Step | Hyp | Ref
| Expression |
1 | | cnmf 30471 |
. 2
class
normfn |
2 | | vt |
. . 3
setvar 𝑡 |
3 | | cc 11110 |
. . . 4
class
ℂ |
4 | | chba 30439 |
. . . 4
class
ℋ |
5 | | cmap 8822 |
. . . 4
class
↑m |
6 | 3, 4, 5 | co 7411 |
. . 3
class (ℂ
↑m ℋ) |
7 | | vz |
. . . . . . . . . 10
setvar 𝑧 |
8 | 7 | cv 1538 |
. . . . . . . . 9
class 𝑧 |
9 | | cno 30443 |
. . . . . . . . 9
class
normℎ |
10 | 8, 9 | cfv 6542 |
. . . . . . . 8
class
(normℎ‘𝑧) |
11 | | c1 11113 |
. . . . . . . 8
class
1 |
12 | | cle 11253 |
. . . . . . . 8
class
≤ |
13 | 10, 11, 12 | wbr 5147 |
. . . . . . 7
wff
(normℎ‘𝑧) ≤ 1 |
14 | | vx |
. . . . . . . . 9
setvar 𝑥 |
15 | 14 | cv 1538 |
. . . . . . . 8
class 𝑥 |
16 | 2 | cv 1538 |
. . . . . . . . . 10
class 𝑡 |
17 | 8, 16 | cfv 6542 |
. . . . . . . . 9
class (𝑡‘𝑧) |
18 | | cabs 15185 |
. . . . . . . . 9
class
abs |
19 | 17, 18 | cfv 6542 |
. . . . . . . 8
class
(abs‘(𝑡‘𝑧)) |
20 | 15, 19 | wceq 1539 |
. . . . . . 7
wff 𝑥 = (abs‘(𝑡‘𝑧)) |
21 | 13, 20 | wa 394 |
. . . . . 6
wff
((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧))) |
22 | 21, 7, 4 | wrex 3068 |
. . . . 5
wff
∃𝑧 ∈
ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧))) |
23 | 22, 14 | cab 2707 |
. . . 4
class {𝑥 ∣ ∃𝑧 ∈ ℋ
((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))} |
24 | | cxr 11251 |
. . . 4
class
ℝ* |
25 | | clt 11252 |
. . . 4
class
< |
26 | 23, 24, 25 | csup 9437 |
. . 3
class
sup({𝑥 ∣
∃𝑧 ∈ ℋ
((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ*, <
) |
27 | 2, 6, 26 | cmpt 5230 |
. 2
class (𝑡 ∈ (ℂ
↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ
((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ*, <
)) |
28 | 1, 27 | wceq 1539 |
1
wff
normfn = (𝑡 ∈ (ℂ ↑m ℋ)
↦ sup({𝑥 ∣
∃𝑧 ∈ ℋ
((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ*, <
)) |