Step |
Hyp |
Ref |
Expression |
1 |
|
tendof.h |
|- H = ( LHyp ` K ) |
2 |
|
tendof.t |
|- T = ( ( LTrn ` K ) ` W ) |
3 |
|
tendof.e |
|- E = ( ( TEndo ` K ) ` W ) |
4 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
5 |
|
eqid |
|- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W ) |
6 |
|
id |
|- ( ( K e. HL /\ W e. H ) -> ( K e. HL /\ W e. H ) ) |
7 |
|
f1oi |
|- ( _I |` T ) : T -1-1-onto-> T |
8 |
|
f1of |
|- ( ( _I |` T ) : T -1-1-onto-> T -> ( _I |` T ) : T --> T ) |
9 |
7 8
|
mp1i |
|- ( ( K e. HL /\ W e. H ) -> ( _I |` T ) : T --> T ) |
10 |
1 2
|
ltrnco |
|- ( ( ( K e. HL /\ W e. H ) /\ f e. T /\ g e. T ) -> ( f o. g ) e. T ) |
11 |
|
fvresi |
|- ( ( f o. g ) e. T -> ( ( _I |` T ) ` ( f o. g ) ) = ( f o. g ) ) |
12 |
10 11
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ f e. T /\ g e. T ) -> ( ( _I |` T ) ` ( f o. g ) ) = ( f o. g ) ) |
13 |
|
fvresi |
|- ( f e. T -> ( ( _I |` T ) ` f ) = f ) |
14 |
13
|
3ad2ant2 |
|- ( ( ( K e. HL /\ W e. H ) /\ f e. T /\ g e. T ) -> ( ( _I |` T ) ` f ) = f ) |
15 |
|
fvresi |
|- ( g e. T -> ( ( _I |` T ) ` g ) = g ) |
16 |
15
|
3ad2ant3 |
|- ( ( ( K e. HL /\ W e. H ) /\ f e. T /\ g e. T ) -> ( ( _I |` T ) ` g ) = g ) |
17 |
14 16
|
coeq12d |
|- ( ( ( K e. HL /\ W e. H ) /\ f e. T /\ g e. T ) -> ( ( ( _I |` T ) ` f ) o. ( ( _I |` T ) ` g ) ) = ( f o. g ) ) |
18 |
12 17
|
eqtr4d |
|- ( ( ( K e. HL /\ W e. H ) /\ f e. T /\ g e. T ) -> ( ( _I |` T ) ` ( f o. g ) ) = ( ( ( _I |` T ) ` f ) o. ( ( _I |` T ) ` g ) ) ) |
19 |
13
|
adantl |
|- ( ( ( K e. HL /\ W e. H ) /\ f e. T ) -> ( ( _I |` T ) ` f ) = f ) |
20 |
19
|
fveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ f e. T ) -> ( ( ( trL ` K ) ` W ) ` ( ( _I |` T ) ` f ) ) = ( ( ( trL ` K ) ` W ) ` f ) ) |
21 |
|
hllat |
|- ( K e. HL -> K e. Lat ) |
22 |
21
|
ad2antrr |
|- ( ( ( K e. HL /\ W e. H ) /\ f e. T ) -> K e. Lat ) |
23 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
24 |
23 1 2 5
|
trlcl |
|- ( ( ( K e. HL /\ W e. H ) /\ f e. T ) -> ( ( ( trL ` K ) ` W ) ` f ) e. ( Base ` K ) ) |
25 |
23 4
|
latref |
|- ( ( K e. Lat /\ ( ( ( trL ` K ) ` W ) ` f ) e. ( Base ` K ) ) -> ( ( ( trL ` K ) ` W ) ` f ) ( le ` K ) ( ( ( trL ` K ) ` W ) ` f ) ) |
26 |
22 24 25
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ f e. T ) -> ( ( ( trL ` K ) ` W ) ` f ) ( le ` K ) ( ( ( trL ` K ) ` W ) ` f ) ) |
27 |
20 26
|
eqbrtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ f e. T ) -> ( ( ( trL ` K ) ` W ) ` ( ( _I |` T ) ` f ) ) ( le ` K ) ( ( ( trL ` K ) ` W ) ` f ) ) |
28 |
4 1 2 5 3 6 9 18 27
|
istendod |
|- ( ( K e. HL /\ W e. H ) -> ( _I |` T ) e. E ) |