Proof of Theorem icorempo
Step | Hyp | Ref
| Expression |
1 | | icorempo.1 |
. 2
⊢ 𝐹 = ([,) ↾ (ℝ ×
ℝ)) |
2 | | df-ico 12827 |
. . . 4
⊢ [,) =
(𝑥 ∈
ℝ*, 𝑦
∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) |
3 | 2 | reseq1i 5821 |
. . 3
⊢ ([,)
↾ (ℝ × ℝ)) = ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ*
↦ {𝑧 ∈
ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ↾ (ℝ ×
ℝ)) |
4 | | ressxr 10763 |
. . . 4
⊢ ℝ
⊆ ℝ* |
5 | | resmpo 7286 |
. . . 4
⊢ ((ℝ
⊆ ℝ* ∧ ℝ ⊆ ℝ*) →
((𝑥 ∈
ℝ*, 𝑦
∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ↾ (ℝ × ℝ)) =
(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ*
∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})) |
6 | 4, 4, 5 | mp2an 692 |
. . 3
⊢ ((𝑥 ∈ ℝ*,
𝑦 ∈
ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ↾ (ℝ × ℝ)) =
(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ*
∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) |
7 | 3, 6 | eqtri 2761 |
. 2
⊢ ([,)
↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) |
8 | | nfv 1921 |
. . . 4
⊢
Ⅎ𝑧(𝑥 ∈ ℝ ∧ 𝑦 ∈
ℝ) |
9 | | nfrab1 3287 |
. . . 4
⊢
Ⅎ𝑧{𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} |
10 | | nfrab1 3287 |
. . . 4
⊢
Ⅎ𝑧{𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} |
11 | | rabid 3281 |
. . . . . . . 8
⊢ (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ↔ (𝑧 ∈ ℝ* ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦))) |
12 | | rexr 10765 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℝ*) |
13 | | nltmnf 12607 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ℝ*
→ ¬ 𝑥 <
-∞) |
14 | 12, 13 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℝ → ¬
𝑥 <
-∞) |
15 | | renemnf 10768 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ℝ → 𝑥 ≠ -∞) |
16 | 15 | neneqd 2939 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℝ → ¬
𝑥 =
-∞) |
17 | 14, 16 | jca 515 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ℝ → (¬
𝑥 < -∞ ∧ ¬
𝑥 =
-∞)) |
18 | | pm4.56 988 |
. . . . . . . . . . . . . . 15
⊢ ((¬
𝑥 < -∞ ∧ ¬
𝑥 = -∞) ↔ ¬
(𝑥 < -∞ ∨ 𝑥 = -∞)) |
19 | 17, 18 | sylib 221 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ → ¬
(𝑥 < -∞ ∨ 𝑥 = -∞)) |
20 | | mnfxr 10776 |
. . . . . . . . . . . . . . 15
⊢ -∞
∈ ℝ* |
21 | | xrleloe 12620 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ*
∧ -∞ ∈ ℝ*) → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞))) |
22 | 12, 20, 21 | sylancl 589 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞))) |
23 | 19, 22 | mtbird 328 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ → ¬
𝑥 ≤
-∞) |
24 | | breq2 5034 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = -∞ → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ -∞)) |
25 | 24 | notbid 321 |
. . . . . . . . . . . . 13
⊢ (𝑧 = -∞ → (¬ 𝑥 ≤ 𝑧 ↔ ¬ 𝑥 ≤ -∞)) |
26 | 23, 25 | syl5ibrcom 250 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ → (𝑧 = -∞ → ¬ 𝑥 ≤ 𝑧)) |
27 | 26 | con2d 136 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ → (𝑥 ≤ 𝑧 → ¬ 𝑧 = -∞)) |
28 | | rexr 10765 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℝ → 𝑦 ∈
ℝ*) |
29 | | pnfnlt 12606 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ ℝ*
→ ¬ +∞ < 𝑦) |
30 | | breq1 5033 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = +∞ → (𝑧 < 𝑦 ↔ +∞ < 𝑦)) |
31 | 30 | notbid 321 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = +∞ → (¬ 𝑧 < 𝑦 ↔ ¬ +∞ < 𝑦)) |
32 | 29, 31 | syl5ibrcom 250 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℝ*
→ (𝑧 = +∞ →
¬ 𝑧 < 𝑦)) |
33 | 32 | con2d 136 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℝ*
→ (𝑧 < 𝑦 → ¬ 𝑧 = +∞)) |
34 | 28, 33 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℝ → (𝑧 < 𝑦 → ¬ 𝑧 = +∞)) |
35 | 27, 34 | im2anan9 623 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦) → (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞))) |
36 | 35 | anim2d 615 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ*
∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)) → (𝑧 ∈ ℝ* ∧ (¬
𝑧 = -∞ ∧ ¬
𝑧 =
+∞)))) |
37 | | renepnf 10767 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ℝ → 𝑧 ≠ +∞) |
38 | 37 | neneqd 2939 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ℝ → ¬
𝑧 =
+∞) |
39 | 38 | pm4.71i 563 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ℝ ↔ (𝑧 ∈ ℝ ∧ ¬
𝑧 =
+∞)) |
40 | | xrnemnf 12595 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℝ*
∧ 𝑧 ≠ -∞)
↔ (𝑧 ∈ ℝ
∨ 𝑧 =
+∞)) |
41 | 40 | anbi1i 627 |
. . . . . . . . . . 11
⊢ (((𝑧 ∈ ℝ*
∧ 𝑧 ≠ -∞)
∧ ¬ 𝑧 = +∞)
↔ ((𝑧 ∈ ℝ
∨ 𝑧 = +∞) ∧
¬ 𝑧 =
+∞)) |
42 | | df-ne 2935 |
. . . . . . . . . . . . 13
⊢ (𝑧 ≠ -∞ ↔ ¬
𝑧 =
-∞) |
43 | 42 | anbi2i 626 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℝ*
∧ 𝑧 ≠ -∞)
↔ (𝑧 ∈
ℝ* ∧ ¬ 𝑧 = -∞)) |
44 | 43 | anbi1i 627 |
. . . . . . . . . . 11
⊢ (((𝑧 ∈ ℝ*
∧ 𝑧 ≠ -∞)
∧ ¬ 𝑧 = +∞)
↔ ((𝑧 ∈
ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞)) |
45 | | pm5.61 1000 |
. . . . . . . . . . 11
⊢ (((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ ∧ ¬
𝑧 =
+∞)) |
46 | 41, 44, 45 | 3bitr3i 304 |
. . . . . . . . . 10
⊢ (((𝑧 ∈ ℝ*
∧ ¬ 𝑧 = -∞)
∧ ¬ 𝑧 = +∞)
↔ (𝑧 ∈ ℝ
∧ ¬ 𝑧 =
+∞)) |
47 | | anass 472 |
. . . . . . . . . 10
⊢ (((𝑧 ∈ ℝ*
∧ ¬ 𝑧 = -∞)
∧ ¬ 𝑧 = +∞)
↔ (𝑧 ∈
ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞))) |
48 | 39, 46, 47 | 3bitr2ri 303 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℝ*
∧ (¬ 𝑧 = -∞
∧ ¬ 𝑧 = +∞))
↔ 𝑧 ∈
ℝ) |
49 | 36, 48 | syl6ib 254 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ*
∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)) → 𝑧 ∈ ℝ)) |
50 | 11, 49 | syl5bi 245 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → 𝑧 ∈ ℝ)) |
51 | 11 | simprbi 500 |
. . . . . . . 8
⊢ (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)) |
52 | 51 | a1i 11 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦))) |
53 | 50, 52 | jcad 516 |
. . . . . 6
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → (𝑧 ∈ ℝ ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)))) |
54 | | rabid 3281 |
. . . . . 6
⊢ (𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ↔ (𝑧 ∈ ℝ ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦))) |
55 | 53, 54 | syl6ibr 255 |
. . . . 5
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → 𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})) |
56 | | rabss2 3967 |
. . . . . . 7
⊢ (ℝ
⊆ ℝ* → {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) |
57 | 4, 56 | ax-mp 5 |
. . . . . 6
⊢ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} |
58 | 57 | sseli 3873 |
. . . . 5
⊢ (𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → 𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) |
59 | 55, 58 | impbid1 228 |
. . . 4
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ↔ 𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})) |
60 | 8, 9, 10, 59 | eqrd 3896 |
. . 3
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → {𝑧 ∈ ℝ*
∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) |
61 | 60 | mpoeq3ia 7246 |
. 2
⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ*
∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) |
62 | 1, 7, 61 | 3eqtri 2765 |
1
⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) |