Step |
Hyp |
Ref |
Expression |
1 |
|
dian0.b |
|- B = ( Base ` K ) |
2 |
|
dian0.l |
|- .<_ = ( le ` K ) |
3 |
|
dian0.h |
|- H = ( LHyp ` K ) |
4 |
|
dian0.i |
|- I = ( ( DIsoA ` K ) ` W ) |
5 |
|
eqid |
|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W ) |
6 |
1 3 5
|
idltrn |
|- ( ( K e. HL /\ W e. H ) -> ( _I |` B ) e. ( ( LTrn ` K ) ` W ) ) |
7 |
6
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( _I |` B ) e. ( ( LTrn ` K ) ` W ) ) |
8 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
9 |
|
eqid |
|- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W ) |
10 |
1 8 3 9
|
trlid0 |
|- ( ( K e. HL /\ W e. H ) -> ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) = ( 0. ` K ) ) |
11 |
10
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) = ( 0. ` K ) ) |
12 |
|
hlatl |
|- ( K e. HL -> K e. AtLat ) |
13 |
12
|
adantr |
|- ( ( K e. HL /\ W e. H ) -> K e. AtLat ) |
14 |
|
simpl |
|- ( ( X e. B /\ X .<_ W ) -> X e. B ) |
15 |
1 2 8
|
atl0le |
|- ( ( K e. AtLat /\ X e. B ) -> ( 0. ` K ) .<_ X ) |
16 |
13 14 15
|
syl2an |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( 0. ` K ) .<_ X ) |
17 |
11 16
|
eqbrtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) .<_ X ) |
18 |
1 2 3 5 9 4
|
diaelval |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( _I |` B ) e. ( I ` X ) <-> ( ( _I |` B ) e. ( ( LTrn ` K ) ` W ) /\ ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) .<_ X ) ) ) |
19 |
7 17 18
|
mpbir2and |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( _I |` B ) e. ( I ` X ) ) |
20 |
19
|
ne0d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) =/= (/) ) |