Step |
Hyp |
Ref |
Expression |
1 |
|
docaval.j |
|- .\/ = ( join ` K ) |
2 |
|
docaval.m |
|- ./\ = ( meet ` K ) |
3 |
|
docaval.o |
|- ._|_ = ( oc ` K ) |
4 |
|
docaval.h |
|- H = ( LHyp ` K ) |
5 |
|
docaval.t |
|- T = ( ( LTrn ` K ) ` W ) |
6 |
|
docaval.i |
|- I = ( ( DIsoA ` K ) ` W ) |
7 |
|
docaval.n |
|- N = ( ( ocA ` K ) ` W ) |
8 |
1 2 3 4 5 6 7
|
docafvalN |
|- ( ( K e. HL /\ W e. H ) -> N = ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) ) |
9 |
8
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> N = ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) ) |
10 |
9
|
fveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( N ` X ) = ( ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) ` X ) ) |
11 |
5
|
fvexi |
|- T e. _V |
12 |
11
|
elpw2 |
|- ( X e. ~P T <-> X C_ T ) |
13 |
12
|
biimpri |
|- ( X C_ T -> X e. ~P T ) |
14 |
13
|
adantl |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> X e. ~P T ) |
15 |
|
fvex |
|- ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) e. _V |
16 |
|
sseq1 |
|- ( x = X -> ( x C_ z <-> X C_ z ) ) |
17 |
16
|
rabbidv |
|- ( x = X -> { z e. ran I | x C_ z } = { z e. ran I | X C_ z } ) |
18 |
17
|
inteqd |
|- ( x = X -> |^| { z e. ran I | x C_ z } = |^| { z e. ran I | X C_ z } ) |
19 |
18
|
fveq2d |
|- ( x = X -> ( `' I ` |^| { z e. ran I | x C_ z } ) = ( `' I ` |^| { z e. ran I | X C_ z } ) ) |
20 |
19
|
fveq2d |
|- ( x = X -> ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) = ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ) |
21 |
20
|
oveq1d |
|- ( x = X -> ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) = ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) ) |
22 |
21
|
fvoveq1d |
|- ( x = X -> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) = ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) |
23 |
|
eqid |
|- ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) = ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) |
24 |
22 23
|
fvmptg |
|- ( ( X e. ~P T /\ ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) e. _V ) -> ( ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) ` X ) = ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) |
25 |
14 15 24
|
sylancl |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) ` X ) = ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) |
26 |
10 25
|
eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( N ` X ) = ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) |