Step |
Hyp |
Ref |
Expression |
0 |
|
ctrl |
|- trL |
1 |
|
vk |
|- k |
2 |
|
cvv |
|- _V |
3 |
|
vw |
|- w |
4 |
|
clh |
|- LHyp |
5 |
1
|
cv |
|- k |
6 |
5 4
|
cfv |
|- ( LHyp ` k ) |
7 |
|
vf |
|- f |
8 |
|
cltrn |
|- LTrn |
9 |
5 8
|
cfv |
|- ( LTrn ` k ) |
10 |
3
|
cv |
|- w |
11 |
10 9
|
cfv |
|- ( ( LTrn ` k ) ` w ) |
12 |
|
vx |
|- x |
13 |
|
cbs |
|- Base |
14 |
5 13
|
cfv |
|- ( Base ` k ) |
15 |
|
vp |
|- p |
16 |
|
catm |
|- Atoms |
17 |
5 16
|
cfv |
|- ( Atoms ` k ) |
18 |
15
|
cv |
|- p |
19 |
|
cple |
|- le |
20 |
5 19
|
cfv |
|- ( le ` k ) |
21 |
18 10 20
|
wbr |
|- p ( le ` k ) w |
22 |
21
|
wn |
|- -. p ( le ` k ) w |
23 |
12
|
cv |
|- x |
24 |
|
cjn |
|- join |
25 |
5 24
|
cfv |
|- ( join ` k ) |
26 |
7
|
cv |
|- f |
27 |
18 26
|
cfv |
|- ( f ` p ) |
28 |
18 27 25
|
co |
|- ( p ( join ` k ) ( f ` p ) ) |
29 |
|
cmee |
|- meet |
30 |
5 29
|
cfv |
|- ( meet ` k ) |
31 |
28 10 30
|
co |
|- ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) |
32 |
23 31
|
wceq |
|- x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) |
33 |
22 32
|
wi |
|- ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) |
34 |
33 15 17
|
wral |
|- A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) |
35 |
34 12 14
|
crio |
|- ( iota_ x e. ( Base ` k ) A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) ) |
36 |
7 11 35
|
cmpt |
|- ( f e. ( ( LTrn ` k ) ` w ) |-> ( iota_ x e. ( Base ` k ) A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) ) ) |
37 |
3 6 36
|
cmpt |
|- ( w e. ( LHyp ` k ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( iota_ x e. ( Base ` k ) A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) ) ) ) |
38 |
1 2 37
|
cmpt |
|- ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( iota_ x e. ( Base ` k ) A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) ) ) ) ) |
39 |
0 38
|
wceq |
|- trL = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( iota_ x e. ( Base ` k ) A. p e. ( Atoms ` k ) ( -. p ( le ` k ) w -> x = ( ( p ( join ` k ) ( f ` p ) ) ( meet ` k ) w ) ) ) ) ) ) |