Metamath Proof Explorer

Theorem dvh0g

Description: The zero vector of vector space H has the zero translation as its first member and the zero trace-preserving endomorphism as the second. (Contributed by NM, 9-Mar-2014) (Revised by Mario Carneiro, 24-Jun-2014)

Ref Expression
Hypotheses dvh0g.b 
|- B = ( Base ` K )
dvh0g.h 
|- H = ( LHyp ` K )
dvh0g.t 
|- T = ( ( LTrn ` K ) ` W )
dvh0g.u 
|- U = ( ( DVecH ` K ) ` W )
dvh0g.z 
|- .0. = ( 0g ` U )
dvh0g.o 
|- O = ( f e. T |-> ( _I |` B ) )
Assertion dvh0g 
|- ( ( K e. HL /\ W e. H ) -> .0. = <. ( _I |` B ) , O >. )

Proof

Step Hyp Ref Expression
1 dvh0g.b 
 |-  B = ( Base ` K )
2 dvh0g.h 
 |-  H = ( LHyp ` K )
3 dvh0g.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 dvh0g.u 
 |-  U = ( ( DVecH ` K ) ` W )
5 dvh0g.z 
 |-  .0. = ( 0g ` U )
6 dvh0g.o 
 |-  O = ( f e. T |-> ( _I |` B ) )
7 id 
 |-  ( ( K e. HL /\ W e. H ) -> ( K e. HL /\ W e. H ) )
8 1 2 3 idltrn 
 |-  ( ( K e. HL /\ W e. H ) -> ( _I |` B ) e. T )
9 eqid 
 |-  ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
10 1 2 3 9 6 tendo0cl 
 |-  ( ( K e. HL /\ W e. H ) -> O e. ( ( TEndo ` K ) ` W ) )
11 eqid 
 |-  ( Scalar ` U ) = ( Scalar ` U )
12 eqid 
 |-  ( +g ` U ) = ( +g ` U )
13 eqid 
 |-  ( +g ` ( Scalar ` U ) ) = ( +g ` ( Scalar ` U ) )
14 2 3 9 4 11 12 13 dvhopvadd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( _I |` B ) e. T /\ O e. ( ( TEndo ` K ) ` W ) ) /\ ( ( _I |` B ) e. T /\ O e. ( ( TEndo ` K ) ` W ) ) ) -> ( <. ( _I |` B ) , O >. ( +g ` U ) <. ( _I |` B ) , O >. ) = <. ( ( _I |` B ) o. ( _I |` B ) ) , ( O ( +g ` ( Scalar ` U ) ) O ) >. )
15 7 8 10 8 10 14 syl122anc 
 |-  ( ( K e. HL /\ W e. H ) -> ( <. ( _I |` B ) , O >. ( +g ` U ) <. ( _I |` B ) , O >. ) = <. ( ( _I |` B ) o. ( _I |` B ) ) , ( O ( +g ` ( Scalar ` U ) ) O ) >. )
16 f1oi 
 |-  ( _I |` B ) : B -1-1-onto-> B
17 f1of 
 |-  ( ( _I |` B ) : B -1-1-onto-> B -> ( _I |` B ) : B --> B )
18 fcoi2 
 |-  ( ( _I |` B ) : B --> B -> ( ( _I |` B ) o. ( _I |` B ) ) = ( _I |` B ) )
19 16 17 18 mp2b 
 |-  ( ( _I |` B ) o. ( _I |` B ) ) = ( _I |` B )
20 19 a1i 
 |-  ( ( K e. HL /\ W e. H ) -> ( ( _I |` B ) o. ( _I |` B ) ) = ( _I |` B ) )
21 eqid 
 |-  ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) = ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )
22 2 3 9 4 11 21 13 dvhfplusr 
 |-  ( ( K e. HL /\ W e. H ) -> ( +g ` ( Scalar ` U ) ) = ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) )
23 22 oveqd 
 |-  ( ( K e. HL /\ W e. H ) -> ( O ( +g ` ( Scalar ` U ) ) O ) = ( O ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) O ) )
24 1 2 3 9 6 21 tendo0pl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ O e. ( ( TEndo ` K ) ` W ) ) -> ( O ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) O ) = O )
25 10 24 mpdan 
 |-  ( ( K e. HL /\ W e. H ) -> ( O ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) O ) = O )
26 23 25 eqtrd 
 |-  ( ( K e. HL /\ W e. H ) -> ( O ( +g ` ( Scalar ` U ) ) O ) = O )
27 20 26 opeq12d 
 |-  ( ( K e. HL /\ W e. H ) -> <. ( ( _I |` B ) o. ( _I |` B ) ) , ( O ( +g ` ( Scalar ` U ) ) O ) >. = <. ( _I |` B ) , O >. )
28 15 27 eqtrd 
 |-  ( ( K e. HL /\ W e. H ) -> ( <. ( _I |` B ) , O >. ( +g ` U ) <. ( _I |` B ) , O >. ) = <. ( _I |` B ) , O >. )
29 2 4 7 dvhlmod 
 |-  ( ( K e. HL /\ W e. H ) -> U e. LMod )
30 eqid 
 |-  ( Base ` U ) = ( Base ` U )
31 2 3 9 4 30 dvhelvbasei 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( _I |` B ) e. T /\ O e. ( ( TEndo ` K ) ` W ) ) ) -> <. ( _I |` B ) , O >. e. ( Base ` U ) )
32 7 8 10 31 syl12anc 
 |-  ( ( K e. HL /\ W e. H ) -> <. ( _I |` B ) , O >. e. ( Base ` U ) )
33 30 12 5 lmod0vid 
 |-  ( ( U e. LMod /\ <. ( _I |` B ) , O >. e. ( Base ` U ) ) -> ( ( <. ( _I |` B ) , O >. ( +g ` U ) <. ( _I |` B ) , O >. ) = <. ( _I |` B ) , O >. <-> .0. = <. ( _I |` B ) , O >. ) )
34 29 32 33 syl2anc 
 |-  ( ( K e. HL /\ W e. H ) -> ( ( <. ( _I |` B ) , O >. ( +g ` U ) <. ( _I |` B ) , O >. ) = <. ( _I |` B ) , O >. <-> .0. = <. ( _I |` B ) , O >. ) )
35 28 34 mpbid 
 |-  ( ( K e. HL /\ W e. H ) -> .0. = <. ( _I |` B ) , O >. )