| 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 7464 |
. . . . . . . . . 10
⊢
((⌊‘((𝐵
− 𝑥) / 𝑇)) · 𝑇) ∈ V |
| 7 | | fourierdlem49.z |
. . . . . . . . . . 11
⊢ 𝑍 = (𝑥 ∈ ℝ ↦
((⌊‘((𝐵 −
𝑥) / 𝑇)) · 𝑇)) |
| 8 | 7 | fvmpt2 7027 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧
((⌊‘((𝐵 −
𝑥) / 𝑇)) · 𝑇) ∈ V) → (𝑍‘𝑥) = ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) |
| 9 | 6, 8 | mpan2 691 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ → (𝑍‘𝑥) = ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) |
| 10 | 9 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ → (𝑥 + (𝑍‘𝑥)) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) |
| 11 | 10 | mpteq2ia 5245 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍‘𝑥))) = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) |
| 12 | 5, 11 | eqtri 2765 |
. . . . . 6
⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) |
| 13 | 1, 2, 3, 4, 12 | fourierdlem4 46126 |
. . . . 5
⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵)) |
| 14 | | fourierdlem49.x |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ ℝ) |
| 15 | 13, 14 | ffvelcdmd 7105 |
. . . 4
⊢ (𝜑 → (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) |
| 16 | | simpr 484 |
. . . . . . 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 46124 |
. . . . . . . . . . . . . 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 232 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑄 ∈ (ℝ ↑m
(0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))) |
| 23 | 22 | simpld 494 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑄 ∈ (ℝ ↑m
(0...𝑀))) |
| 24 | | elmapi 8889 |
. . . . . . . . . . 11
⊢ (𝑄 ∈ (ℝ
↑m (0...𝑀))
→ 𝑄:(0...𝑀)⟶ℝ) |
| 25 | 23, 24 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ) |
| 26 | | ffn 6736 |
. . . . . . . . . 10
⊢ (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀)) |
| 27 | 25, 26 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑄 Fn (0...𝑀)) |
| 28 | 27 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → 𝑄 Fn (0...𝑀)) |
| 29 | | fvelrnb 6969 |
. . . . . . . 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 232 |
. . . . . 6
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑗 ∈ (0...𝑀)(𝑄‘𝑗) = (𝐸‘𝑋)) |
| 32 | | 1zzd 12648 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 1 ∈ ℤ) |
| 33 | | elfzelz 13564 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ) |
| 34 | 33 | ad2antlr 727 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ∈ ℤ) |
| 35 | | 1e0p1 12775 |
. . . . . . . . . . . . . . . . 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 12722 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ∈ ℝ) |
| 38 | | elfzle1 13567 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ (0...𝑀) → 0 ≤ 𝑗) |
| 39 | 38 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . 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 2743 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑄‘𝑗) = (𝐸‘𝑋) → (𝐸‘𝑋) = (𝑄‘𝑗)) |
| 42 | 41 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝐸‘𝑋) = (𝑄‘𝑗)) |
| 43 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 = 0 → (𝑄‘𝑗) = (𝑄‘0)) |
| 44 | 43 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝑄‘𝑗) = (𝑄‘0)) |
| 45 | 22 | simprld 772 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵)) |
| 46 | 45 | simpld 494 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑄‘0) = 𝐴) |
| 47 | 46 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝑄‘0) = 𝐴) |
| 48 | 42, 44, 47 | 3eqtrd 2781 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝐸‘𝑋) = 𝐴) |
| 49 | 48 | adantllr 719 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝐸‘𝑋) = 𝐴) |
| 50 | 49 | adantllr 719 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → (𝐸‘𝑋) = 𝐴) |
| 51 | 1 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → 𝐴 ∈ ℝ) |
| 52 | 1 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 53 | 52 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → 𝐴 ∈
ℝ*) |
| 54 | 2 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 55 | 54 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → 𝐵 ∈
ℝ*) |
| 56 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) |
| 57 | | iocgtlb 45515 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → 𝐴 < (𝐸‘𝑋)) |
| 58 | 53, 55, 56, 57 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → 𝐴 < (𝐸‘𝑋)) |
| 59 | 51, 58 | gtned 11396 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → (𝐸‘𝑋) ≠ 𝐴) |
| 60 | 59 | neneqd 2945 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → ¬ (𝐸‘𝑋) = 𝐴) |
| 61 | 60 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) ∧ 𝑗 = 0) → ¬ (𝐸‘𝑋) = 𝐴) |
| 62 | 50, 61 | pm2.65da 817 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → ¬ 𝑗 = 0) |
| 63 | 62 | neqned 2947 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ≠ 0) |
| 64 | 37, 39, 63 | ne0gt0d 11398 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 0 < 𝑗) |
| 65 | | 0zd 12625 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 0 ∈ ℤ) |
| 66 | | zltp1le 12667 |
. . . . . . . . . . . . . . . . . 18
⊢ ((0
∈ ℤ ∧ 𝑗
∈ ℤ) → (0 < 𝑗 ↔ (0 + 1) ≤ 𝑗)) |
| 67 | 65, 34, 66 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (0 < 𝑗 ↔ (0 + 1) ≤ 𝑗)) |
| 68 | 64, 67 | mpbid 232 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (0 + 1) ≤ 𝑗) |
| 69 | 36, 68 | eqbrtrd 5165 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 1 ≤ 𝑗) |
| 70 | | eluz2 12884 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈
(ℤ≥‘1) ↔ (1 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 1 ≤
𝑗)) |
| 71 | 32, 34, 69, 70 | syl3anbrc 1344 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ∈
(ℤ≥‘1)) |
| 72 | | nnuz 12921 |
. . . . . . . . . . . . . 14
⊢ ℕ =
(ℤ≥‘1) |
| 73 | 71, 72 | eleqtrrdi 2852 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑗 ∈ ℕ) |
| 74 | | nnm1nn0 12567 |
. . . . . . . . . . . . 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 12920 |
. . . . . . . . . . . . 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 2843 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈
(ℤ≥‘0)) |
| 79 | 18 | nnzd 12640 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 80 | 79 | ad3antrrr 730 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑀 ∈ ℤ) |
| 81 | | peano2zm 12660 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℤ → (𝑗 − 1) ∈
ℤ) |
| 82 | 33, 81 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ∈ ℤ) |
| 83 | 82 | zred 12722 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ∈ ℝ) |
| 84 | 33 | zred 12722 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ) |
| 85 | | elfzel2 13562 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℤ) |
| 86 | 85 | zred 12722 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ ℝ) |
| 87 | 84 | ltm1d 12200 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) < 𝑗) |
| 88 | | elfzle2 13568 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ≤ 𝑀) |
| 89 | 83, 84, 86, 87, 88 | ltletrd 11421 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) < 𝑀) |
| 90 | 89 | ad2antlr 727 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) < 𝑀) |
| 91 | | elfzo2 13702 |
. . . . . . . . . . 11
⊢ ((𝑗 − 1) ∈ (0..^𝑀) ↔ ((𝑗 − 1) ∈
(ℤ≥‘0) ∧ 𝑀 ∈ ℤ ∧ (𝑗 − 1) < 𝑀)) |
| 92 | 78, 80, 90, 91 | syl3anbrc 1344 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈ (0..^𝑀)) |
| 93 | 25 | ad3antrrr 730 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝑄:(0...𝑀)⟶ℝ) |
| 94 | 34, 81 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈ ℤ) |
| 95 | 75 | nn0ge0d 12590 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 0 ≤ (𝑗 − 1)) |
| 96 | 83, 86, 89 | ltled 11409 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 − 1) ≤ 𝑀) |
| 97 | 96 | ad2antlr 727 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ≤ 𝑀) |
| 98 | 65, 80, 94, 95, 97 | elfzd 13555 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑗 − 1) ∈ (0...𝑀)) |
| 99 | 93, 98 | ffvelcdmd 7105 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) ∈ ℝ) |
| 100 | 99 | rexrd 11311 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) ∈
ℝ*) |
| 101 | 25 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈ ℝ) |
| 102 | 101 | rexrd 11311 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈
ℝ*) |
| 103 | 102 | adantlr 715 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈
ℝ*) |
| 104 | 103 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘𝑗) ∈
ℝ*) |
| 105 | | iocssre 13467 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈ ℝ)
→ (𝐴(,]𝐵) ⊆
ℝ) |
| 106 | 52, 2, 105 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ) |
| 107 | 106 | sselda 3983 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → (𝐸‘𝑋) ∈ ℝ) |
| 108 | 107 | rexrd 11311 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → (𝐸‘𝑋) ∈
ℝ*) |
| 109 | 108 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈
ℝ*) |
| 110 | | simplll 775 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → 𝜑) |
| 111 | | ovex 7464 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 − 1) ∈
V |
| 112 | | eleq1 2829 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = (𝑗 − 1) → (𝑖 ∈ (0..^𝑀) ↔ (𝑗 − 1) ∈ (0..^𝑀))) |
| 113 | 112 | anbi2d 630 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = (𝑗 − 1) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ (𝑗 − 1) ∈ (0..^𝑀)))) |
| 114 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = (𝑗 − 1) → (𝑄‘𝑖) = (𝑄‘(𝑗 − 1))) |
| 115 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = (𝑗 − 1) → (𝑖 + 1) = ((𝑗 − 1) + 1)) |
| 116 | 115 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = (𝑗 − 1) → (𝑄‘(𝑖 + 1)) = (𝑄‘((𝑗 − 1) + 1))) |
| 117 | 114, 116 | breq12d 5156 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = (𝑗 − 1) → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘(𝑗 − 1)) < (𝑄‘((𝑗 − 1) + 1)))) |
| 118 | 113, 117 | imbi12d 344 |
. . . . . . . . . . . . . . . 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))))) |
| 119 | 22 | simprrd 774 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) |
| 120 | 119 | r19.21bi 3251 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) |
| 121 | 111, 118,
120 | vtocl 3558 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑗 − 1) ∈ (0..^𝑀)) → (𝑄‘(𝑗 − 1)) < (𝑄‘((𝑗 − 1) + 1))) |
| 122 | 110, 92, 121 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) < (𝑄‘((𝑗 − 1) + 1))) |
| 123 | 33 | zcnd 12723 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℂ) |
| 124 | | 1cnd 11256 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ (0...𝑀) → 1 ∈ ℂ) |
| 125 | 123, 124 | npcand 11624 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 − 1) + 1) = 𝑗) |
| 126 | 125 | eqcomd 2743 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 = ((𝑗 − 1) + 1)) |
| 127 | 126 | fveq2d 6910 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ (0...𝑀) → (𝑄‘𝑗) = (𝑄‘((𝑗 − 1) + 1))) |
| 128 | 127 | eqcomd 2743 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ (0...𝑀) → (𝑄‘((𝑗 − 1) + 1)) = (𝑄‘𝑗)) |
| 129 | 128 | ad2antlr 727 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘((𝑗 − 1) + 1)) = (𝑄‘𝑗)) |
| 130 | 122, 129 | breqtrd 5169 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) < (𝑄‘𝑗)) |
| 131 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘𝑗) = (𝐸‘𝑋)) |
| 132 | 130, 131 | breqtrd 5169 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝑄‘(𝑗 − 1)) < (𝐸‘𝑋)) |
| 133 | 106, 15 | sseldd 3984 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐸‘𝑋) ∈ ℝ) |
| 134 | 133 | leidd 11829 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐸‘𝑋) ≤ (𝐸‘𝑋)) |
| 135 | 134 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ≤ (𝐸‘𝑋)) |
| 136 | 41 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) = (𝑄‘𝑗)) |
| 137 | 135, 136 | breqtrd 5169 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ≤ (𝑄‘𝑗)) |
| 138 | 137 | adantllr 719 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ≤ (𝑄‘𝑗)) |
| 139 | 100, 104,
109, 132, 138 | eliocd 45520 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ((𝑄‘(𝑗 − 1))(,](𝑄‘𝑗))) |
| 140 | 127 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ (0...𝑀) → ((𝑄‘(𝑗 − 1))(,](𝑄‘𝑗)) = ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1)))) |
| 141 | 140 | ad2antlr 727 |
. . . . . . . . . . 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)))) |
| 142 | 139, 141 | eleqtrd 2843 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1)))) |
| 143 | 114, 116 | oveq12d 7449 |
. . . . . . . . . . . 12
⊢ (𝑖 = (𝑗 − 1) → ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) = ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1)))) |
| 144 | 143 | eleq2d 2827 |
. . . . . . . . . . 11
⊢ (𝑖 = (𝑗 − 1) → ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) ↔ (𝐸‘𝑋) ∈ ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1))))) |
| 145 | 144 | rspcev 3622 |
. . . . . . . . . 10
⊢ (((𝑗 − 1) ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘(𝑗 − 1))(,](𝑄‘((𝑗 − 1) + 1)))) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
| 146 | 92, 142, 145 | syl2anc 584 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑗) = (𝐸‘𝑋)) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
| 147 | 146 | ex 412 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑄‘𝑗) = (𝐸‘𝑋) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))))) |
| 148 | 147 | adantlr 715 |
. . . . . . 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))))) |
| 149 | 148 | rexlimdva 3155 |
. . . . . 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))))) |
| 150 | 31, 149 | mpd 15 |
. . . . 5
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
| 151 | 18 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → 𝑀 ∈ ℕ) |
| 152 | 25 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ) |
| 153 | | iocssicc 13477 |
. . . . . . . . . 10
⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵) |
| 154 | 46 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 = (𝑄‘0)) |
| 155 | 45 | simprd 495 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑄‘𝑀) = 𝐵) |
| 156 | 155 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 = (𝑄‘𝑀)) |
| 157 | 154, 156 | oveq12d 7449 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀))) |
| 158 | 153, 157 | sseqtrid 4026 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ((𝑄‘0)[,](𝑄‘𝑀))) |
| 159 | 158 | sselda 3983 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → (𝐸‘𝑋) ∈ ((𝑄‘0)[,](𝑄‘𝑀))) |
| 160 | 159 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → (𝐸‘𝑋) ∈ ((𝑄‘0)[,](𝑄‘𝑀))) |
| 161 | | simpr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → ¬ (𝐸‘𝑋) ∈ ran 𝑄) |
| 162 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑗 → (𝑄‘𝑘) = (𝑄‘𝑗)) |
| 163 | 162 | breq1d 5153 |
. . . . . . . . 9
⊢ (𝑘 = 𝑗 → ((𝑄‘𝑘) < (𝐸‘𝑋) ↔ (𝑄‘𝑗) < (𝐸‘𝑋))) |
| 164 | 163 | cbvrabv 3447 |
. . . . . . . 8
⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < (𝐸‘𝑋)} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < (𝐸‘𝑋)} |
| 165 | 164 | supeq1i 9487 |
. . . . . . 7
⊢
sup({𝑘 ∈
(0..^𝑀) ∣ (𝑄‘𝑘) < (𝐸‘𝑋)}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < (𝐸‘𝑋)}, ℝ, < ) |
| 166 | 151, 152,
160, 161, 165 | fourierdlem25 46147 |
. . . . . 6
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
| 167 | | ioossioc 45505 |
. . . . . . . . 9
⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) |
| 168 | 167 | sseli 3979 |
. . . . . . . 8
⊢ ((𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
| 169 | 168 | 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))))) |
| 170 | 169 | reximdva 3168 |
. . . . . 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))))) |
| 171 | 166, 170 | mpd 15 |
. . . . 5
⊢ (((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) ∧ ¬ (𝐸‘𝑋) ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
| 172 | 150, 171 | pm2.61dan 813 |
. . . 4
⊢ ((𝜑 ∧ (𝐸‘𝑋) ∈ (𝐴(,]𝐵)) → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
| 173 | 15, 172 | mpdan 687 |
. . 3
⊢ (𝜑 → ∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
| 174 | | fourierdlem49.f |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
| 175 | | frel 6741 |
. . . . . . . . . . 11
⊢ (𝐹:𝐷⟶ℝ → Rel 𝐹) |
| 176 | 174, 175 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → Rel 𝐹) |
| 177 | | resindm 6048 |
. . . . . . . . . . 11
⊢ (Rel
𝐹 → (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ dom 𝐹)) = (𝐹 ↾ (-∞(,)(𝐸‘𝑋)))) |
| 178 | 177 | eqcomd 2743 |
. . . . . . . . . 10
⊢ (Rel
𝐹 → (𝐹 ↾ (-∞(,)(𝐸‘𝑋))) = (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ dom 𝐹))) |
| 179 | 176, 178 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 ↾ (-∞(,)(𝐸‘𝑋))) = (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ dom 𝐹))) |
| 180 | | fdm 6745 |
. . . . . . . . . . . 12
⊢ (𝐹:𝐷⟶ℝ → dom 𝐹 = 𝐷) |
| 181 | 174, 180 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → dom 𝐹 = 𝐷) |
| 182 | 181 | ineq2d 4220 |
. . . . . . . . . 10
⊢ (𝜑 → ((-∞(,)(𝐸‘𝑋)) ∩ dom 𝐹) = ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) |
| 183 | 182 | reseq2d 5997 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ dom 𝐹)) = (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷))) |
| 184 | 179, 183 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 → (𝐹 ↾ (-∞(,)(𝐸‘𝑋))) = (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷))) |
| 185 | 184 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐹 ↾ (-∞(,)(𝐸‘𝑋))) = (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷))) |
| 186 | 185 | oveq1d 7446 |
. . . . . 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))) |
| 187 | | ax-resscn 11212 |
. . . . . . . . . . 11
⊢ ℝ
⊆ ℂ |
| 188 | 187 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℝ ⊆
ℂ) |
| 189 | 174, 188 | fssd 6753 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:𝐷⟶ℂ) |
| 190 | 189 | 3ad2ant1 1134 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝐹:𝐷⟶ℂ) |
| 191 | | inss2 4238 |
. . . . . . . . 9
⊢
((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ⊆ 𝐷 |
| 192 | 191 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ⊆ 𝐷) |
| 193 | 190, 192 | fssresd 6775 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐹 ↾ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)):((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)⟶ℂ) |
| 194 | | mnfxr 11318 |
. . . . . . . . . 10
⊢ -∞
∈ ℝ* |
| 195 | 194 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -∞ ∈
ℝ*) |
| 196 | 25 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ) |
| 197 | | elfzofz 13715 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀)) |
| 198 | 197 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀)) |
| 199 | 196, 198 | ffvelcdmd 7105 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ) |
| 200 | 199 | rexrd 11311 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈
ℝ*) |
| 201 | 199 | mnfltd 13166 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -∞ < (𝑄‘𝑖)) |
| 202 | 195, 200,
201 | xrltled 13192 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -∞ ≤ (𝑄‘𝑖)) |
| 203 | | iooss1 13422 |
. . . . . . . . . 10
⊢
((-∞ ∈ ℝ* ∧ -∞ ≤ (𝑄‘𝑖)) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ (-∞(,)(𝐸‘𝑋))) |
| 204 | 194, 202,
203 | sylancr 587 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ (-∞(,)(𝐸‘𝑋))) |
| 205 | 204 | 3adant3 1133 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ (-∞(,)(𝐸‘𝑋))) |
| 206 | | fzofzp1 13803 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀)) |
| 207 | 206 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀)) |
| 208 | 196, 207 | ffvelcdmd 7105 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
| 209 | 208 | 3adant3 1133 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
| 210 | 209 | rexrd 11311 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘(𝑖 + 1)) ∈
ℝ*) |
| 211 | 199 | 3adant3 1133 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈ ℝ) |
| 212 | 211 | rexrd 11311 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) ∈
ℝ*) |
| 213 | | simp3 1139 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) |
| 214 | | iocleub 45516 |
. . . . . . . . . . 11
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ* ∧
(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ≤ (𝑄‘(𝑖 + 1))) |
| 215 | 212, 210,
213, 214 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ≤ (𝑄‘(𝑖 + 1))) |
| 216 | | iooss2 13423 |
. . . . . . . . . 10
⊢ (((𝑄‘(𝑖 + 1)) ∈ ℝ* ∧
(𝐸‘𝑋) ≤ (𝑄‘(𝑖 + 1))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
| 217 | 210, 215,
216 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
| 218 | | fourierdlem49.cn |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) |
| 219 | | cncff 24919 |
. . . . . . . . . . . . 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) |
| 220 | | fdm 6745 |
. . . . . . . . . . . . 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)))) |
| 221 | 218, 219,
220 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
| 222 | | ssdmres 6031 |
. . . . . . . . . . . 12
⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) |
| 223 | 221, 222 | sylibr 234 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹) |
| 224 | 181 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom 𝐹 = 𝐷) |
| 225 | 223, 224 | sseqtrd 4020 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷) |
| 226 | 225 | 3adant3 1133 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷) |
| 227 | 217, 226 | sstrd 3994 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ 𝐷) |
| 228 | 205, 227 | ssind 4241 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) |
| 229 | | fourierdlem49.d |
. . . . . . . . . 10
⊢ (𝜑 → 𝐷 ⊆ ℝ) |
| 230 | 229, 188 | sstrd 3994 |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 ⊆ ℂ) |
| 231 | 230 | 3ad2ant1 1134 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝐷 ⊆ ℂ) |
| 232 | 191, 231 | sstrid 3995 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ⊆ ℂ) |
| 233 | | eqid 2737 |
. . . . . . 7
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
| 234 | | eqid 2737 |
. . . . . . 7
⊢
((TopOpen‘ℂfld) ↾t
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) =
((TopOpen‘ℂfld) ↾t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) |
| 235 | 133 | 3ad2ant1 1134 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ℝ) |
| 236 | 235 | rexrd 11311 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈
ℝ*) |
| 237 | | iocgtlb 45515 |
. . . . . . . . . 10
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝑄‘(𝑖 + 1)) ∈ ℝ* ∧
(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝐸‘𝑋)) |
| 238 | 212, 210,
213, 237 | syl3anc 1373 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝐸‘𝑋)) |
| 239 | 235 | leidd 11829 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ≤ (𝐸‘𝑋)) |
| 240 | 212, 236,
236, 238, 239 | eliocd 45520 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) |
| 241 | | ioounsn 13517 |
. . . . . . . . . . 11
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝐸‘𝑋) ∈ ℝ* ∧ (𝑄‘𝑖) < (𝐸‘𝑋)) → (((𝑄‘𝑖)(,)(𝐸‘𝑋)) ∪ {(𝐸‘𝑋)}) = ((𝑄‘𝑖)(,](𝐸‘𝑋))) |
| 242 | 212, 236,
238, 241 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖)(,)(𝐸‘𝑋)) ∪ {(𝐸‘𝑋)}) = ((𝑄‘𝑖)(,](𝐸‘𝑋))) |
| 243 | 242 | fveq2d 6910 |
. . . . . . . . 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)))) |
| 244 | 233 | cnfldtop 24804 |
. . . . . . . . . . 11
⊢
(TopOpen‘ℂfld) ∈ Top |
| 245 | | ovex 7464 |
. . . . . . . . . . . . 13
⊢
(-∞(,)(𝐸‘𝑋)) ∈ V |
| 246 | 245 | inex1 5317 |
. . . . . . . . . . . 12
⊢
((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∈ V |
| 247 | | snex 5436 |
. . . . . . . . . . . 12
⊢ {(𝐸‘𝑋)} ∈ V |
| 248 | 246, 247 | unex 7764 |
. . . . . . . . . . 11
⊢
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ∈ V |
| 249 | | resttop 23168 |
. . . . . . . . . . 11
⊢
(((TopOpen‘ℂfld) ∈ Top ∧
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ∈ V) →
((TopOpen‘ℂfld) ↾t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∈ Top) |
| 250 | 244, 248,
249 | mp2an 692 |
. . . . . . . . . 10
⊢
((TopOpen‘ℂfld) ↾t
(((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∈ Top |
| 251 | | retop 24782 |
. . . . . . . . . . . . 13
⊢
(topGen‘ran (,)) ∈ Top |
| 252 | 251 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (topGen‘ran (,)) ∈
Top) |
| 253 | 248 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ∈ V) |
| 254 | | iooretop 24786 |
. . . . . . . . . . . . 13
⊢ ((𝑄‘𝑖)(,)+∞) ∈ (topGen‘ran
(,)) |
| 255 | 254 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)+∞) ∈ (topGen‘ran
(,))) |
| 256 | | elrestr 17473 |
. . . . . . . . . . . 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)}))) |
| 257 | 252, 253,
255, 256 | syl3anc 1373 |
. . . . . . . . . . 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)}))) |
| 258 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 = (𝐸‘𝑋)) → 𝑥 = (𝐸‘𝑋)) |
| 259 | | pnfxr 11315 |
. . . . . . . . . . . . . . . . . . . 20
⊢ +∞
∈ ℝ* |
| 260 | 259 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → +∞ ∈
ℝ*) |
| 261 | 235 | ltpnfd 13163 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) < +∞) |
| 262 | 212, 260,
235, 238, 261 | eliood 45511 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,)+∞)) |
| 263 | | snidg 4660 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐸‘𝑋) ∈ ℝ → (𝐸‘𝑋) ∈ {(𝐸‘𝑋)}) |
| 264 | | elun2 4183 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐸‘𝑋) ∈ {(𝐸‘𝑋)} → (𝐸‘𝑋) ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) |
| 265 | 263, 264 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐸‘𝑋) ∈ ℝ → (𝐸‘𝑋) ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) |
| 266 | 133, 265 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐸‘𝑋) ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) |
| 267 | 266 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) |
| 268 | 262, 267 | elind 4200 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
| 269 | 268 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
| 270 | 258, 269 | eqeltrd 2841 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
| 271 | 270 | adantlr 715 |
. . . . . . . . . . . . . 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)}))) |
| 272 | 212 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝑄‘𝑖) ∈
ℝ*) |
| 273 | 259 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → +∞ ∈
ℝ*) |
| 274 | 200 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝑄‘𝑖) ∈
ℝ*) |
| 275 | 133 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐸‘𝑋) ∈ ℝ) |
| 276 | 275 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝐸‘𝑋) ∈ ℝ) |
| 277 | | iocssre 13467 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝐸‘𝑋) ∈ ℝ) → ((𝑄‘𝑖)(,](𝐸‘𝑋)) ⊆ ℝ) |
| 278 | 274, 276,
277 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → ((𝑄‘𝑖)(,](𝐸‘𝑋)) ⊆ ℝ) |
| 279 | | simpr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) |
| 280 | 278, 279 | sseldd 3984 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ∈ ℝ) |
| 281 | 280 | 3adantl3 1169 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ∈ ℝ) |
| 282 | 276 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝐸‘𝑋) ∈
ℝ*) |
| 283 | | iocgtlb 45515 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝐸‘𝑋) ∈ ℝ* ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝑄‘𝑖) < 𝑥) |
| 284 | 274, 282,
279, 283 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝑄‘𝑖) < 𝑥) |
| 285 | 284 | 3adantl3 1169 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → (𝑄‘𝑖) < 𝑥) |
| 286 | 281 | ltpnfd 13163 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 < +∞) |
| 287 | 272, 273,
281, 285, 286 | eliood 45511 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ∈ ((𝑄‘𝑖)(,)+∞)) |
| 288 | 287 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ ((𝑄‘𝑖)(,)+∞)) |
| 289 | 194 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → -∞ ∈
ℝ*) |
| 290 | 282 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈
ℝ*) |
| 291 | 280 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ ℝ) |
| 292 | 291 | mnfltd 13166 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → -∞ < 𝑥) |
| 293 | 133 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ℝ) |
| 294 | | iocleub 45516 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ (𝐸‘𝑋) ∈ ℝ* ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ≤ (𝐸‘𝑋)) |
| 295 | 274, 282,
279, 294 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) → 𝑥 ≤ (𝐸‘𝑋)) |
| 296 | 295 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ≤ (𝐸‘𝑋)) |
| 297 | | neqne 2948 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (¬
𝑥 = (𝐸‘𝑋) → 𝑥 ≠ (𝐸‘𝑋)) |
| 298 | 297 | necomd 2996 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (¬
𝑥 = (𝐸‘𝑋) → (𝐸‘𝑋) ≠ 𝑥) |
| 299 | 298 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ≠ 𝑥) |
| 300 | 291, 293,
296, 299 | leneltd 11415 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 < (𝐸‘𝑋)) |
| 301 | 289, 290,
291, 292, 300 | eliood 45511 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) |
| 302 | 301 | 3adantll3 45047 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) |
| 303 | 226 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 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) |
| 304 | 272 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝑄‘𝑖) ∈
ℝ*) |
| 305 | 210 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝑄‘(𝑖 + 1)) ∈
ℝ*) |
| 306 | 281 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ ℝ) |
| 307 | 285 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝑄‘𝑖) < 𝑥) |
| 308 | 235 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝐸‘𝑋) ∈ ℝ) |
| 309 | 209 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → (𝑄‘(𝑖 + 1)) ∈ ℝ) |
| 310 | 300 | 3adantll3 45047 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 < (𝐸‘𝑋)) |
| 311 | 215 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . 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))) |
| 312 | 306, 308,
309, 310, 311 | ltletrd 11421 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 < (𝑄‘(𝑖 + 1))) |
| 313 | 304, 305,
306, 307, 312 | eliood 45511 |
. . . . . . . . . . . . . . . . . 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)))) |
| 314 | 303, 313 | sseldd 3984 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ 𝐷) |
| 315 | 302, 314 | elind 4200 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,](𝐸‘𝑋))) ∧ ¬ 𝑥 = (𝐸‘𝑋)) → 𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) |
| 316 | | elun1 4182 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) → 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) |
| 317 | 315, 316 | 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)})) |
| 318 | 288, 317 | elind 4200 |
. . . . . . . . . . . . . 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)}))) |
| 319 | 271, 318 | pm2.61dan 813 |
. . . . . . . . . . . . 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)}))) |
| 320 | 212 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → (𝑄‘𝑖) ∈
ℝ*) |
| 321 | 236 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → (𝐸‘𝑋) ∈
ℝ*) |
| 322 | | elinel1 4201 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) → 𝑥 ∈ ((𝑄‘𝑖)(,)+∞)) |
| 323 | | elioore 13417 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)+∞) → 𝑥 ∈ ℝ) |
| 324 | 323 | rexrd 11311 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)+∞) → 𝑥 ∈ ℝ*) |
| 325 | 322, 324 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) → 𝑥 ∈ ℝ*) |
| 326 | 325 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → 𝑥 ∈ ℝ*) |
| 327 | 200 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → (𝑄‘𝑖) ∈
ℝ*) |
| 328 | 259 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → +∞ ∈
ℝ*) |
| 329 | 322 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → 𝑥 ∈ ((𝑄‘𝑖)(,)+∞)) |
| 330 | | ioogtlb 45508 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑄‘𝑖) ∈ ℝ* ∧ +∞
∈ ℝ* ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)+∞)) → (𝑄‘𝑖) < 𝑥) |
| 331 | 327, 328,
329, 330 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → (𝑄‘𝑖) < 𝑥) |
| 332 | 331 | 3adantl3 1169 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → (𝑄‘𝑖) < 𝑥) |
| 333 | | elinel2 4202 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) → 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) |
| 334 | | elsni 4643 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ {(𝐸‘𝑋)} → 𝑥 = (𝐸‘𝑋)) |
| 335 | 334 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 = (𝐸‘𝑋)) |
| 336 | 134 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ {(𝐸‘𝑋)}) → (𝐸‘𝑋) ≤ (𝐸‘𝑋)) |
| 337 | 335, 336 | eqbrtrd 5165 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 ≤ (𝐸‘𝑋)) |
| 338 | 337 | adantlr 715 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∧ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 ≤ (𝐸‘𝑋)) |
| 339 | | simpll 767 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∧ ¬ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝜑) |
| 340 | | elunnel2 4155 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ∧ ¬ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) |
| 341 | 340 | adantll 714 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∧ ¬ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) |
| 342 | | elinel1 4201 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) → 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) |
| 343 | | elioore 13417 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ (-∞(,)(𝐸‘𝑋)) → 𝑥 ∈ ℝ) |
| 344 | 343 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → 𝑥 ∈ ℝ) |
| 345 | 133 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → (𝐸‘𝑋) ∈ ℝ) |
| 346 | 194 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → -∞ ∈
ℝ*) |
| 347 | 345 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → (𝐸‘𝑋) ∈
ℝ*) |
| 348 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) |
| 349 | | iooltub 45523 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((-∞ ∈ ℝ* ∧ (𝐸‘𝑋) ∈ ℝ* ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → 𝑥 < (𝐸‘𝑋)) |
| 350 | 346, 347,
348, 349 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → 𝑥 < (𝐸‘𝑋)) |
| 351 | 344, 345,
350 | ltled 11409 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)(𝐸‘𝑋))) → 𝑥 ≤ (𝐸‘𝑋)) |
| 352 | 342, 351 | sylan2 593 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷)) → 𝑥 ≤ (𝐸‘𝑋)) |
| 353 | 339, 341,
352 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) ∧ ¬ 𝑥 ∈ {(𝐸‘𝑋)}) → 𝑥 ≤ (𝐸‘𝑋)) |
| 354 | 338, 353 | pm2.61dan 813 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) → 𝑥 ≤ (𝐸‘𝑋)) |
| 355 | 354 | adantlr 715 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) → 𝑥 ≤ (𝐸‘𝑋)) |
| 356 | 333, 355 | sylan2 593 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → 𝑥 ≤ (𝐸‘𝑋)) |
| 357 | 356 | 3adantl3 1169 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) → 𝑥 ≤ (𝐸‘𝑋)) |
| 358 | 320, 321,
326, 332, 357 | eliocd 45520 |
. . . . . . . . . . . . 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))) |
| 359 | 319, 358 | impbida 801 |
. . . . . . . . . . . 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)})))) |
| 360 | 359 | eqrdv 2735 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,](𝐸‘𝑋)) = (((𝑄‘𝑖)(,)+∞) ∩ (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
| 361 | | ioossre 13448 |
. . . . . . . . . . . . . 14
⊢
(-∞(,)(𝐸‘𝑋)) ⊆ ℝ |
| 362 | | ssinss1 4246 |
. . . . . . . . . . . . . 14
⊢
((-∞(,)(𝐸‘𝑋)) ⊆ ℝ →
((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ⊆ ℝ) |
| 363 | 361, 362 | mp1i 13 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ⊆ ℝ) |
| 364 | 235 | snssd 4809 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → {(𝐸‘𝑋)} ⊆ ℝ) |
| 365 | 363, 364 | unssd 4192 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ⊆ ℝ) |
| 366 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢
(topGen‘ran (,)) = (topGen‘ran (,)) |
| 367 | 233, 366 | rerest 24825 |
. . . . . . . . . . . 12
⊢
((((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}) ⊆ ℝ →
((TopOpen‘ℂfld) ↾t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)})) = ((topGen‘ran (,))
↾t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
| 368 | 365, 367 | 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)}))) |
| 369 | 257, 360,
368 | 3eltr4d 2856 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,](𝐸‘𝑋)) ∈
((TopOpen‘ℂfld) ↾t (((-∞(,)(𝐸‘𝑋)) ∩ 𝐷) ∪ {(𝐸‘𝑋)}))) |
| 370 | | isopn3i 23090 |
. . . . . . . . . 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))) |
| 371 | 250, 369,
370 | sylancr 587 |
. . . . . . . . 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))) |
| 372 | 243, 371 | eqtr2d 2778 |
. . . . . . . 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)}))) |
| 373 | 240, 372 | eleqtrd 2843 |
. . . . . . 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)}))) |
| 374 | 193, 228,
232, 233, 234, 373 | limcres 25921 |
. . . . . 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))) |
| 375 | 228 | resabs1d 6026 |
. . . . . . 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)))) |
| 376 | 375 | oveq1d 7446 |
. . . . . 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))) |
| 377 | 186, 374,
376 | 3eqtr2d 2783 |
. . . . 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))) |
| 378 | 181 | feq2d 6722 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ↔ 𝐹:𝐷⟶ℂ)) |
| 379 | 189, 378 | mpbird 257 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ) |
| 380 | 379 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → 𝐹:dom 𝐹⟶ℂ) |
| 381 | 380 | 3ad2antl1 1186 |
. . . . . . . . 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) |
| 382 | | ioosscn 13449 |
. . . . . . . . . 10
⊢ ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ ℂ |
| 383 | 382 | 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) |
| 384 | 181 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐷 = dom 𝐹) |
| 385 | 384 | 3ad2ant1 1134 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝐷 = dom 𝐹) |
| 386 | 227, 385 | sseqtrd 4020 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ dom 𝐹) |
| 387 | 386 | adantr 480 |
. . . . . . . . 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) |
| 388 | 7 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑍 = (𝑥 ∈ ℝ ↦
((⌊‘((𝐵 −
𝑥) / 𝑇)) · 𝑇))) |
| 389 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑋 → (𝐵 − 𝑥) = (𝐵 − 𝑋)) |
| 390 | 389 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑋 → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − 𝑋) / 𝑇)) |
| 391 | 390 | fveq2d 6910 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑋 → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − 𝑋) / 𝑇))) |
| 392 | 391 | oveq1d 7446 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑋 → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
| 393 | 392 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
| 394 | 2, 14 | resubcld 11691 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ) |
| 395 | 2, 1 | resubcld 11691 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
| 396 | 4, 395 | eqeltrid 2845 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑇 ∈ ℝ) |
| 397 | 1, 2 | posdifd 11850 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
| 398 | 3, 397 | mpbid 232 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 0 < (𝐵 − 𝐴)) |
| 399 | 4 | eqcomi 2746 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐵 − 𝐴) = 𝑇 |
| 400 | 399 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐵 − 𝐴) = 𝑇) |
| 401 | 398, 400 | breqtrd 5169 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 0 < 𝑇) |
| 402 | 401 | gt0ne0d 11827 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑇 ≠ 0) |
| 403 | 394, 396,
402 | redivcld 12095 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝐵 − 𝑋) / 𝑇) ∈ ℝ) |
| 404 | 403 | flcld 13838 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
| 405 | 404 | zred 12722 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℝ) |
| 406 | 405, 396 | remulcld 11291 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) ∈ ℝ) |
| 407 | 388, 393,
14, 406 | fvmptd 7023 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑍‘𝑋) = ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
| 408 | 407, 406 | eqeltrd 2841 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑍‘𝑋) ∈ ℝ) |
| 409 | 408 | recnd 11289 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑍‘𝑋) ∈ ℂ) |
| 410 | 409 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → (𝑍‘𝑋) ∈ ℂ) |
| 411 | 410 | 3ad2antl1 1186 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → (𝑍‘𝑋) ∈ ℂ) |
| 412 | 411 | negcld 11607 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) → -(𝑍‘𝑋) ∈ ℂ) |
| 413 | | eqid 2737 |
. . . . . . . . 9
⊢ {𝑧 ∈ ℂ ∣
∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))} = {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))} |
| 414 | | ioosscn 13449 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ ℂ |
| 415 | 414 | sseli 3979 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) → 𝑦 ∈ ℂ) |
| 416 | 415 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ ℂ) |
| 417 | 409 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑍‘𝑋) ∈ ℂ) |
| 418 | 416, 417 | pncand 11621 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = 𝑦) |
| 419 | 418 | eqcomd 2743 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 = ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋))) |
| 420 | 419 | 3ad2antl1 1186 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 = ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋))) |
| 421 | 407 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = ((𝑦 + (𝑍‘𝑋)) − ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
| 422 | 421 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = ((𝑦 + (𝑍‘𝑋)) − ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
| 423 | 416, 417 | addcld 11280 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ℂ) |
| 424 | 406 | recnd 11289 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) ∈ ℂ) |
| 425 | 424 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) ∈ ℂ) |
| 426 | 423, 425 | negsubd 11626 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) + -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) − ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
| 427 | 404 | zcnd 12723 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℂ) |
| 428 | 396 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑇 ∈ ℂ) |
| 429 | 427, 428 | mulneg1d 11716 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) = -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
| 430 | 429 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇) = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
| 431 | 430 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑦 + (𝑍‘𝑋)) + -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
| 432 | 431 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) + -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
| 433 | 422, 426,
432 | 3eqtr2d 2783 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
| 434 | 433 | 3ad2antl1 1186 |
. . . . . . . . . . . . . . 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))) |
| 435 | 404 | znegcld 12724 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
| 436 | 435 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
| 437 | 436 | 3ad2antl1 1186 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
| 438 | | simpl1 1192 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝜑) |
| 439 | 227 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) ⊆ 𝐷) |
| 440 | 200 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘𝑖) ∈
ℝ*) |
| 441 | 133 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐸‘𝑋) ∈
ℝ*) |
| 442 | 441 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐸‘𝑋) ∈
ℝ*) |
| 443 | | elioore 13417 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) → 𝑦 ∈ ℝ) |
| 444 | 443 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ ℝ) |
| 445 | 408 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑍‘𝑋) ∈ ℝ) |
| 446 | 444, 445 | readdcld 11290 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ℝ) |
| 447 | 446 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ℝ) |
| 448 | 408 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑍‘𝑋) ∈ ℝ) |
| 449 | 199, 448 | resubcld 11691 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ) |
| 450 | 449 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈
ℝ*) |
| 451 | 450 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈
ℝ*) |
| 452 | 14 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝑋 ∈
ℝ*) |
| 453 | 452 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑋 ∈
ℝ*) |
| 454 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) |
| 455 | | ioogtlb 45508 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ* ∧ 𝑋 ∈ ℝ*
∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑦) |
| 456 | 451, 453,
454, 455 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑦) |
| 457 | 199 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘𝑖) ∈ ℝ) |
| 458 | 448 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑍‘𝑋) ∈ ℝ) |
| 459 | 443 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ ℝ) |
| 460 | 457, 458,
459 | ltsubaddd 11859 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑦 ↔ (𝑄‘𝑖) < (𝑦 + (𝑍‘𝑋)))) |
| 461 | 456, 460 | mpbid 232 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑄‘𝑖) < (𝑦 + (𝑍‘𝑋))) |
| 462 | 14 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑋 ∈ ℝ) |
| 463 | | iooltub 45523 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ* ∧ 𝑋 ∈ ℝ*
∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 < 𝑋) |
| 464 | 451, 453,
454, 463 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 < 𝑋) |
| 465 | 459, 462,
458, 464 | ltadd1dd 11874 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) < (𝑋 + (𝑍‘𝑋))) |
| 466 | 5 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍‘𝑥)))) |
| 467 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) |
| 468 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 = 𝑋 → (𝑍‘𝑥) = (𝑍‘𝑋)) |
| 469 | 467, 468 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = 𝑋 → (𝑥 + (𝑍‘𝑥)) = (𝑋 + (𝑍‘𝑋))) |
| 470 | 469 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥 + (𝑍‘𝑥)) = (𝑋 + (𝑍‘𝑋))) |
| 471 | 14, 408 | readdcld 11290 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑋 + (𝑍‘𝑋)) ∈ ℝ) |
| 472 | 466, 470,
14, 471 | fvmptd 7023 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝐸‘𝑋) = (𝑋 + (𝑍‘𝑋))) |
| 473 | 472 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑋 + (𝑍‘𝑋)) = (𝐸‘𝑋)) |
| 474 | 473 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑋 + (𝑍‘𝑋)) = (𝐸‘𝑋)) |
| 475 | 465, 474 | breqtrd 5169 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) < (𝐸‘𝑋)) |
| 476 | 440, 442,
447, 461, 475 | eliood 45511 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) |
| 477 | 476 | 3adantl3 1169 |
. . . . . . . . . . . . . . . . . 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))) |
| 478 | 439, 477 | sseldd 3984 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝑦 + (𝑍‘𝑋)) ∈ 𝐷) |
| 479 | 438, 478,
437 | 3jca 1129 |
. . . . . . . . . . . . . . . 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)) |
| 480 | | eleq1 2829 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑘 ∈ ℤ ↔
-(⌊‘((𝐵 −
𝑋) / 𝑇)) ∈ ℤ)) |
| 481 | 480 | 3anbi3d 1444 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ))) |
| 482 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑘 · 𝑇) = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
| 483 | 482 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
| 484 | 483 | eleq1d 2826 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷 ↔ ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ 𝐷)) |
| 485 | 481, 484 | imbi12d 344 |
. . . . . . . . . . . . . . . . 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))) |
| 486 | | ovex 7464 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 + (𝑍‘𝑋)) ∈ V |
| 487 | | eleq1 2829 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → (𝑥 ∈ 𝐷 ↔ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷)) |
| 488 | 487 | 3anbi2d 1443 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ))) |
| 489 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → (𝑥 + (𝑘 · 𝑇)) = ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇))) |
| 490 | 489 | eleq1d 2826 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = (𝑦 + (𝑍‘𝑋)) → ((𝑥 + (𝑘 · 𝑇)) ∈ 𝐷 ↔ ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷)) |
| 491 | 488, 490 | imbi12d 344 |
. . . . . . . . . . . . . . . . . 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))) |
| 492 | | fourierdlem49.dper |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷) |
| 493 | 486, 491,
492 | vtocl 3558 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (𝑘 · 𝑇)) ∈ 𝐷) |
| 494 | 485, 493 | vtoclg 3554 |
. . . . . . . . . . . . . . . 16
⊢
(-(⌊‘((𝐵
− 𝑋) / 𝑇)) ∈ ℤ → ((𝜑 ∧ (𝑦 + (𝑍‘𝑋)) ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → ((𝑦 + (𝑍‘𝑋)) + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) ∈ 𝐷)) |
| 495 | 437, 479,
494 | 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) |
| 496 | 434, 495 | eqeltrd 2841 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → ((𝑦 + (𝑍‘𝑋)) − (𝑍‘𝑋)) ∈ 𝐷) |
| 497 | 420, 496 | eqeltrd 2841 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑦 ∈ 𝐷) |
| 498 | 497 | ralrimiva 3146 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ∀𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑦 ∈ 𝐷) |
| 499 | | dfss3 3972 |
. . . . . . . . . . . 12
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ 𝐷 ↔ ∀𝑦 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑦 ∈ 𝐷) |
| 500 | 498, 499 | sylibr 234 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ 𝐷) |
| 501 | 199 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ) |
| 502 | 409 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑍‘𝑋) ∈ ℂ) |
| 503 | 501, 502 | negsubd 11626 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + -(𝑍‘𝑋)) = ((𝑄‘𝑖) − (𝑍‘𝑋))) |
| 504 | 503 | eqcomd 2743 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) = ((𝑄‘𝑖) + -(𝑍‘𝑋))) |
| 505 | 472 | oveq1d 7446 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝐸‘𝑋) + -(𝑍‘𝑋)) = ((𝑋 + (𝑍‘𝑋)) + -(𝑍‘𝑋))) |
| 506 | 471 | recnd 11289 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑋 + (𝑍‘𝑋)) ∈ ℂ) |
| 507 | 506, 409 | negsubd 11626 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑋 + (𝑍‘𝑋)) + -(𝑍‘𝑋)) = ((𝑋 + (𝑍‘𝑋)) − (𝑍‘𝑋))) |
| 508 | 14 | recnd 11289 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑋 ∈ ℂ) |
| 509 | 508, 409 | pncand 11621 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑋 + (𝑍‘𝑋)) − (𝑍‘𝑋)) = 𝑋) |
| 510 | 505, 507,
509 | 3eqtrrd 2782 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑋 = ((𝐸‘𝑋) + -(𝑍‘𝑋))) |
| 511 | 510 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 = ((𝐸‘𝑋) + -(𝑍‘𝑋))) |
| 512 | 504, 511 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) = (((𝑄‘𝑖) + -(𝑍‘𝑋))(,)((𝐸‘𝑋) + -(𝑍‘𝑋)))) |
| 513 | 448 | renegcld 11690 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -(𝑍‘𝑋) ∈ ℝ) |
| 514 | 199, 275,
513 | iooshift 45535 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + -(𝑍‘𝑋))(,)((𝐸‘𝑋) + -(𝑍‘𝑋))) = {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))}) |
| 515 | 512, 514 | eqtr2d 2778 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))} = (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) |
| 516 | 515 | 3adant3 1133 |
. . . . . . . . . . 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)) |
| 517 | 181 | 3ad2ant1 1134 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → dom 𝐹 = 𝐷) |
| 518 | 500, 516,
517 | 3sstr4d 4039 |
. . . . . . . . . 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) |
| 519 | 518 | adantr 480 |
. . . . . . . . 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) |
| 520 | 407 | negeqd 11502 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → -(𝑍‘𝑋) = -((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
| 521 | 520, 430 | eqtrd 2777 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → -(𝑍‘𝑋) = (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
| 522 | 521 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑥 + -(𝑍‘𝑋)) = (𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
| 523 | 522 | fveq2d 6910 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐹‘(𝑥 + -(𝑍‘𝑋))) = (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))) |
| 524 | 523 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → (𝐹‘(𝑥 + -(𝑍‘𝑋))) = (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))) |
| 525 | 524 | 3ad2antl1 1186 |
. . . . . . . . . . 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)))) |
| 526 | 435 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
| 527 | 526 | 3ad2antl1 1186 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
| 528 | | simpl1 1192 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → 𝜑) |
| 529 | 227 | sselda 3983 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → 𝑥 ∈ 𝐷) |
| 530 | 528, 529,
527 | 3jca 1129 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → (𝜑 ∧ 𝑥 ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)) |
| 531 | 480 | 3anbi3d 1444 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ 𝑥 ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ))) |
| 532 | 482 | oveq2d 7447 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑥 + (𝑘 · 𝑇)) = (𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
| 533 | 532 | fveq2d 6910 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))) |
| 534 | 533 | eqeq1d 2739 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = -(⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥) ↔ (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))) |
| 535 | 531, 534 | imbi12d 344 |
. . . . . . . . . . . . 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)))) |
| 536 | | fourierdlem49.per |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥)) |
| 537 | 535, 536 | vtoclg 3554 |
. . . . . . . . . . . 12
⊢
(-(⌊‘((𝐵
− 𝑋) / 𝑇)) ∈ ℤ → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ -(⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + (-(⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))) |
| 538 | 527, 530,
537 | 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)) |
| 539 | 525, 538 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) → (𝐹‘(𝑥 + -(𝑍‘𝑋))) = (𝐹‘𝑥)) |
| 540 | 539 | adantlr 715 |
. . . . . . . . 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)) |
| 541 | | simpr 484 |
. . . . . . . . 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))) |
| 542 | 381, 383,
387, 412, 413, 519, 540, 541 | limcperiod 45643 |
. . . . . . . 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)))) |
| 543 | 515 | reseq2d 5997 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ ((𝑄‘𝑖)(,)(𝐸‘𝑋))𝑧 = (𝑥 + -(𝑍‘𝑋))}) = (𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋))) |
| 544 | 511 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐸‘𝑋) + -(𝑍‘𝑋)) = 𝑋) |
| 545 | 543, 544 | oveq12d 7449 |
. . . . . . . . . 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)) |
| 546 | 545 | 3adant3 1133 |
. . . . . . . . 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)) |
| 547 | 546 | adantr 480 |
. . . . . . . 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)) |
| 548 | 542, 547 | eleqtrd 2843 |
. . . . . . 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)) |
| 549 | 379 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) → 𝐹:dom 𝐹⟶ℂ) |
| 550 | 549 | 3ad2antl1 1186 |
. . . . . . . . 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) |
| 551 | 414 | 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) |
| 552 | 500, 517 | sseqtrrd 4021 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ dom 𝐹) |
| 553 | 552 | adantr 480 |
. . . . . . . . 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) |
| 554 | 409 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) → (𝑍‘𝑋) ∈ ℂ) |
| 555 | 554 | 3ad2antl1 1186 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋)) → (𝑍‘𝑋) ∈ ℂ) |
| 556 | | eqid 2737 |
. . . . . . . . 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))} |
| 557 | 501, 502 | npcand 11624 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) − (𝑍‘𝑋)) + (𝑍‘𝑋)) = (𝑄‘𝑖)) |
| 558 | 557 | eqcomd 2743 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = (((𝑄‘𝑖) − (𝑍‘𝑋)) + (𝑍‘𝑋))) |
| 559 | 472 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐸‘𝑋) = (𝑋 + (𝑍‘𝑋))) |
| 560 | 558, 559 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝐸‘𝑋)) = ((((𝑄‘𝑖) − (𝑍‘𝑋)) + (𝑍‘𝑋))(,)(𝑋 + (𝑍‘𝑋)))) |
| 561 | 14 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ) |
| 562 | 449, 561,
448 | iooshift 45535 |
. . . . . . . . . . . . 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))}) |
| 563 | 560, 562 | eqtr2d 2778 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))} = ((𝑄‘𝑖)(,)(𝐸‘𝑋))) |
| 564 | 563 | 3adant3 1133 |
. . . . . . . . . . 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))) |
| 565 | 227, 564,
517 | 3sstr4d 4039 |
. . . . . . . . . 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) |
| 566 | 565 | adantr 480 |
. . . . . . . . 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) |
| 567 | 407 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑥 + (𝑍‘𝑋)) = (𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
| 568 | 567 | fveq2d 6910 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐹‘(𝑥 + (𝑍‘𝑋))) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))) |
| 569 | 568 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐹‘(𝑥 + (𝑍‘𝑋))) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))) |
| 570 | 569 | 3ad2antl1 1186 |
. . . . . . . . . . 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)))) |
| 571 | 404 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
| 572 | 571 | 3ad2antl1 1186 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) |
| 573 | | simpl1 1192 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝜑) |
| 574 | 500 | sselda 3983 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑥 ∈ 𝐷) |
| 575 | 573, 574,
572 | 3jca 1129 |
. . . . . . . . . . . 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)) |
| 576 | | eleq1 2829 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑘 ∈ ℤ ↔ (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ)) |
| 577 | 576 | 3anbi3d 1444 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ))) |
| 578 | | oveq1 7438 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑘 · 𝑇) = ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)) |
| 579 | 578 | oveq2d 7447 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝑥 + (𝑘 · 𝑇)) = (𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) |
| 580 | 579 | fveq2d 6910 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇)))) |
| 581 | 580 | eqeq1d 2739 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = (⌊‘((𝐵 − 𝑋) / 𝑇)) → ((𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥) ↔ (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))) |
| 582 | 577, 581 | imbi12d 344 |
. . . . . . . . . . . . 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)))) |
| 583 | 582, 536 | vtoclg 3554 |
. . . . . . . . . . . 12
⊢
((⌊‘((𝐵
− 𝑋) / 𝑇)) ∈ ℤ → ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ (⌊‘((𝐵 − 𝑋) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑋) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))) |
| 584 | 572, 575,
583 | 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)) |
| 585 | 570, 584 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → (𝐹‘(𝑥 + (𝑍‘𝑋))) = (𝐹‘𝑥)) |
| 586 | 585 | adantlr 715 |
. . . . . . . . 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)) |
| 587 | | simpr 484 |
. . . . . . . . 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)) |
| 588 | 550, 551,
553, 555, 556, 566, 586, 587 | limcperiod 45643 |
. . . . . . . 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)))) |
| 589 | 563 | reseq2d 5997 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ {𝑧 ∈ ℂ ∣ ∃𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)𝑧 = (𝑥 + (𝑍‘𝑋))}) = (𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋)))) |
| 590 | 473 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑋 + (𝑍‘𝑋)) = (𝐸‘𝑋)) |
| 591 | 589, 590 | oveq12d 7449 |
. . . . . . . . . 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))) |
| 592 | 591 | 3adant3 1133 |
. . . . . . . . 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))) |
| 593 | 592 | adantr 480 |
. . . . . . . 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))) |
| 594 | 588, 593 | eleqtrd 2843 |
. . . . . . 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))) |
| 595 | 548, 594 | impbida 801 |
. . . . . 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))) |
| 596 | 595 | eqrdv 2735 |
. . . . 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)) |
| 597 | | resindm 6048 |
. . . . . . . . . . 11
⊢ (Rel
𝐹 → (𝐹 ↾ ((-∞(,)𝑋) ∩ dom 𝐹)) = (𝐹 ↾ (-∞(,)𝑋))) |
| 598 | 597 | eqcomd 2743 |
. . . . . . . . . 10
⊢ (Rel
𝐹 → (𝐹 ↾ (-∞(,)𝑋)) = (𝐹 ↾ ((-∞(,)𝑋) ∩ dom 𝐹))) |
| 599 | 176, 598 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 ↾ (-∞(,)𝑋)) = (𝐹 ↾ ((-∞(,)𝑋) ∩ dom 𝐹))) |
| 600 | 181 | ineq2d 4220 |
. . . . . . . . . 10
⊢ (𝜑 → ((-∞(,)𝑋) ∩ dom 𝐹) = ((-∞(,)𝑋) ∩ 𝐷)) |
| 601 | 600 | reseq2d 5997 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 ↾ ((-∞(,)𝑋) ∩ dom 𝐹)) = (𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷))) |
| 602 | 599, 601 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 → (𝐹 ↾ (-∞(,)𝑋)) = (𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷))) |
| 603 | 602 | oveq1d 7446 |
. . . . . . 7
⊢ (𝜑 → ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋) = ((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) limℂ 𝑋)) |
| 604 | 603 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋) = ((𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)) limℂ 𝑋)) |
| 605 | | inss2 4238 |
. . . . . . . . . 10
⊢
((-∞(,)𝑋)
∩ 𝐷) ⊆ 𝐷 |
| 606 | 605 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((-∞(,)𝑋) ∩ 𝐷) ⊆ 𝐷) |
| 607 | 190, 606 | fssresd 6775 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐹 ↾ ((-∞(,)𝑋) ∩ 𝐷)):((-∞(,)𝑋) ∩ 𝐷)⟶ℂ) |
| 608 | 449 | mnfltd 13166 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -∞ < ((𝑄‘𝑖) − (𝑍‘𝑋))) |
| 609 | 195, 450,
608 | xrltled 13192 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → -∞ ≤ ((𝑄‘𝑖) − (𝑍‘𝑋))) |
| 610 | | iooss1 13422 |
. . . . . . . . . . 11
⊢
((-∞ ∈ ℝ* ∧ -∞ ≤ ((𝑄‘𝑖) − (𝑍‘𝑋))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ (-∞(,)𝑋)) |
| 611 | 194, 609,
610 | sylancr 587 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ (-∞(,)𝑋)) |
| 612 | 611 | 3adant3 1133 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ (-∞(,)𝑋)) |
| 613 | 612, 500 | ssind 4241 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ⊆ ((-∞(,)𝑋) ∩ 𝐷)) |
| 614 | 605, 231 | sstrid 3995 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((-∞(,)𝑋) ∩ 𝐷) ⊆ ℂ) |
| 615 | | eqid 2737 |
. . . . . . . 8
⊢
((TopOpen‘ℂfld) ↾t
(((-∞(,)𝑋) ∩
𝐷) ∪ {𝑋})) = ((TopOpen‘ℂfld)
↾t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
| 616 | 450 | 3adant3 1133 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈
ℝ*) |
| 617 | 452 | 3ad2ant1 1134 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝑋 ∈
ℝ*) |
| 618 | 472 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝐸‘𝑋) = (𝑋 + (𝑍‘𝑋))) |
| 619 | 238, 618 | breqtrd 5169 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝑋 + (𝑍‘𝑋))) |
| 620 | 408 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑍‘𝑋) ∈ ℝ) |
| 621 | 14 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝑋 ∈ ℝ) |
| 622 | 211, 620,
621 | ltsubaddd 11859 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑋 ↔ (𝑄‘𝑖) < (𝑋 + (𝑍‘𝑋)))) |
| 623 | 619, 622 | mpbird 257 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑋) |
| 624 | 14 | leidd 11829 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 ≤ 𝑋) |
| 625 | 624 | 3ad2ant1 1134 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝑋 ≤ 𝑋) |
| 626 | 616, 617,
617, 623, 625 | eliocd 45520 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝑋 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) |
| 627 | | ioounsn 13517 |
. . . . . . . . . . . 12
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ* ∧ 𝑋 ∈ ℝ*
∧ ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑋) → ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ∪ {𝑋}) = (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) |
| 628 | 616, 617,
623, 627 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋) ∪ {𝑋}) = (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) |
| 629 | 628 | fveq2d 6910 |
. . . . . . . . . 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))) |
| 630 | | ovex 7464 |
. . . . . . . . . . . . . 14
⊢
(-∞(,)𝑋)
∈ V |
| 631 | 630 | inex1 5317 |
. . . . . . . . . . . . 13
⊢
((-∞(,)𝑋)
∩ 𝐷) ∈
V |
| 632 | | snex 5436 |
. . . . . . . . . . . . 13
⊢ {𝑋} ∈ V |
| 633 | 631, 632 | unex 7764 |
. . . . . . . . . . . 12
⊢
(((-∞(,)𝑋)
∩ 𝐷) ∪ {𝑋}) ∈ V |
| 634 | | resttop 23168 |
. . . . . . . . . . . 12
⊢
(((TopOpen‘ℂfld) ∈ Top ∧
(((-∞(,)𝑋) ∩
𝐷) ∪ {𝑋}) ∈ V) →
((TopOpen‘ℂfld) ↾t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∈ Top) |
| 635 | 244, 633,
634 | mp2an 692 |
. . . . . . . . . . 11
⊢
((TopOpen‘ℂfld) ↾t
(((-∞(,)𝑋) ∩
𝐷) ∪ {𝑋})) ∈ Top |
| 636 | 633 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ∈ V) |
| 637 | | iooretop 24786 |
. . . . . . . . . . . . . 14
⊢ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∈ (topGen‘ran
(,)) |
| 638 | 637 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∈ (topGen‘ran
(,))) |
| 639 | | elrestr 17473 |
. . . . . . . . . . . . 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}))) |
| 640 | 252, 636,
638, 639 | syl3anc 1373 |
. . . . . . . . . . . 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}))) |
| 641 | 450 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈
ℝ*) |
| 642 | 259 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → +∞ ∈
ℝ*) |
| 643 | 14 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑋 ∈ ℝ) |
| 644 | | iocssre 13467 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ* ∧ 𝑋 ∈ ℝ) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) ⊆ ℝ) |
| 645 | 641, 643,
644 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) ⊆ ℝ) |
| 646 | | simpr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) |
| 647 | 645, 646 | sseldd 3984 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ ℝ) |
| 648 | 452 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑋 ∈
ℝ*) |
| 649 | | iocgtlb 45515 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ* ∧ 𝑋 ∈ ℝ*
∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑥) |
| 650 | 641, 648,
646, 649 | syl3anc 1373 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑥) |
| 651 | 647 | ltpnfd 13163 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 < +∞) |
| 652 | 641, 642,
647, 650, 651 | eliood 45511 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞)) |
| 653 | 652 | 3adantl3 1169 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞)) |
| 654 | | eqvisset 3500 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 𝑋 → 𝑋 ∈ V) |
| 655 | | snidg 4660 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑋 ∈ V → 𝑋 ∈ {𝑋}) |
| 656 | 654, 655 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑋 → 𝑋 ∈ {𝑋}) |
| 657 | 467, 656 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑋 → 𝑥 ∈ {𝑋}) |
| 658 | | elun2 4183 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ {𝑋} → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
| 659 | 657, 658 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑋 → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
| 660 | 659 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ 𝑥 = 𝑋) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
| 661 | | simpll 767 |
. . . . . . . . . . . . . . . . 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))))) |
| 662 | 641 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈
ℝ*) |
| 663 | 452 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑋 ∈
ℝ*) |
| 664 | 647 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 ∈ ℝ) |
| 665 | 650 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑥) |
| 666 | 14 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑋 ∈ ℝ) |
| 667 | | iocleub 45516 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ* ∧ 𝑋 ∈ ℝ*
∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ≤ 𝑋) |
| 668 | 641, 648,
646, 667 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ≤ 𝑋) |
| 669 | 668 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 ≤ 𝑋) |
| 670 | 467 | eqcoms 2745 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑋 = 𝑥 → 𝑥 = 𝑋) |
| 671 | 670 | necon3bi 2967 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (¬
𝑥 = 𝑋 → 𝑋 ≠ 𝑥) |
| 672 | 671 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑋 ≠ 𝑥) |
| 673 | 664, 666,
669, 672 | leneltd 11415 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 < 𝑋) |
| 674 | 662, 663,
664, 665, 673 | eliood 45511 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) |
| 675 | 674 | 3adantll3 45047 |
. . . . . . . . . . . . . . . . 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)) |
| 676 | 613 | sselda 3983 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑥 ∈ ((-∞(,)𝑋) ∩ 𝐷)) |
| 677 | | elun1 4182 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ((-∞(,)𝑋) ∩ 𝐷) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
| 678 | 676, 677 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
| 679 | 661, 675,
678 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) ∧ ¬ 𝑥 = 𝑋) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
| 680 | 660, 679 | pm2.61dan 813 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋)) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
| 681 | 653, 680 | elind 4200 |
. . . . . . . . . . . . . 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}))) |
| 682 | 616 | adantr 480 |
. . . . . . . . . . . . . . 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*) |
| 683 | 617 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑋 ∈
ℝ*) |
| 684 | | elinel1 4201 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞)) |
| 685 | | elioore 13417 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) → 𝑥 ∈ ℝ) |
| 686 | 684, 685 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) → 𝑥 ∈ ℝ) |
| 687 | 686 | rexrd 11311 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) → 𝑥 ∈ ℝ*) |
| 688 | 687 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑥 ∈ ℝ*) |
| 689 | 450 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) ∈
ℝ*) |
| 690 | 259 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → +∞ ∈
ℝ*) |
| 691 | 684 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞)) |
| 692 | | ioogtlb 45508 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑄‘𝑖) − (𝑍‘𝑋)) ∈ ℝ* ∧ +∞
∈ ℝ* ∧ 𝑥 ∈ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞)) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑥) |
| 693 | 689, 690,
691, 692 | syl3anc 1373 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → ((𝑄‘𝑖) − (𝑍‘𝑋)) < 𝑥) |
| 694 | 693 | 3adantl3 1169 |
. . . . . . . . . . . . . . 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) |
| 695 | | elinel2 4202 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) → 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) |
| 696 | | elsni 4643 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ {𝑋} → 𝑥 = 𝑋) |
| 697 | 696 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑋}) → 𝑥 = 𝑋) |
| 698 | 624 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑋}) → 𝑋 ≤ 𝑋) |
| 699 | 697, 698 | eqbrtrd 5165 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑋}) → 𝑥 ≤ 𝑋) |
| 700 | 699 | adantlr 715 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∧ 𝑥 ∈ {𝑋}) → 𝑥 ≤ 𝑋) |
| 701 | | simpll 767 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∧ ¬ 𝑥 ∈ {𝑋}) → 𝜑) |
| 702 | | elunnel2 4155 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ∧ ¬ 𝑥 ∈ {𝑋}) → 𝑥 ∈ ((-∞(,)𝑋) ∩ 𝐷)) |
| 703 | 702 | adantll 714 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∧ ¬ 𝑥 ∈ {𝑋}) → 𝑥 ∈ ((-∞(,)𝑋) ∩ 𝐷)) |
| 704 | | elinel1 4201 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ((-∞(,)𝑋) ∩ 𝐷) → 𝑥 ∈ (-∞(,)𝑋)) |
| 705 | 703, 704 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∧ ¬ 𝑥 ∈ {𝑋}) → 𝑥 ∈ (-∞(,)𝑋)) |
| 706 | | elioore 13417 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (-∞(,)𝑋) → 𝑥 ∈ ℝ) |
| 707 | 706 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑥 ∈ ℝ) |
| 708 | 14 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑋 ∈ ℝ) |
| 709 | 194 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → -∞ ∈
ℝ*) |
| 710 | 452 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑋 ∈
ℝ*) |
| 711 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑥 ∈ (-∞(,)𝑋)) |
| 712 | | iooltub 45523 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((-∞ ∈ ℝ* ∧ 𝑋 ∈ ℝ* ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑥 < 𝑋) |
| 713 | 709, 710,
711, 712 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑥 < 𝑋) |
| 714 | 707, 708,
713 | ltled 11409 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (-∞(,)𝑋)) → 𝑥 ≤ 𝑋) |
| 715 | 701, 705,
714 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) ∧ ¬ 𝑥 ∈ {𝑋}) → 𝑥 ≤ 𝑋) |
| 716 | 700, 715 | pm2.61dan 813 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) → 𝑥 ≤ 𝑋) |
| 717 | 695, 716 | sylan2 593 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑥 ≤ 𝑋) |
| 718 | 717 | 3ad2antl1 1186 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) ∧ 𝑥 ∈ ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) → 𝑥 ≤ 𝑋) |
| 719 | 682, 683,
688, 694, 718 | eliocd 45520 |
. . . . . . . . . . . . . 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)) |
| 720 | 681, 719 | impbida 801 |
. . . . . . . . . . . . 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})))) |
| 721 | 720 | eqrdv 2735 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) = ((((𝑄‘𝑖) − (𝑍‘𝑋))(,)+∞) ∩ (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) |
| 722 | 605, 229 | sstrid 3995 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((-∞(,)𝑋) ∩ 𝐷) ⊆ ℝ) |
| 723 | 14 | snssd 4809 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → {𝑋} ⊆ ℝ) |
| 724 | 722, 723 | unssd 4192 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ⊆ ℝ) |
| 725 | 724 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}) ⊆ ℝ) |
| 726 | 233, 366 | rerest 24825 |
. . . . . . . . . . . . 13
⊢
((((-∞(,)𝑋)
∩ 𝐷) ∪ {𝑋}) ⊆ ℝ →
((TopOpen‘ℂfld) ↾t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋})) = ((topGen‘ran (,))
↾t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) |
| 727 | 725, 726 | 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}))) |
| 728 | 640, 721,
727 | 3eltr4d 2856 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (((𝑄‘𝑖) − (𝑍‘𝑋))(,]𝑋) ∈
((TopOpen‘ℂfld) ↾t (((-∞(,)𝑋) ∩ 𝐷) ∪ {𝑋}))) |
| 729 | | isopn3i 23090 |
. . . . . . . . . . 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)) |
| 730 | 635, 728,
729 | sylancr 587 |
. . . . . . . . . 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)) |
| 731 | 629, 730 | eqtr2d 2778 |
. . . . . . . . 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}))) |
| 732 | 626, 731 | eleqtrd 2843 |
. . . . . . . 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}))) |
| 733 | 607, 613,
614, 233, 615, 732 | limcres 25921 |
. . . . . . 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)) |
| 734 | 733 | eqcomd 2743 |
. . . . . 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)) |
| 735 | 613 | resabs1d 6026 |
. . . . . . 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))) |
| 736 | 735 | oveq1d 7446 |
. . . . . 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)) |
| 737 | 604, 734,
736 | 3eqtrrd 2782 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (((𝑄‘𝑖) − (𝑍‘𝑋))(,)𝑋)) limℂ 𝑋) = ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋)) |
| 738 | 377, 596,
737 | 3eqtrrd 2782 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋) = ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) |
| 739 | 738 | rexlimdv3a 3159 |
. . 3
⊢ (𝜑 → (∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) → ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋) = ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)))) |
| 740 | 173, 739 | mpd 15 |
. 2
⊢ (𝜑 → ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋) = ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋))) |
| 741 | 120 | 3adant3 1133 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) |
| 742 | 218 | 3adant3 1133 |
. . . . . . . 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)) |
| 743 | | fourierdlem49.l |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) |
| 744 | 743 | 3adant3 1133 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1)))) |
| 745 | | eqid 2737 |
. . . . . . . 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))) |
| 746 | | eqid 2737 |
. . . . . . . 8
⊢
((TopOpen‘ℂfld) ↾t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) =
((TopOpen‘ℂfld) ↾t (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∪ {(𝑄‘(𝑖 + 1))})) |
| 747 | 211, 209,
741, 742, 744, 211, 235, 238, 217, 745, 746 | fourierdlem33 46155 |
. . . . . . 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))) |
| 748 | 217 | resabs1d 6026 |
. . . . . . . 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)))) |
| 749 | 748 | oveq1d 7446 |
. . . . . . 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))) |
| 750 | 747, 749 | eleqtrd 2843 |
. . . . . 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))) |
| 751 | | ne0i 4341 |
. . . . . 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)) ≠ ∅) |
| 752 | 750, 751 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) ≠ ∅) |
| 753 | 377, 752 | eqnetrd 3008 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1)))) → ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) ≠ ∅) |
| 754 | 753 | rexlimdv3a 3159 |
. . 3
⊢ (𝜑 → (∃𝑖 ∈ (0..^𝑀)(𝐸‘𝑋) ∈ ((𝑄‘𝑖)(,](𝑄‘(𝑖 + 1))) → ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) ≠ ∅)) |
| 755 | 173, 754 | mpd 15 |
. 2
⊢ (𝜑 → ((𝐹 ↾ (-∞(,)(𝐸‘𝑋))) limℂ (𝐸‘𝑋)) ≠ ∅) |
| 756 | 740, 755 | eqnetrd 3008 |
1
⊢ (𝜑 → ((𝐹 ↾ (-∞(,)𝑋)) limℂ 𝑋) ≠ ∅) |