Step |
Hyp |
Ref |
Expression |
1 |
|
dihord3.b |
|- B = ( Base ` K ) |
2 |
|
dihord3.l |
|- .<_ = ( le ` K ) |
3 |
|
dihord3.h |
|- H = ( LHyp ` K ) |
4 |
|
dihord3.i |
|- I = ( ( DIsoH ` K ) ` W ) |
5 |
|
eqid |
|- ( ( DIsoB ` K ) ` W ) = ( ( DIsoB ` K ) ` W ) |
6 |
1 2 3 4 5
|
dihvalb |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( ( DIsoB ` K ) ` W ) ` X ) ) |
7 |
6
|
3adant3 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( I ` X ) = ( ( ( DIsoB ` K ) ` W ) ` X ) ) |
8 |
1 2 3 4 5
|
dihvalb |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( I ` Y ) = ( ( ( DIsoB ` K ) ` W ) ` Y ) ) |
9 |
8
|
3adant2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( I ` Y ) = ( ( ( DIsoB ` K ) ` W ) ` Y ) ) |
10 |
7 9
|
sseq12d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( I ` X ) C_ ( I ` Y ) <-> ( ( ( DIsoB ` K ) ` W ) ` X ) C_ ( ( ( DIsoB ` K ) ` W ) ` Y ) ) ) |
11 |
1 2 3 5
|
dibord |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( ( ( DIsoB ` K ) ` W ) ` X ) C_ ( ( ( DIsoB ` K ) ` W ) ` Y ) <-> X .<_ Y ) ) |
12 |
10 11
|
bitrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( I ` X ) C_ ( I ` Y ) <-> X .<_ Y ) ) |