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Metamath Proof Explorer

Theorem dvhfvadd

Description: The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013) (Revised by Mario Carneiro, 23-Jun-2014)

Ref Expression
Hypotheses dvhfvadd.h 
|- H = ( LHyp ` K )
dvhfvadd.t 
|- T = ( ( LTrn ` K ) ` W )
dvhfvadd.e 
|- E = ( ( TEndo ` K ) ` W )
dvhfvadd.u 
|- U = ( ( DVecH ` K ) ` W )
dvhfvadd.f 
|- D = ( Scalar ` U )
dvhfvadd.p 
|- .+^ = ( +g ` D )
dvhfvadd.a 
|- .+b = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. )
dvhfvadd.s 
|- .+ = ( +g ` U )
Assertion dvhfvadd 
|- ( ( K e. HL /\ W e. H ) -> .+ = .+b )

Proof

Step Hyp Ref Expression
1 dvhfvadd.h 
 |-  H = ( LHyp ` K )
2 dvhfvadd.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 dvhfvadd.e 
 |-  E = ( ( TEndo ` K ) ` W )
4 dvhfvadd.u 
 |-  U = ( ( DVecH ` K ) ` W )
5 dvhfvadd.f 
 |-  D = ( Scalar ` U )
6 dvhfvadd.p 
 |-  .+^ = ( +g ` D )
7 dvhfvadd.a 
 |-  .+b = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. )
8 dvhfvadd.s 
 |-  .+ = ( +g ` U )
9 eqid 
 |-  ( ( EDRing ` K ) ` W ) = ( ( EDRing ` K ) ` W )
10 1 2 3 9 4 dvhset 
 |-  ( ( K e. HL /\ W e. H ) -> U = ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) )
11 10 fveq2d 
 |-  ( ( K e. HL /\ W e. H ) -> ( +g ` U ) = ( +g ` ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )
12 1 9 4 5 dvhsca 
 |-  ( ( K e. HL /\ W e. H ) -> D = ( ( EDRing ` K ) ` W ) )
13 12 fveq2d 
 |-  ( ( K e. HL /\ W e. H ) -> ( +g ` D ) = ( +g ` ( ( EDRing ` K ) ` W ) ) )
14 6 13 eqtrid 
 |-  ( ( K e. HL /\ W e. H ) -> .+^ = ( +g ` ( ( EDRing ` K ) ` W ) ) )
15 14 oveqd 
 |-  ( ( K e. HL /\ W e. H ) -> ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) = ( ( 2nd ` f ) ( +g ` ( ( EDRing ` K ) ` W ) ) ( 2nd ` g ) ) )
16 15 3ad2ant1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) /\ g e. ( T X. E ) ) -> ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) = ( ( 2nd ` f ) ( +g ` ( ( EDRing ` K ) ` W ) ) ( 2nd ` g ) ) )
17 xp2nd 
 |-  ( f e. ( T X. E ) -> ( 2nd ` f ) e. E )
18 xp2nd 
 |-  ( g e. ( T X. E ) -> ( 2nd ` g ) e. E )
19 17 18 anim12i 
 |-  ( ( f e. ( T X. E ) /\ g e. ( T X. E ) ) -> ( ( 2nd ` f ) e. E /\ ( 2nd ` g ) e. E ) )
20 eqid 
 |-  ( +g ` ( ( EDRing ` K ) ` W ) ) = ( +g ` ( ( EDRing ` K ) ` W ) )
21 1 2 3 9 20 erngplus 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( 2nd ` f ) e. E /\ ( 2nd ` g ) e. E ) ) -> ( ( 2nd ` f ) ( +g ` ( ( EDRing ` K ) ` W ) ) ( 2nd ` g ) ) = ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) )
22 19 21 sylan2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( T X. E ) /\ g e. ( T X. E ) ) ) -> ( ( 2nd ` f ) ( +g ` ( ( EDRing ` K ) ` W ) ) ( 2nd ` g ) ) = ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) )
23 22 3impb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) /\ g e. ( T X. E ) ) -> ( ( 2nd ` f ) ( +g ` ( ( EDRing ` K ) ` W ) ) ( 2nd ` g ) ) = ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) )
24 16 23 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) /\ g e. ( T X. E ) ) -> ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) = ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) )
25 24 opeq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) /\ g e. ( T X. E ) ) -> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. = <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. )
26 25 mpoeq3dva 
 |-  ( ( K e. HL /\ W e. H ) -> ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. ) = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) )
27 2 fvexi 
 |-  T e. _V
28 3 fvexi 
 |-  E e. _V
29 27 28 xpex 
 |-  ( T X. E ) e. _V
30 29 29 mpoex 
 |-  ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) e. _V
31 eqid 
 |-  ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) = ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } )
32 31 lmodplusg 
 |-  ( ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) e. _V -> ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) = ( +g ` ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )
33 30 32 ax-mp 
 |-  ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) = ( +g ` ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) )
34 26 33 eqtr2di 
 |-  ( ( K e. HL /\ W e. H ) -> ( +g ` ( { <. ( Base ` ndx ) , ( T X. E ) >. , <. ( +g ` ndx ) , ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. T |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. ( Scalar ` ndx ) , ( ( EDRing ` K ) ` W ) >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. ) )
35 11 34 eqtrd 
 |-  ( ( K e. HL /\ W e. H ) -> ( +g ` U ) = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. ) )
36 35 8 7 3eqtr4g 
 |-  ( ( K e. HL /\ W e. H ) -> .+ = .+b )