Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemg46.b |
|- B = ( Base ` K ) |
2 |
|
cdlemg46.h |
|- H = ( LHyp ` K ) |
3 |
|
cdlemg46.t |
|- T = ( ( LTrn ` K ) ` W ) |
4 |
|
cdlemg46.r |
|- R = ( ( trL ` K ) ` W ) |
5 |
1 2 3 4
|
cdlemftr1 |
|- ( ( K e. HL /\ W e. H ) -> E. h e. T ( h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) |
6 |
5
|
3ad2ant1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= ( _I |` B ) /\ ( R ` F ) = ( R ` G ) ) ) -> E. h e. T ( h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) |
7 |
|
simp11 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= ( _I |` B ) /\ ( R ` F ) = ( R ` G ) ) ) /\ h e. T /\ ( h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( K e. HL /\ W e. H ) ) |
8 |
|
simp12l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= ( _I |` B ) /\ ( R ` F ) = ( R ` G ) ) ) /\ h e. T /\ ( h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> F e. T ) |
9 |
|
simp12r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= ( _I |` B ) /\ ( R ` F ) = ( R ` G ) ) ) /\ h e. T /\ ( h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> G e. T ) |
10 |
|
simp2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= ( _I |` B ) /\ ( R ` F ) = ( R ` G ) ) ) /\ h e. T /\ ( h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> h e. T ) |
11 |
|
simp13r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= ( _I |` B ) /\ ( R ` F ) = ( R ` G ) ) ) /\ h e. T /\ ( h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( R ` F ) = ( R ` G ) ) |
12 |
|
simp13l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= ( _I |` B ) /\ ( R ` F ) = ( R ` G ) ) ) /\ h e. T /\ ( h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> F =/= ( _I |` B ) ) |
13 |
|
simp3l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= ( _I |` B ) /\ ( R ` F ) = ( R ` G ) ) ) /\ h e. T /\ ( h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> h =/= ( _I |` B ) ) |
14 |
|
simp3r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= ( _I |` B ) /\ ( R ` F ) = ( R ` G ) ) ) /\ h e. T /\ ( h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( R ` h ) =/= ( R ` F ) ) |
15 |
1 2 3 4
|
cdlemg47 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( h e. T /\ ( R ` F ) = ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( F o. G ) = ( G o. F ) ) |
16 |
7 8 9 10 11 12 13 14 15
|
syl323anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= ( _I |` B ) /\ ( R ` F ) = ( R ` G ) ) ) /\ h e. T /\ ( h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( F o. G ) = ( G o. F ) ) |
17 |
16
|
rexlimdv3a |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= ( _I |` B ) /\ ( R ` F ) = ( R ` G ) ) ) -> ( E. h e. T ( h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) -> ( F o. G ) = ( G o. F ) ) ) |
18 |
6 17
|
mpd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= ( _I |` B ) /\ ( R ` F ) = ( R ` G ) ) ) -> ( F o. G ) = ( G o. F ) ) |