Metamath Proof Explorer

Theorem dvhvscacl

Description: Closure of the scalar product operation for the constructed full vector space H. (Contributed by NM, 12-Feb-2014)

Ref Expression
Hypotheses dvhfvsca.h 
|- H = ( LHyp ` K )
dvhfvsca.t 
|- T = ( ( LTrn ` K ) ` W )
dvhfvsca.e 
|- E = ( ( TEndo ` K ) ` W )
dvhfvsca.u 
|- U = ( ( DVecH ` K ) ` W )
dvhfvsca.s 
|- .x. = ( .s ` U )
Assertion dvhvscacl 
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( R .x. F ) e. ( T X. E ) )

Proof

Step Hyp Ref Expression
1 dvhfvsca.h 
 |-  H = ( LHyp ` K )
2 dvhfvsca.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 dvhfvsca.e 
 |-  E = ( ( TEndo ` K ) ` W )
4 dvhfvsca.u 
 |-  U = ( ( DVecH ` K ) ` W )
5 dvhfvsca.s 
 |-  .x. = ( .s ` U )
6 1 2 3 4 5 dvhvsca 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( R .x. F ) = <. ( R ` ( 1st ` F ) ) , ( R o. ( 2nd ` F ) ) >. )
7 simpl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( K e. HL /\ W e. H ) )
8 simprl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> R e. E )
9 xp1st 
 |-  ( F e. ( T X. E ) -> ( 1st ` F ) e. T )
10 9 ad2antll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( 1st ` F ) e. T )
11 1 2 3 tendocl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ R e. E /\ ( 1st ` F ) e. T ) -> ( R ` ( 1st ` F ) ) e. T )
12 7 8 10 11 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( R ` ( 1st ` F ) ) e. T )
13 xp2nd 
 |-  ( F e. ( T X. E ) -> ( 2nd ` F ) e. E )
14 13 ad2antll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( 2nd ` F ) e. E )
15 1 3 tendococl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ R e. E /\ ( 2nd ` F ) e. E ) -> ( R o. ( 2nd ` F ) ) e. E )
16 7 8 14 15 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( R o. ( 2nd ` F ) ) e. E )
17 opelxpi 
 |-  ( ( ( R ` ( 1st ` F ) ) e. T /\ ( R o. ( 2nd ` F ) ) e. E ) -> <. ( R ` ( 1st ` F ) ) , ( R o. ( 2nd ` F ) ) >. e. ( T X. E ) )
18 12 16 17 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> <. ( R ` ( 1st ` F ) ) , ( R o. ( 2nd ` F ) ) >. e. ( T X. E ) )
19 6 18 eqeltrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. E /\ F e. ( T X. E ) ) ) -> ( R .x. F ) e. ( T X. E ) )