Detailed syntax breakdown of Definition df-cvlat
Step | Hyp | Ref
| Expression |
1 | | clc 37258 |
. 2
class
CvLat |
2 | | va |
. . . . . . . . . . 11
setvar 𝑎 |
3 | 2 | cv 1540 |
. . . . . . . . . 10
class 𝑎 |
4 | | vc |
. . . . . . . . . . 11
setvar 𝑐 |
5 | 4 | cv 1540 |
. . . . . . . . . 10
class 𝑐 |
6 | | vk |
. . . . . . . . . . . 12
setvar 𝑘 |
7 | 6 | cv 1540 |
. . . . . . . . . . 11
class 𝑘 |
8 | | cple 16950 |
. . . . . . . . . . 11
class
le |
9 | 7, 8 | cfv 6430 |
. . . . . . . . . 10
class
(le‘𝑘) |
10 | 3, 5, 9 | wbr 5078 |
. . . . . . . . 9
wff 𝑎(le‘𝑘)𝑐 |
11 | 10 | wn 3 |
. . . . . . . 8
wff ¬
𝑎(le‘𝑘)𝑐 |
12 | | vb |
. . . . . . . . . . 11
setvar 𝑏 |
13 | 12 | cv 1540 |
. . . . . . . . . 10
class 𝑏 |
14 | | cjn 18010 |
. . . . . . . . . . 11
class
join |
15 | 7, 14 | cfv 6430 |
. . . . . . . . . 10
class
(join‘𝑘) |
16 | 5, 13, 15 | co 7268 |
. . . . . . . . 9
class (𝑐(join‘𝑘)𝑏) |
17 | 3, 16, 9 | wbr 5078 |
. . . . . . . 8
wff 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏) |
18 | 11, 17 | wa 395 |
. . . . . . 7
wff (¬
𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) |
19 | 5, 3, 15 | co 7268 |
. . . . . . . 8
class (𝑐(join‘𝑘)𝑎) |
20 | 13, 19, 9 | wbr 5078 |
. . . . . . 7
wff 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎) |
21 | 18, 20 | wi 4 |
. . . . . 6
wff ((¬
𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎)) |
22 | | cbs 16893 |
. . . . . . 7
class
Base |
23 | 7, 22 | cfv 6430 |
. . . . . 6
class
(Base‘𝑘) |
24 | 21, 4, 23 | wral 3065 |
. . . . 5
wff
∀𝑐 ∈
(Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎)) |
25 | | catm 37256 |
. . . . . 6
class
Atoms |
26 | 7, 25 | cfv 6430 |
. . . . 5
class
(Atoms‘𝑘) |
27 | 24, 12, 26 | wral 3065 |
. . . 4
wff
∀𝑏 ∈
(Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎)) |
28 | 27, 2, 26 | wral 3065 |
. . 3
wff
∀𝑎 ∈
(Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎)) |
29 | | cal 37257 |
. . 3
class
AtLat |
30 | 28, 6, 29 | crab 3069 |
. 2
class {𝑘 ∈ AtLat ∣
∀𝑎 ∈
(Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))} |
31 | 1, 30 | wceq 1541 |
1
wff CvLat =
{𝑘 ∈ AtLat ∣
∀𝑎 ∈
(Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))} |