Step |
Hyp |
Ref |
Expression |
1 |
|
diaoc.j |
|- .\/ = ( join ` K ) |
2 |
|
diaoc.m |
|- ./\ = ( meet ` K ) |
3 |
|
diaoc.o |
|- ._|_ = ( oc ` K ) |
4 |
|
diaoc.h |
|- H = ( LHyp ` K ) |
5 |
|
diaoc.t |
|- T = ( ( LTrn ` K ) ` W ) |
6 |
|
diaoc.i |
|- I = ( ( DIsoA ` K ) ` W ) |
7 |
|
diaoc.n |
|- N = ( ( ocA ` K ) ` W ) |
8 |
|
simpl |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> ( K e. HL /\ W e. H ) ) |
9 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
10 |
9 4 6
|
diadmclN |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> X e. ( Base ` K ) ) |
11 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
12 |
11 4 6
|
diadmleN |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> X ( le ` K ) W ) |
13 |
9 11 4 5 6
|
diass |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. ( Base ` K ) /\ X ( le ` K ) W ) ) -> ( I ` X ) C_ T ) |
14 |
8 10 12 13
|
syl12anc |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> ( I ` X ) C_ T ) |
15 |
1 2 3 4 5 6 7
|
docavalN |
|- ( ( ( K e. HL /\ W e. H ) /\ ( I ` X ) C_ T ) -> ( N ` ( I ` X ) ) = ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | ( I ` X ) C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) |
16 |
14 15
|
syldan |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> ( N ` ( I ` X ) ) = ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | ( I ` X ) C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) |
17 |
4 6
|
diaclN |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> ( I ` X ) e. ran I ) |
18 |
|
intmin |
|- ( ( I ` X ) e. ran I -> |^| { z e. ran I | ( I ` X ) C_ z } = ( I ` X ) ) |
19 |
17 18
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> |^| { z e. ran I | ( I ` X ) C_ z } = ( I ` X ) ) |
20 |
19
|
fveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> ( `' I ` |^| { z e. ran I | ( I ` X ) C_ z } ) = ( `' I ` ( I ` X ) ) ) |
21 |
4 6
|
diaf11N |
|- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> ran I ) |
22 |
|
f1ocnvfv1 |
|- ( ( I : dom I -1-1-onto-> ran I /\ X e. dom I ) -> ( `' I ` ( I ` X ) ) = X ) |
23 |
21 22
|
sylan |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> ( `' I ` ( I ` X ) ) = X ) |
24 |
20 23
|
eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> ( `' I ` |^| { z e. ran I | ( I ` X ) C_ z } ) = X ) |
25 |
24
|
fveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> ( ._|_ ` ( `' I ` |^| { z e. ran I | ( I ` X ) C_ z } ) ) = ( ._|_ ` X ) ) |
26 |
25
|
oveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> ( ( ._|_ ` ( `' I ` |^| { z e. ran I | ( I ` X ) C_ z } ) ) .\/ ( ._|_ ` W ) ) = ( ( ._|_ ` X ) .\/ ( ._|_ ` W ) ) ) |
27 |
26
|
fvoveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | ( I ` X ) C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) = ( I ` ( ( ( ._|_ ` X ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) |
28 |
16 27
|
eqtr2d |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> ( I ` ( ( ( ._|_ ` X ) .\/ ( ._|_ ` W ) ) ./\ W ) ) = ( N ` ( I ` X ) ) ) |