Step | Hyp | Ref
| Expression |
1 | | citg2 25365 |
. 2
class
∫2 |
2 | | vf |
. . 3
setvar 𝑓 |
3 | | cc0 11112 |
. . . . 5
class
0 |
4 | | cpnf 11249 |
. . . . 5
class
+∞ |
5 | | cicc 13331 |
. . . . 5
class
[,] |
6 | 3, 4, 5 | co 7411 |
. . . 4
class
(0[,]+∞) |
7 | | cr 11111 |
. . . 4
class
ℝ |
8 | | cmap 8822 |
. . . 4
class
↑m |
9 | 6, 7, 8 | co 7411 |
. . 3
class
((0[,]+∞) ↑m ℝ) |
10 | | vg |
. . . . . . . . 9
setvar 𝑔 |
11 | 10 | cv 1538 |
. . . . . . . 8
class 𝑔 |
12 | 2 | cv 1538 |
. . . . . . . 8
class 𝑓 |
13 | | cle 11253 |
. . . . . . . . 9
class
≤ |
14 | 13 | cofr 7671 |
. . . . . . . 8
class
∘r ≤ |
15 | 11, 12, 14 | wbr 5147 |
. . . . . . 7
wff 𝑔 ∘r ≤ 𝑓 |
16 | | vx |
. . . . . . . . 9
setvar 𝑥 |
17 | 16 | cv 1538 |
. . . . . . . 8
class 𝑥 |
18 | | citg1 25364 |
. . . . . . . . 9
class
∫1 |
19 | 11, 18 | cfv 6542 |
. . . . . . . 8
class
(∫1‘𝑔) |
20 | 17, 19 | wceq 1539 |
. . . . . . 7
wff 𝑥 =
(∫1‘𝑔) |
21 | 15, 20 | wa 394 |
. . . . . 6
wff (𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔)) |
22 | 18 | cdm 5675 |
. . . . . 6
class dom
∫1 |
23 | 21, 10, 22 | wrex 3068 |
. . . . 5
wff
∃𝑔 ∈ dom
∫1(𝑔
∘r ≤ 𝑓
∧ 𝑥 =
(∫1‘𝑔)) |
24 | 23, 16 | cab 2707 |
. . . 4
class {𝑥 ∣ ∃𝑔 ∈ dom
∫1(𝑔
∘r ≤ 𝑓
∧ 𝑥 =
(∫1‘𝑔))} |
25 | | cxr 11251 |
. . . 4
class
ℝ* |
26 | | clt 11252 |
. . . 4
class
< |
27 | 24, 25, 26 | csup 9437 |
. . 3
class
sup({𝑥 ∣
∃𝑔 ∈ dom
∫1(𝑔
∘r ≤ 𝑓
∧ 𝑥 =
(∫1‘𝑔))}, ℝ*, <
) |
28 | 2, 9, 27 | cmpt 5230 |
. 2
class (𝑓 ∈ ((0[,]+∞)
↑m ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, <
)) |
29 | 1, 28 | wceq 1539 |
1
wff
∫2 = (𝑓 ∈ ((0[,]+∞) ↑m
ℝ) ↦ sup({𝑥
∣ ∃𝑔 ∈ dom
∫1(𝑔
∘r ≤ 𝑓
∧ 𝑥 =
(∫1‘𝑔))}, ℝ*, <
)) |