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Metamath Proof Explorer

Theorem tendo1ne0

Description: The identity (unity) is not equal to the zero trace-preserving endomorphism. (Contributed by NM, 8-Aug-2013)

Ref Expression
Hypotheses tendoid0.b 
|- B = ( Base ` K )
tendoid0.h 
|- H = ( LHyp ` K )
tendoid0.t 
|- T = ( ( LTrn ` K ) ` W )
tendoid0.e 
|- E = ( ( TEndo ` K ) ` W )
tendoid0.o 
|- O = ( f e. T |-> ( _I |` B ) )
Assertion tendo1ne0 
|- ( ( K e. HL /\ W e. H ) -> ( _I |` T ) =/= O )

Proof

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 )