Step |
Hyp |
Ref |
Expression |
1 |
|
ltrnnidn.b |
|- B = ( Base ` K ) |
2 |
|
ltrnnidn.l |
|- .<_ = ( le ` K ) |
3 |
|
ltrnnidn.a |
|- A = ( Atoms ` K ) |
4 |
|
ltrnnidn.h |
|- H = ( LHyp ` K ) |
5 |
|
ltrnnidn.t |
|- T = ( ( LTrn ` K ) ` W ) |
6 |
|
simpr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ F = ( _I |` B ) ) -> F = ( _I |` B ) ) |
7 |
6
|
fveq1d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ F = ( _I |` B ) ) -> ( F ` P ) = ( ( _I |` B ) ` P ) ) |
8 |
|
simpl3l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ F = ( _I |` B ) ) -> P e. A ) |
9 |
1 3
|
atbase |
|- ( P e. A -> P e. B ) |
10 |
|
fvresi |
|- ( P e. B -> ( ( _I |` B ) ` P ) = P ) |
11 |
8 9 10
|
3syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ F = ( _I |` B ) ) -> ( ( _I |` B ) ` P ) = P ) |
12 |
7 11
|
eqtrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ F = ( _I |` B ) ) -> ( F ` P ) = P ) |
13 |
12
|
ex |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F = ( _I |` B ) -> ( F ` P ) = P ) ) |
14 |
|
simpl1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= ( _I |` B ) ) -> ( K e. HL /\ W e. H ) ) |
15 |
|
simpl2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= ( _I |` B ) ) -> F e. T ) |
16 |
|
simpr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= ( _I |` B ) ) -> F =/= ( _I |` B ) ) |
17 |
|
simpl3 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= ( _I |` B ) ) -> ( P e. A /\ -. P .<_ W ) ) |
18 |
1 2 3 4 5
|
ltrnnidn |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` P ) =/= P ) |
19 |
14 15 16 17 18
|
syl121anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= ( _I |` B ) ) -> ( F ` P ) =/= P ) |
20 |
19
|
ex |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F =/= ( _I |` B ) -> ( F ` P ) =/= P ) ) |
21 |
20
|
necon4d |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( F ` P ) = P -> F = ( _I |` B ) ) ) |
22 |
13 21
|
impbid |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F = ( _I |` B ) <-> ( F ` P ) = P ) ) |