Detailed syntax breakdown of Definition df-hst
Step | Hyp | Ref
| Expression |
1 | | chst 29321 |
. 2
class
CHStates |
2 | | chba 29277 |
. . . . . . 7
class
ℋ |
3 | | vf |
. . . . . . . 8
setvar 𝑓 |
4 | 3 | cv 1541 |
. . . . . . 7
class 𝑓 |
5 | 2, 4 | cfv 6432 |
. . . . . 6
class (𝑓‘
ℋ) |
6 | | cno 29281 |
. . . . . 6
class
normℎ |
7 | 5, 6 | cfv 6432 |
. . . . 5
class
(normℎ‘(𝑓‘ ℋ)) |
8 | | c1 10873 |
. . . . 5
class
1 |
9 | 7, 8 | wceq 1542 |
. . . 4
wff
(normℎ‘(𝑓‘ ℋ)) = 1 |
10 | | vx |
. . . . . . . . 9
setvar 𝑥 |
11 | 10 | cv 1541 |
. . . . . . . 8
class 𝑥 |
12 | | vy |
. . . . . . . . . 10
setvar 𝑦 |
13 | 12 | cv 1541 |
. . . . . . . . 9
class 𝑦 |
14 | | cort 29288 |
. . . . . . . . 9
class
⊥ |
15 | 13, 14 | cfv 6432 |
. . . . . . . 8
class
(⊥‘𝑦) |
16 | 11, 15 | wss 3892 |
. . . . . . 7
wff 𝑥 ⊆ (⊥‘𝑦) |
17 | 11, 4 | cfv 6432 |
. . . . . . . . . 10
class (𝑓‘𝑥) |
18 | 13, 4 | cfv 6432 |
. . . . . . . . . 10
class (𝑓‘𝑦) |
19 | | csp 29280 |
. . . . . . . . . 10
class
·ih |
20 | 17, 18, 19 | co 7271 |
. . . . . . . . 9
class ((𝑓‘𝑥) ·ih (𝑓‘𝑦)) |
21 | | cc0 10872 |
. . . . . . . . 9
class
0 |
22 | 20, 21 | wceq 1542 |
. . . . . . . 8
wff ((𝑓‘𝑥) ·ih (𝑓‘𝑦)) = 0 |
23 | | chj 29291 |
. . . . . . . . . . 11
class
∨ℋ |
24 | 11, 13, 23 | co 7271 |
. . . . . . . . . 10
class (𝑥 ∨ℋ 𝑦) |
25 | 24, 4 | cfv 6432 |
. . . . . . . . 9
class (𝑓‘(𝑥 ∨ℋ 𝑦)) |
26 | | cva 29278 |
. . . . . . . . . 10
class
+ℎ |
27 | 17, 18, 26 | co 7271 |
. . . . . . . . 9
class ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)) |
28 | 25, 27 | wceq 1542 |
. . . . . . . 8
wff (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)) |
29 | 22, 28 | wa 396 |
. . . . . . 7
wff (((𝑓‘𝑥) ·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦))) |
30 | 16, 29 | wi 4 |
. . . . . 6
wff (𝑥 ⊆ (⊥‘𝑦) → (((𝑓‘𝑥) ·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)))) |
31 | | cch 29287 |
. . . . . 6
class
Cℋ |
32 | 30, 12, 31 | wral 3066 |
. . . . 5
wff
∀𝑦 ∈
Cℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑓‘𝑥) ·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)))) |
33 | 32, 10, 31 | wral 3066 |
. . . 4
wff
∀𝑥 ∈
Cℋ ∀𝑦 ∈ Cℋ
(𝑥 ⊆
(⊥‘𝑦) →
(((𝑓‘𝑥)
·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)))) |
34 | 9, 33 | wa 396 |
. . 3
wff
((normℎ‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥 ∈
Cℋ ∀𝑦 ∈ Cℋ
(𝑥 ⊆
(⊥‘𝑦) →
(((𝑓‘𝑥)
·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦))))) |
35 | | cmap 8598 |
. . . 4
class
↑m |
36 | 2, 31, 35 | co 7271 |
. . 3
class ( ℋ
↑m Cℋ ) |
37 | 34, 3, 36 | crab 3070 |
. 2
class {𝑓 ∈ ( ℋ
↑m Cℋ ) ∣
((normℎ‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥 ∈
Cℋ ∀𝑦 ∈ Cℋ
(𝑥 ⊆
(⊥‘𝑦) →
(((𝑓‘𝑥)
·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)))))} |
38 | 1, 37 | wceq 1542 |
1
wff CHStates =
{𝑓 ∈ ( ℋ
↑m Cℋ ) ∣
((normℎ‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥 ∈
Cℋ ∀𝑦 ∈ Cℋ
(𝑥 ⊆
(⊥‘𝑦) →
(((𝑓‘𝑥)
·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)))))} |