Step |
Hyp |
Ref |
Expression |
1 |
|
trlnid.b |
|- B = ( Base ` K ) |
2 |
|
trlnid.h |
|- H = ( LHyp ` K ) |
3 |
|
trlnid.t |
|- T = ( ( LTrn ` K ) ` W ) |
4 |
|
trlnid.r |
|- R = ( ( trL ` K ) ` W ) |
5 |
|
simp3l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> F =/= G ) |
6 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> ( K e. HL /\ W e. H ) ) |
7 |
|
simp2l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> F e. T ) |
8 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
9 |
1 8 2 3 4
|
trlid0b |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F = ( _I |` B ) <-> ( R ` F ) = ( 0. ` K ) ) ) |
10 |
6 7 9
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> ( F = ( _I |` B ) <-> ( R ` F ) = ( 0. ` K ) ) ) |
11 |
10
|
biimpar |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) /\ ( R ` F ) = ( 0. ` K ) ) -> F = ( _I |` B ) ) |
12 |
|
simp3r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> ( R ` F ) = ( R ` G ) ) |
13 |
12
|
eqeq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> ( ( R ` F ) = ( 0. ` K ) <-> ( R ` G ) = ( 0. ` K ) ) ) |
14 |
13
|
biimpa |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) /\ ( R ` F ) = ( 0. ` K ) ) -> ( R ` G ) = ( 0. ` K ) ) |
15 |
|
simpl1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) /\ ( R ` F ) = ( 0. ` K ) ) -> ( K e. HL /\ W e. H ) ) |
16 |
|
simpl2r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) /\ ( R ` F ) = ( 0. ` K ) ) -> G e. T ) |
17 |
1 8 2 3 4
|
trlid0b |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( G = ( _I |` B ) <-> ( R ` G ) = ( 0. ` K ) ) ) |
18 |
15 16 17
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) /\ ( R ` F ) = ( 0. ` K ) ) -> ( G = ( _I |` B ) <-> ( R ` G ) = ( 0. ` K ) ) ) |
19 |
14 18
|
mpbird |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) /\ ( R ` F ) = ( 0. ` K ) ) -> G = ( _I |` B ) ) |
20 |
11 19
|
eqtr4d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) /\ ( R ` F ) = ( 0. ` K ) ) -> F = G ) |
21 |
20
|
ex |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> ( ( R ` F ) = ( 0. ` K ) -> F = G ) ) |
22 |
10 21
|
sylbid |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> ( F = ( _I |` B ) -> F = G ) ) |
23 |
22
|
necon3d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> ( F =/= G -> F =/= ( _I |` B ) ) ) |
24 |
5 23
|
mpd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> F =/= ( _I |` B ) ) |