Step |
Hyp |
Ref |
Expression |
1 |
|
tendoid0.b |
|- B = ( Base ` K ) |
2 |
|
tendoid0.h |
|- H = ( LHyp ` K ) |
3 |
|
tendoid0.t |
|- T = ( ( LTrn ` K ) ` W ) |
4 |
|
tendoid0.e |
|- E = ( ( TEndo ` K ) ` W ) |
5 |
|
tendoid0.o |
|- O = ( f e. T |-> ( _I |` B ) ) |
6 |
1 2 3
|
cdlemftr0 |
|- ( ( K e. HL /\ W e. H ) -> E. g e. T g =/= ( _I |` B ) ) |
7 |
|
simp3 |
|- ( ( ( K e. HL /\ W e. H ) /\ g e. T /\ g =/= ( _I |` B ) ) -> g =/= ( _I |` B ) ) |
8 |
|
fveq1 |
|- ( ( _I |` T ) = O -> ( ( _I |` T ) ` g ) = ( O ` g ) ) |
9 |
8
|
adantl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( _I |` T ) = O ) -> ( ( _I |` T ) ` g ) = ( O ` g ) ) |
10 |
|
simpl2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( _I |` T ) = O ) -> g e. T ) |
11 |
|
fvresi |
|- ( g e. T -> ( ( _I |` T ) ` g ) = g ) |
12 |
10 11
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( _I |` T ) = O ) -> ( ( _I |` T ) ` g ) = g ) |
13 |
5 1
|
tendo02 |
|- ( g e. T -> ( O ` g ) = ( _I |` B ) ) |
14 |
10 13
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( _I |` T ) = O ) -> ( O ` g ) = ( _I |` B ) ) |
15 |
9 12 14
|
3eqtr3d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ g e. T /\ g =/= ( _I |` B ) ) /\ ( _I |` T ) = O ) -> g = ( _I |` B ) ) |
16 |
15
|
ex |
|- ( ( ( K e. HL /\ W e. H ) /\ g e. T /\ g =/= ( _I |` B ) ) -> ( ( _I |` T ) = O -> g = ( _I |` B ) ) ) |
17 |
16
|
necon3d |
|- ( ( ( K e. HL /\ W e. H ) /\ g e. T /\ g =/= ( _I |` B ) ) -> ( g =/= ( _I |` B ) -> ( _I |` T ) =/= O ) ) |
18 |
7 17
|
mpd |
|- ( ( ( K e. HL /\ W e. H ) /\ g e. T /\ g =/= ( _I |` B ) ) -> ( _I |` T ) =/= O ) |
19 |
18
|
rexlimdv3a |
|- ( ( K e. HL /\ W e. H ) -> ( E. g e. T g =/= ( _I |` B ) -> ( _I |` T ) =/= O ) ) |
20 |
6 19
|
mpd |
|- ( ( K e. HL /\ W e. H ) -> ( _I |` T ) =/= O ) |