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Theorem dvhfvadd 35888
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.)
Hypotheses
Ref Expression
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
Ref Expression
dvhfvadd  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .+  =  .+b  )
Distinct variable groups:    f, g, E    f, H, g    f, K, g    T, f, g   
f, W, g
Allowed substitution hints:    D( f, g)    .+ ( f, g)    .+^ ( f, g)    .+b ( f, g)    U( f, g)

Proof of Theorem dvhfvadd
Dummy variables  h  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvhfvadd.h . . . . 5  |-  H  =  ( LHyp `  K
)
2 dvhfvadd.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
3 dvhfvadd.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
4 eqid 2467 . . . . 5  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
5 dvhfvadd.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
61, 2, 3, 4, 5dvhset 35878 . . . 4  |-  ( ( 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
) ) >. ) >. } ) )
76fveq2d 5868 . . 3  |-  ( ( 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
) ) >. ) >. } ) ) )
8 dvhfvadd.p . . . . . . . . . 10  |-  .+^  =  ( +g  `  D )
9 dvhfvadd.f . . . . . . . . . . . 12  |-  D  =  (Scalar `  U )
101, 4, 5, 9dvhsca 35879 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
1110fveq2d 5868 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( +g  `  D
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
128, 11syl5eq 2520 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
.+^  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
1312oveqd 6299 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( 2nd `  f
)  .+^  ( 2nd `  g
) )  =  ( ( 2nd `  f
) ( +g  `  (
( EDRing `  K ) `  W ) ) ( 2nd `  g ) ) )
14133ad2ant1 1017 . . . . . . 7  |-  ( ( ( 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 ) ) )
15 xp2nd 6812 . . . . . . . . . 10  |-  ( f  e.  ( T  X.  E )  ->  ( 2nd `  f )  e.  E )
16 xp2nd 6812 . . . . . . . . . 10  |-  ( g  e.  ( T  X.  E )  ->  ( 2nd `  g )  e.  E )
1715, 16anim12i 566 . . . . . . . . 9  |-  ( ( f  e.  ( T  X.  E )  /\  g  e.  ( T  X.  E ) )  -> 
( ( 2nd `  f
)  e.  E  /\  ( 2nd `  g )  e.  E ) )
18 eqid 2467 . . . . . . . . . 10  |-  ( +g  `  ( ( EDRing `  K
) `  W )
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
)
191, 2, 3, 4, 18erngplus 35599 . . . . . . . . 9  |-  ( ( ( 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 )
) ) )
2017, 19sylan2 474 . . . . . . . 8  |-  ( ( ( 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 )
) ) )
21203impb 1192 . . . . . . 7  |-  ( ( ( 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 )
) ) )
2214, 21eqtrd 2508 . . . . . 6  |-  ( ( ( 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 )
) ) )
2322opeq2d 4220 . . . . 5  |-  ( ( ( 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 ) ) ) >. )
2423mpt2eq3dva 6343 . . . 4  |-  ( ( 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 ) ) ) >. ) )
25 fvex 5874 . . . . . . . 8  |-  ( (
LTrn `  K ) `  W )  e.  _V
262, 25eqeltri 2551 . . . . . . 7  |-  T  e. 
_V
27 fvex 5874 . . . . . . . 8  |-  ( (
TEndo `  K ) `  W )  e.  _V
283, 27eqeltri 2551 . . . . . . 7  |-  E  e. 
_V
2926, 28xpex 6711 . . . . . 6  |-  ( T  X.  E )  e. 
_V
3029, 29mpt2ex 6857 . . . . 5  |-  ( 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 2467 . . . . . 6  |-  ( {
<. ( 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
) ) >. ) >. } )
3231lmodplusg 14614 . . . . 5  |-  ( ( 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
) ) >. ) >. } ) ) )
3330, 32ax-mp 5 . . . 4  |-  ( 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
) ) >. ) >. } ) )
3424, 33syl6req 2525 . . 3  |-  ( ( 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 ) ) >.
) )
357, 34eqtrd 2508 . 2  |-  ( ( 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 dvhfvadd.s . 2  |-  .+  =  ( +g  `  U )
37 dvhfvadd.a . 2  |-  .+b  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E ) 
|->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f )  .+^  ( 2nd `  g ) ) >.
)
3835, 36, 373eqtr4g 2533 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .+  =  .+b  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    u. cun 3474   {csn 4027   {ctp 4031   <.cop 4033    |-> cmpt 4505    X. cxp 4997    o. ccom 5003   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780   ndxcnx 14480   Basecbs 14483   +g cplusg 14548  Scalarcsca 14551   .scvsca 14552   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   TEndoctendo 35548   EDRingcedring 35549   DVecHcdvh 35875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-plusg 14561  df-mulr 14562  df-sca 14564  df-vsca 14565  df-edring 35553  df-dvech 35876
This theorem is referenced by:  dvhvadd  35889
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