 | 436 | 3ad2antl1 1242 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
438 | 407 | znegcld 11812 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
439 | 438 | adantr 474 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
440 | 439 | 3ad2antl1 1242 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
441 | | simpl1 1248 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝜑) |
442 | 230 | adantr 474 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ 𝐷) |
443 | 203 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘𝑖) ∈
ℝ*) |
444 | 136 | rexrd 10406 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐸‘𝑋) ∈
ℝ*) |
445 | 444 | ad2antrr 719 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐸‘𝑋) ∈
ℝ*) |
446 | | elioore 12493 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) → 𝑦 ∈ ℝ) |
447 | 446 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ ℝ) |
448 | 411 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑍‘𝑋) ∈ ℝ) |
449 | 447, 448 | readdcld 10386 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ℝ) |
450 | 449 | adantlr 708 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ℝ) |
451 | 411 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑍‘𝑋) ∈ ℝ) |
452 | 202, 451 | resubcld 10782 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ) |
453 | 452 | rexrd 10406 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈
ℝ*) |
454 | 453 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈
ℝ*) |
455 | 14 | rexrd 10406 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝑋 ∈
ℝ*) |
456 | 455 | ad2antrr 719 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑋 ∈
ℝ*) |
457 | | simpr 479 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) |
458 | | ioogtlb 40516 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ* ∧ 𝑋 ∈ ℝ*
∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑦) |
459 | 454, 456,
457, 458 | syl3anc 1496 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑦) |
460 | 202 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘𝑖) ∈ ℝ) |
461 | 451 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑍‘𝑋) ∈ ℝ) |
462 | 446 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ ℝ) |
463 | 460, 461,
462 | ltsubaddd 10948 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑦 ↔ (𝑄‘𝑖) < (𝑦 + (𝑍‘𝑋)))) |
464 | 459, 463 | mpbid 224 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘𝑖) < (𝑦 + (𝑍‘𝑋))) |
465 | 14 | ad2antrr 719 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑋 ∈ ℝ) |
466 | | iooltub 40532 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ* ∧ 𝑋 ∈ ℝ*
∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 < 𝑋) |
467 | 454, 456,
457, 466 | syl3anc 1496 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 < 𝑋) |
468 | 462, 465,
461, 467 | ltadd1dd 10963 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) < (𝑋 + (𝑍‘𝑋))) |
469 | 5 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍‘𝑥)))) |
470 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) |
471 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 = 𝑋 → (𝑍‘𝑥) = (𝑍‘𝑋)) |
472 | 470, 471 | oveq12d 6923 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = 𝑋 → (𝑥 + (𝑍‘𝑥)) = (𝑋 + (𝑍‘𝑋))) |
473 | 472 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥 + (𝑍‘𝑥)) = (𝑋 + (𝑍‘𝑋))) |
474 | 14, 411 | readdcld 10386 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑋 + (𝑍‘𝑋)) ∈ ℝ) |
475 | 469, 473,
14, 474 | fvmptd 6535 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝐸‘𝑋) = (𝑋 + (𝑍‘𝑋))) |
476 | 475 | eqcomd 2831 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑋 + (𝑍‘𝑋)) = (𝐸‘𝑋)) |
477 | 476 | ad2antrr 719 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑋 + (𝑍‘𝑋)) = (𝐸‘𝑋)) |
478 | 468, 477 | breqtrd 4899 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) < (𝐸‘𝑋)) |
479 | 443, 445,
450, 464, 478 | eliood 40519 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) |
480 | 479 | 3adantl3 1215 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) |
481 | 442, 480 | sseldd 3828 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ 𝐷) |
482 | 441, 481,
440 | 3jca 1164 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)) |
483 | | eleq1 2894 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑘 ∈ ℤ ↔
-(⌊‘((𝐵 −
𝑋) / 𝑇)) ∈ ℤ)) |
484 | 483 | 3anbi3d 1572 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ))) |
485 | | oveq1 6912 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑘 · 𝑇) = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
486 | 485 | oveq2d 6921 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
487 | 486 | eleq1d 2891 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷 ↔ ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ 𝐷)) |
488 | 484, 487 | imbi12d 336 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷) ↔ ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ 𝐷))) |
489 | | ovex 6937 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 + (𝑍‘𝑋)) ∈ V |
490 | | eleq1 2894 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → (𝑥 ∈ 𝐷 ↔ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷)) |
491 | 490 | 3anbi2d 1571 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ))) |
492 | | oveq1 6912 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → (𝑥 + (𝑘 · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇))) |
493 | 492 | eleq1d 2891 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → ((𝑥 + (𝑘 · 𝑇)) ∈ 𝐷 ↔ ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷)) |
494 | 491, 493 | imbi12d 336 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → (((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷) ↔ ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷))) |
495 | | fourierdlem49.dper |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷) |
496 | 489, 494,
495 | vtocl 3475 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷) |
497 | 488, 496 | vtoclg 3482 |
. . . . . . . . . . . . . . . 16
⊢
(-(⌊‘((𝐵
− 𝑋) / 𝑇)) ∈ ℤ → ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ 𝐷)) |
498 | 440, 482,
497 | sylc 65 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ 𝐷) |
499 | 437, 498 | eqeltrd 2906 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) ∈ 𝐷) |
500 | 423, 499 | eqeltrd 2906 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ 𝐷) |
501 | 500 | ralrimiva 3175 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ∀𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑦 ∈ 𝐷) |
502 | | dfss3 3816 |
. . . . . . . . . . . 12
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ 𝐷 ↔ ∀𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑦 ∈ 𝐷) |
503 | 501, 502 | sylibr 226 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ 𝐷) |
504 | 202 | recnd 10385 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ) |
505 | 412 | adantr 474 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑍‘𝑋) ∈ ℂ) |
506 | 504, 505 | negsubd 10719 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + -(𝑍‘𝑋)) = ((𝑄‘𝑖) − (𝑍‘𝑋))) |
507 | 506 | eqcomd 2831 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) = ((𝑄‘𝑖) + -(𝑍‘𝑋))) |
508 | 475 | oveq1d 6920 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝐸‘𝑋) + -(𝑍‘𝑋)) = ((𝑋 + (𝑍‘𝑋)) + -(𝑍‘𝑋))) |
509 | 474 | recnd 10385 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑋 + (𝑍‘𝑋)) ∈ ℂ) |
510 | 509, 412 | negsubd 10719 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑋 + (𝑍‘𝑋)) + -(𝑍‘𝑋)) = ((𝑋 + (𝑍‘𝑋)) − (𝑍‘𝑋))) |
511 | 14 | recnd 10385 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑋 ∈ ℂ) |
512 | 511, 412 | pncand 10714 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑋 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = 𝑋) |
513 | 508, 510,
512 | 3eqtrrd 2866 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑋 = ((𝐸‘𝑋) + -(𝑍‘𝑋))) |
514 | 513 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 = ((𝐸‘𝑋) + -(𝑍‘𝑋))) |
515 | 507, 514 | oveq12d 6923 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) = (((𝑄‘𝑖) + -(𝑍‘𝑋))(,)((𝐸‘𝑋) + -(𝑍‘𝑋)))) |
516 | 451 | renegcld 10781 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -(𝑍‘𝑋) ∈ ℝ) |
517 | 202, 278,
516 | iooshift 40544 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + -(𝑍‘𝑋))(,)((𝐸‘𝑋) + -(𝑍‘𝑋))) = {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))}) |
518 | 515, 517 | eqtr2d 2862 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))} = (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) |
519 | 518 | 3adant3 1168 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))} = (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) |
520 | 184 | 3ad2ant1 1169 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → dom 𝐹 = 𝐷) |
521 | 503, 519,
520 | 3sstr4d 3873 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))} ⊆ dom 𝐹) |
522 | 521 | adantr 474 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))} ⊆ dom 𝐹) |
523 | 410 | negeqd 10595 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → -(𝑍‘𝑋) = -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
524 | 523, 433 | eqtrd 2861 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → -(𝑍‘𝑋) = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
525 | 524 | oveq2d 6921 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑥 + -(𝑍‘𝑋)) = (𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
526 | 525 | fveq2d 6437 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐹‘(𝑥 + -(𝑍‘𝑋))) = (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))) |
527 | 526 | adantr 474 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → (𝐹‘(𝑥 + -(𝑍‘𝑋))) = (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))) |
528 | 527 | 3ad2antl1 1242 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → (𝐹‘(𝑥 + -(𝑍‘𝑋))) = (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))) |
529 | 438 | adantr 474 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
530 | 529 | 3ad2antl1 1242 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
531 | | simpl1 1248 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → 𝜑) |
532 | 230 | sselda 3827 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → 𝑥 ∈ 𝐷) |
533 | 531, 532,
530 | 3jca 1164 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → (𝜑 ∧ 𝑥 ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)) |
534 | 483 | 3anbi3d 1572 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ 𝑥 ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ))) |
535 | 485 | oveq2d 6921 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑥 + (𝑘 · 𝑇)) = (𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
536 | 535 | fveq2d 6437 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))) |
537 | 536 | eqeq1d 2827 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥) ↔ (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))) |
538 | 534, 537 | imbi12d 336 |
. . . . . . . . . . . . 13
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥)))) |
539 | | fourierdlem49.per |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥)) |
540 | 538, 539 | vtoclg 3482 |
. . . . . . . . . . . 12
⊢
(-(⌊‘((𝐵
− 𝑋) / 𝑇)) ∈ ℤ → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))) |
541 | 530, 533,
540 | sylc 65 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥)) |
542 | 528, 541 | eqtrd 2861 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → (𝐹‘(𝑥 + -(𝑍‘𝑋))) = (𝐹‘𝑥)) |
543 | 542 | adantlr 708 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → (𝐹‘(𝑥 + -(𝑍‘𝑋))) = (𝐹‘𝑥)) |
544 | | simpr 479 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) |
545 | 384, 386,
390, 415, 416, 522, 543, 544 | limcperiod 40655 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → 𝑦 ∈ ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))}) limℂ ((𝐸‘𝑋) + -(𝑍‘𝑋)))) |
546 | 518 | reseq2d 5629 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))}) = (𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋))) |
547 | 514 | eqcomd 2831 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐸‘𝑋) + -(𝑍‘𝑋)) = 𝑋) |
548 | 546, 547 | oveq12d 6923 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))}) limℂ ((𝐸‘𝑋) + -(𝑍‘𝑋))) = ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) |
549 | 548 | 3adant3 1168 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))}) limℂ ((𝐸‘𝑋) + -(𝑍‘𝑋))) = ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) |
550 | 549 | adantr 474 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))}) limℂ ((𝐸‘𝑋) + -(𝑍‘𝑋))) = ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) |
551 | 545, 550 | eleqtrd 2908 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) |
552 | 382 | adantr 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) → 𝐹:dom 𝐹⟶ℂ) |
553 | 552 | 3ad2antl1 1242 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) → 𝐹:dom 𝐹⟶ℂ) |
554 | 417 | a1i 11 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ ℂ) |
555 | 503, 520 | sseqtr4d 3867 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ dom 𝐹) |
556 | 555 | adantr 474 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ dom 𝐹) |
557 | 412 | adantr 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) → (𝑍‘𝑋) ∈ ℂ) |
558 | 557 | 3ad2antl1 1242 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) → (𝑍‘𝑋) ∈ ℂ) |
559 | | eqid 2825 |
. . . . . . . . 9
⊢ {𝑧 ∈ ℂ ∣
∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))} = {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))} |
560 | 504, 505 | npcand 10717 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) − (𝑍‘𝑋)) + (𝑍‘𝑋)) = (𝑄‘𝑖)) |
561 | 560 | eqcomd 2831 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = (((𝑄‘𝑖) − (𝑍‘𝑋)) + (𝑍‘𝑋))) |
562 | 475 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐸‘𝑋) = (𝑋 + (𝑍‘𝑋))) |
563 | 561, 562 | oveq12d 6923 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) = ((((𝑄‘𝑖) − (𝑍‘𝑋)) + (𝑍‘𝑋))(,)(𝑋 + (𝑍‘𝑋)))) |
564 | 14 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ) |
565 | 452, 564,
451 | iooshift 40544 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((((𝑄‘𝑖) − (𝑍‘𝑋)) + (𝑍‘𝑋))(,)(𝑋 + (𝑍‘𝑋))) = {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))}) |
566 | 563, 565 | eqtr2d 2862 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))} = ((𝑄‘𝑖)(,)(𝐸‘𝑋))) |
567 | 566 | 3adant3 1168 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))} = ((𝑄‘𝑖)(,)(𝐸‘𝑋))) |
568 | 230, 567,
520 | 3sstr4d 3873 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))} ⊆ dom 𝐹) |
569 | 568 | adantr 474 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))} ⊆ dom 𝐹) |
570 | 410 | oveq2d 6921 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑥 + (𝑍‘𝑋)) = (𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
571 | 570 | fveq2d 6437 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐹‘(𝑥 + (𝑍‘𝑋))) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))) |
572 | 571 | adantr 474 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐹‘(𝑥 + (𝑍‘𝑋))) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))) |
573 | 572 | 3ad2antl1 1242 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐹‘(𝑥 + (𝑍‘𝑋))) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))) |
574 | 407 | adantr 474 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
575 | 574 | 3ad2antl1 1242 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
576 | | simpl1 1248 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝜑) |
577 | 503 | sselda 3827 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑥 ∈ 𝐷) |
578 | 576, 577,
575 | 3jca 1164 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)) |
579 | | eleq1 2894 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑘 ∈ ℤ ↔ (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)) |
580 | 579 | 3anbi3d 1572 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ))) |
581 | | oveq1 6912 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑘 · 𝑇) = ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
582 | 581 | oveq2d 6921 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑥 + (𝑘 · 𝑇)) = (𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
583 | 582 | fveq2d 6437 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))) |
584 | 583 | eqeq1d 2827 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥) ↔ (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))) |
585 | 580, 584 | imbi12d 336 |
. . . . . . . . . . . . 13
⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → (((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥)))) |
586 | 585, 539 | vtoclg 3482 |
. . . . . . . . . . . 12
⊢
((⌊‘((𝐵
− 𝑋) / 𝑇)) ∈ ℤ → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))) |
587 | 575, 578,
586 | sylc 65 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥)) |
588 | 573, 587 | eqtrd 2861 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐹‘(𝑥 + (𝑍‘𝑋))) = (𝐹‘𝑥)) |
589 | 588 | adantlr 708 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐹‘(𝑥 + (𝑍‘𝑋))) = (𝐹‘𝑥)) |
590 | | simpr 479 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) → 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) |
591 | 553, 554,
556, 558, 559, 569, 589, 590 | limcperiod 40655 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) → 𝑦 ∈ ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))}) limℂ (𝑋 + (𝑍‘𝑋)))) |
592 | 566 | reseq2d 5629 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))}) = (𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋)))) |
593 | 476 | adantr 474 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑍‘𝑋)) = (𝐸‘𝑋)) |
594 | 592, 593 | oveq12d 6923 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))}) limℂ (𝑋 + (𝑍‘𝑋))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) |
595 | 594 | 3adant3 1168 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))}) limℂ (𝑋 + (𝑍‘𝑋))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) |
596 | 595 | adantr 474 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) → ((𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))}) limℂ (𝑋 + (𝑍‘𝑋))) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) |
597 | 591, 596 | eleqtrd 2908 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) → 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) |
598 | 551, 597 | impbida 837 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) ↔ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋))) |
599 | 598 | eqrdv 2823 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) = ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) |
600 | | resindm 5681 |
. . . . . . . . . . 11
⊢ (Rel
𝐹 → (𝐹 ↾ ((-∞(,)𝑋) ∩ dom 𝐹)) = (𝐹 ↾ (-∞(,)𝑋))) |
601 | 600 | eqcomd 2831 |
. . . . . . . . . 10
⊢ (Rel
𝐹 → (𝐹 ↾ (-∞(,)𝑋)) = (𝐹 ↾ ((-∞(,)𝑋) ∩ dom 𝐹))) |
602 | 179, 601 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 ↾ (-∞(,)𝑋)) = (𝐹 ↾ ((-∞(,)𝑋) ∩ dom 𝐹))) |
603 | 184 | ineq2d 4041 |
. . . . . . . . . 10
⊢ (𝜑 → ((-∞(,)𝑋) ∩ dom 𝐹) = ((-∞(,)𝑋) ∩ 𝐷)) |
604 | 603 | reseq2d 5629 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 ↾ ((-∞(,)𝑋) ∩ dom 𝐹)) = (𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷))) |
605 | 602, 604 | eqtrd 2861 |
. . . . . . . 8
⊢ (𝜑 → (𝐹 ↾ (-∞(,)𝑋)) = (𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷))) |
606 | 605 | oveq1d 6920 |
. . . . . . 7
⊢ (𝜑 → ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋) = ((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) limℂ 𝑋)) |
607 | 606 | 3ad2ant1 1169 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋) = ((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) limℂ 𝑋)) |
608 | | inss2 4058 |
. . . . . . . . . 10
⊢
((-∞(,)𝑋)
∩ 𝐷) ⊆ 𝐷 |
609 | 608 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((-∞(,)𝑋) ∩ 𝐷) ⊆ 𝐷) |
610 | 193, 609 | fssresd 6308 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)):((-∞(,)𝑋) ∩ 𝐷)⟶ℂ) |
611 | 452 | mnfltd 12244 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -∞ < ((𝑄‘𝑖) − (𝑍‘𝑋))) |
612 | 198, 453,
611 | xrltled 12269 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -∞ ≤ ((𝑄‘𝑖) − (𝑍‘𝑋))) |
613 | | iooss1 12498 |
. . . . . . . . . . 11
⊢
((-∞ ∈ ℝ* ∧ -∞ ≤ ((𝑄‘𝑖) − (𝑍‘𝑋))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ (-∞(,)𝑋)) |
614 | 197, 612,
613 | sylancr 583 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ (-∞(,)𝑋)) |
615 | 614 | 3adant3 1168 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ (-∞(,)𝑋)) |
616 | 615, 503 | ssind 4061 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ ((-∞(,)𝑋) ∩ 𝐷)) |
617 | 608, 234 | syl5ss 3838 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((-∞(,)𝑋) ∩ 𝐷) ⊆ ℂ) |
618 | | eqid 2825 |
. . . . . . . 8
⊢
((TopOpen‘ℂfld) ↾t
(((-∞(,)𝑋) ∩
𝐷) ∪ {𝑋})) = ((TopOpen‘ℂfld)
↾t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
619 | 453 | 3adant3 1168 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈
ℝ*) |
620 | 455 | 3ad2ant1 1169 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝑋 ∈
ℝ*) |
621 | 475 | 3ad2ant1 1169 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) = (𝑋 + (𝑍‘𝑋))) |
622 | 241, 621 | breqtrd 4899 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝑋 + (𝑍‘𝑋))) |
623 | 411 | 3ad2ant1 1169 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑍‘𝑋) ∈ ℝ) |
624 | 14 | 3ad2ant1 1169 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ) |
625 | 214, 623,
624 | ltsubaddd 10948 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑋 ↔ (𝑄‘𝑖) < (𝑋 + (𝑍‘𝑋)))) |
626 | 622, 625 | mpbird 249 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑋) |
627 | 14 | leidd 10918 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 ≤ 𝑋) |
628 | 627 | 3ad2ant1 1169 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝑋 ≤ 𝑋) |
629 | 619, 620,
620, 626, 628 | eliocd 40529 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝑋 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) |
630 | | ioounsn 12589 |
. . . . . . . . . . . 12
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ* ∧ 𝑋 ∈ ℝ*
∧ ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑋) → ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ∪ {𝑋}) = (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) |
631 | 619, 620,
626, 630 | syl3anc 1496 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ∪ {𝑋}) = (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) |
632 | 631 | fveq2d 6437 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) →
((int‘((TopOpen‘ℂfld) ↾t
(((-∞(,)𝑋) ∩
𝐷) ∪ {𝑋})))‘((((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ∪ {𝑋})) =
((int‘((TopOpen‘ℂfld) ↾t
(((-∞(,)𝑋) ∩
𝐷) ∪ {𝑋})))‘(((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋))) |
633 | | ovex 6937 |
. . . . . . . . . . . . . 14
⊢
(-∞(,)𝑋)
∈ V |
634 | 633 | inex1 5024 |
. . . . . . . . . . . . 13
⊢
((-∞(,)𝑋)
∩ 𝐷) ∈
V |
635 | | snex 5129 |
. . . . . . . . . . . . 13
⊢ {𝑋} ∈ V |
636 | 634, 635 | unex 7216 |
. . . . . . . . . . . 12
⊢
(((-∞(,)𝑋)
∩ 𝐷) ∪ {𝑋}) ∈ V |
637 | | resttop 21335 |
. . . . . . . . . . . 12
⊢
(((TopOpen‘ℂfld) ∈ Top ∧
(((-∞(,)𝑋) ∩
𝐷) ∪ {𝑋}) ∈ V) →
((TopOpen‘ℂfld) ↾t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∈ Top) |
638 | 247, 636,
637 | mp2an 685 |
. . . . . . . . . . 11
⊢
((TopOpen‘ℂfld) ↾t
(((-∞(,)𝑋) ∩
𝐷) ∪ {𝑋})) ∈ Top |
639 | 636 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ∈ V) |
640 | | iooretop 22939 |
. . . . . . . . . . . . . 14
⊢ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∈ (topGen‘ran
(,)) |
641 | 640 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∈ (topGen‘ran
(,))) |
642 | | elrestr 16442 |
. . . . . . . . . . . . 13
⊢
(((topGen‘ran (,)) ∈ Top ∧ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ∈ V ∧ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∈ (topGen‘ran
(,))) → ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∈ ((topGen‘ran (,))
↾t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) |
643 | 255, 639,
641, 642 | syl3anc 1496 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∈ ((topGen‘ran (,))
↾t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) |
644 | 453 | adantr 474 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈
ℝ*) |
645 | 262 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → +∞ ∈
ℝ*) |
646 | 14 | ad2antrr 719 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑋 ∈ ℝ) |
647 | | iocssre 12541 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ* ∧ 𝑋 ∈ ℝ) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) ⊆ ℝ) |
648 | 644, 646,
647 | syl2anc 581 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) ⊆ ℝ) |
649 | | simpr 479 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) |
650 | 648, 649 | sseldd 3828 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ ℝ) |
651 | 455 | ad2antrr 719 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑋 ∈
ℝ*) |
652 | | iocgtlb 40523 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ* ∧ 𝑋 ∈ ℝ*
∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑥) |
653 | 644, 651,
649, 652 | syl3anc 1496 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑥) |
654 | 650 | ltpnfd 12241 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 < +∞) |
655 | 644, 645,
650, 653, 654 | eliood 40519 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞)) |
656 | 655 | 3adantl3 1215 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞)) |
657 | | eqvisset 3428 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 𝑋 → 𝑋 ∈ V) |
658 | | snidg 4427 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑋 ∈ V → 𝑋 ∈ {𝑋}) |
659 | 657, 658 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑋 → 𝑋 ∈ {𝑋}) |
660 | 470, 659 | eqeltrd 2906 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑋 → 𝑥 ∈ {𝑋}) |
661 | | elun2 4008 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ {𝑋} → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
662 | 660, 661 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑋 → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
663 | 662 | adantl 475 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ 𝑥 = 𝑋) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
664 | | simpll 785 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → (𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))) |
665 | 644 | adantr 474 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈
ℝ*) |
666 | 455 | ad3antrrr 723 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑋 ∈
ℝ*) |
667 | 650 | adantr 474 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 ∈ ℝ) |
668 | 653 | adantr 474 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑥) |
669 | 14 | ad3antrrr 723 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑋 ∈ ℝ) |
670 | | iocleub 40524 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ* ∧ 𝑋 ∈ ℝ*
∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ≤ 𝑋) |
671 | 644, 651,
649, 670 | syl3anc 1496 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ≤ 𝑋) |
672 | 671 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 ≤ 𝑋) |
673 | 470 | eqcoms 2833 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑋 = 𝑥 → 𝑥 = 𝑋) |
674 | 673 | necon3bi 3025 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (¬
𝑥 = 𝑋 → 𝑋 ≠ 𝑥) |
675 | 674 | adantl 475 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑋 ≠ 𝑥) |
676 | 667, 669,
672, 675 | leneltd 10510 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 < 𝑋) |
677 | 665, 666,
667, 668, 676 | eliood 40519 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) |
678 | 677 | 3adantll3 40021 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) |
679 | 616 | sselda 3827 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑥 ∈ ((-∞(,)𝑋) ∩ 𝐷)) |
680 | | elun1 4007 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ((-∞(,)𝑋) ∩ 𝐷) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
681 | 679, 680 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
682 | 664, 678,
681 | syl2anc 581 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
683 | 663, 682 | pm2.61dan 849 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
684 | 656, 683 | elind 4025 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) |
685 | 619 | adantr 474 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈
ℝ*) |
686 | 620 | adantr 474 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑋 ∈
ℝ*) |
687 | | elinel1 4026 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞)) |
688 | | elioore 12493 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) → 𝑥 ∈ ℝ) |
689 | 687, 688 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) → 𝑥 ∈ ℝ) |
690 | 689 | rexrd 10406 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) → 𝑥 ∈ ℝ*) |
691 | 690 | adantl 475 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑥 ∈ ℝ*) |
692 | 453 | adantr 474 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈
ℝ*) |
693 | 262 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → +∞ ∈
ℝ*) |
694 | 687 | adantl 475 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞)) |
695 | | ioogtlb 40516 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ* ∧ +∞
∈ ℝ* ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑥) |
696 | 692, 693,
694, 695 | syl3anc 1496 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑥) |
697 | 696 | 3adantl3 1215 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑥) |
698 | | elinel2 4027 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
699 | | elsni 4414 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ {𝑋} → 𝑥 = 𝑋) |
700 | 699 | adantl 475 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑋}) → 𝑥 = 𝑋) |
701 | 627 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑋}) → 𝑋 ≤ 𝑋) |
702 | 700, 701 | eqbrtrd 4895 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑋}) → 𝑥 ≤ 𝑋) |
703 | 702 | adantlr 708 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∧ 𝑥 ∈ {𝑋}) → 𝑥 ≤ 𝑋) |
704 | | simpll 785 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∧ ¬ 𝑥 ∈ {𝑋}) → 𝜑) |
705 | | elunnel2 40016 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ∧ ¬ 𝑥 ∈ {𝑋}) → 𝑥 ∈ ((-∞(,)𝑋) ∩ 𝐷)) |
706 | 705 | adantll 707 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∧ ¬ 𝑥 ∈ {𝑋}) → 𝑥 ∈ ((-∞(,)𝑋) ∩ 𝐷)) |
707 | | elinel1 4026 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ((-∞(,)𝑋) ∩ 𝐷) → 𝑥 ∈ (-∞(,)𝑋)) |
708 | 706, 707 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∧ ¬ 𝑥 ∈ {𝑋}) → 𝑥 ∈ (-∞(,)𝑋)) |
709 | | elioore 12493 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (-∞(,)𝑋) → 𝑥 ∈ ℝ) |
710 | 709 | adantl 475 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑥 ∈ ℝ) |
711 | 14 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑋 ∈ ℝ) |
712 | 197 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → -∞ ∈
ℝ*) |
713 | 455 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑋 ∈
ℝ*) |
714 | | simpr 479 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑥 ∈ (-∞(,)𝑋)) |
715 | | iooltub 40532 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((-∞ ∈ ℝ* ∧ 𝑋 ∈ ℝ* ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑥 < 𝑋) |
716 | 712, 713,
714, 715 | syl3anc 1496 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑥 < 𝑋) |
717 | 710, 711,
716 | ltled 10504 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑥 ≤ 𝑋) |
718 | 704, 708,
717 | syl2anc 581 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∧ ¬ 𝑥 ∈ {𝑋}) → 𝑥 ≤ 𝑋) |
719 | 703, 718 | pm2.61dan 849 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) → 𝑥 ≤ 𝑋) |
720 | 698, 719 | sylan2 588 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑥 ≤ 𝑋) |
721 | 720 | 3ad2antl1 1242 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑥 ≤ 𝑋) |
722 | 685, 686,
691, 697, 721 | eliocd 40529 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) |
723 | 684, 722 | impbida 837 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) ↔ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})))) |
724 | 723 | eqrdv 2823 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) = ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) |
725 | 608, 232 | syl5ss 3838 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((-∞(,)𝑋) ∩ 𝐷) ⊆ ℝ) |
726 | 14 | snssd 4558 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → {𝑋} ⊆ ℝ) |
727 | 725, 726 | unssd 4016 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ⊆ ℝ) |
728 | 727 | 3ad2ant1 1169 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ⊆ ℝ) |
729 | 236, 369 | rerest 22977 |
. . . . . . . . . . . . 13
⊢
((((-∞(,)𝑋)
∩ 𝐷) ∪ {𝑋}) ⊆ ℝ →
((TopOpen‘ℂfld) ↾t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) = ((topGen‘ran (,))
↾t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) |
730 | 728, 729 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) →
((TopOpen‘ℂfld) ↾t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) = ((topGen‘ran (,))
↾t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) |
731 | 643, 724,
730 | 3eltr4d 2921 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) ∈
((TopOpen‘ℂfld) ↾t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) |
732 | | isopn3i 21257 |
. . . . . . . . . . 11
⊢
((((TopOpen‘ℂfld) ↾t
(((-∞(,)𝑋) ∩
𝐷) ∪ {𝑋})) ∈ Top ∧ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) ∈
((TopOpen‘ℂfld) ↾t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) →
((int‘((TopOpen‘ℂfld) ↾t
(((-∞(,)𝑋) ∩
𝐷) ∪ {𝑋})))‘(((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) = (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) |
733 | 638, 731,
732 | sylancr 583 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) →
((int‘((TopOpen‘ℂfld) ↾t
(((-∞(,)𝑋) ∩
𝐷) ∪ {𝑋})))‘(((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) = (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) |
734 | 632, 733 | eqtr2d 2862 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) =
((int‘((TopOpen‘ℂfld) ↾t
(((-∞(,)𝑋) ∩
𝐷) ∪ {𝑋})))‘((((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ∪ {𝑋}))) |
735 | 629, 734 | eleqtrd 2908 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝑋 ∈
((int‘((TopOpen‘ℂfld) ↾t
(((-∞(,)𝑋) ∩
𝐷) ∪ {𝑋})))‘((((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ∪ {𝑋}))) |
736 | 610, 616,
617, 236, 618, 735 | limcres 24049 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋) = ((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) limℂ 𝑋)) |
737 | 736 | eqcomd 2831 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) limℂ 𝑋) = (((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) |
738 | 616 | resabs1d 5664 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) = (𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋))) |
739 | 738 | oveq1d 6920 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋) = ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) |
740 | 607, 737,
739 | 3eqtrrd 2866 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋) = ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋)) |
741 | 380, 599,
740 | 3eqtrrd 2866 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋) = ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) |
742 | 741 | rexlimdv3a 3242 |
. . 3
⊢ (𝜑 → (∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) → ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋) = ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)))) |
743 | 176, 742 | mpd 15 |
. 2
⊢ (𝜑 → ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋) = ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) |
744 | 123 | 3adant3 1168 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) |
745 | 221 | 3adant3 1168 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
746 | | fourierdlem49.l |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) |
747 | 746 | 3adant3 1168 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) |
748 | | eqid 2825 |
. . . . . . . 8
⊢ if((𝐸‘𝑋) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝐸‘𝑋))) = if((𝐸‘𝑋) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝐸‘𝑋))) |
749 | | eqid 2825 |
. . . . . . . 8
⊢
((TopOpen‘ℂfld) ↾t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) =
((TopOpen‘ℂfld) ↾t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) |
750 | 214, 212,
744, 745, 747, 214, 238, 241, 220, 748, 749 | fourierdlem33 41151 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → if((𝐸‘𝑋) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝐸‘𝑋))) ∈ (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) |
751 | 220 | resabs1d 5664 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) = (𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋)))) |
752 | 751 | oveq1d 6920 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) |
753 | 750, 752 | eleqtrd 2908 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → if((𝐸‘𝑋) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝐸‘𝑋))) ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) |
754 | | ne0i 4150 |
. . . . . 6
⊢
(if((𝐸‘𝑋) = (𝑄‘(𝑖 + 1)), 𝐿, ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝐸‘𝑋))) ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) ≠ ∅) |
755 | 753, 754 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) ≠ ∅) |
756 | 380, 755 | eqnetrd 3066 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) ≠ ∅) |
757 | 756 | rexlimdv3a 3242 |
. . 3
⊢ (𝜑 → (∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) → ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) ≠ ∅)) |
758 | 176, 757 | mpd 15 |
. 2
⊢ (𝜑 → ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) ≠ ∅) |
759 | 743, 758 | eqnetrd 3066 |
1
⊢ (𝜑 → ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋) ≠ ∅) |