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Metamath Proof Explorer

Theorem tendoid

Description: The identity value of a trace-preserving endomorphism. (Contributed by NM, 21-Jun-2013)

Ref Expression
Hypotheses tendoid.b 
|- B = ( Base ` K )
tendoid.h 
|- H = ( LHyp ` K )
tendoid.e 
|- E = ( ( TEndo ` K ) ` W )
Assertion tendoid 
|- ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( S ` ( _I |` B ) ) = ( _I |` B ) )

Proof

Step Hyp Ref Expression
1 tendoid.b 
 |-  B = ( Base ` K )
2 tendoid.h 
 |-  H = ( LHyp ` K )
3 tendoid.e 
 |-  E = ( ( TEndo ` K ) ` W )
4 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
5 1 2 4 idltrn 
 |-  ( ( K e. HL /\ W e. H ) -> ( _I |` B ) e. ( ( LTrn ` K ) ` W ) )
6 5 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( _I |` B ) e. ( ( LTrn ` K ) ` W ) )
7 eqid 
 |-  ( le ` K ) = ( le ` K )
8 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
9 7 2 4 8 3 tendotp 
 |-  ( ( ( K e. HL /\ W e. H ) /\ S e. E /\ ( _I |` B ) e. ( ( LTrn ` K ) ` W ) ) -> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) ( le ` K ) ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) )
10 6 9 mpd3an3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) ( le ` K ) ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) )
11 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
12 1 11 2 8 trlid0 
 |-  ( ( K e. HL /\ W e. H ) -> ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) = ( 0. ` K ) )
13 12 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( ( ( trL ` K ) ` W ) ` ( _I |` B ) ) = ( 0. ` K ) )
14 10 13 breqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) ( le ` K ) ( 0. ` K ) )
15 hlop 
 |-  ( K e. HL -> K e. OP )
16 15 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> K e. OP )
17 2 4 3 tendocl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ S e. E /\ ( _I |` B ) e. ( ( LTrn ` K ) ` W ) ) -> ( S ` ( _I |` B ) ) e. ( ( LTrn ` K ) ` W ) )
18 6 17 mpd3an3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( S ` ( _I |` B ) ) e. ( ( LTrn ` K ) ` W ) )
19 1 2 4 8 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S ` ( _I |` B ) ) e. ( ( LTrn ` K ) ` W ) ) -> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) e. B )
20 18 19 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) e. B )
21 1 7 11 ople0 
 |-  ( ( K e. OP /\ ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) e. B ) -> ( ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) ( le ` K ) ( 0. ` K ) <-> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) = ( 0. ` K ) ) )
22 16 20 21 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) ( le ` K ) ( 0. ` K ) <-> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) = ( 0. ` K ) ) )
23 14 22 mpbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) = ( 0. ` K ) )
24 1 11 2 4 8 trlid0b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S ` ( _I |` B ) ) e. ( ( LTrn ` K ) ` W ) ) -> ( ( S ` ( _I |` B ) ) = ( _I |` B ) <-> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) = ( 0. ` K ) ) )
25 18 24 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( ( S ` ( _I |` B ) ) = ( _I |` B ) <-> ( ( ( trL ` K ) ` W ) ` ( S ` ( _I |` B ) ) ) = ( 0. ` K ) ) )
26 23 25 mpbird 
 |-  ( ( ( K e. HL /\ W e. H ) /\ S e. E ) -> ( S ` ( _I |` B ) ) = ( _I |` B ) )