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Theorem dvhfvadd 31889
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 2436 . . . . 5  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
5 dvhfvadd.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
61, 2, 3, 4, 5dvhset 31879 . . . 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 5732 . . 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 31880 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
1110fveq2d 5732 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( +g  `  D
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
128, 11syl5eq 2480 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
.+^  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
1312oveqd 6098 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( 2nd `  f
)  .+^  ( 2nd `  g
) )  =  ( ( 2nd `  f
) ( +g  `  (
( EDRing `  K ) `  W ) ) ( 2nd `  g ) ) )
14133ad2ant1 978 . . . . . . 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 6377 . . . . . . . . . 10  |-  ( f  e.  ( T  X.  E )  ->  ( 2nd `  f )  e.  E )
16 xp2nd 6377 . . . . . . . . . 10  |-  ( g  e.  ( T  X.  E )  ->  ( 2nd `  g )  e.  E )
1715, 16anim12i 550 . . . . . . . . 9  |-  ( ( f  e.  ( T  X.  E )  /\  g  e.  ( T  X.  E ) )  -> 
( ( 2nd `  f
)  e.  E  /\  ( 2nd `  g )  e.  E ) )
18 eqid 2436 . . . . . . . . . 10  |-  ( +g  `  ( ( EDRing `  K
) `  W )
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
)
191, 2, 3, 4, 18erngplus 31600 . . . . . . . . 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 461 . . . . . . . 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 1149 . . . . . . 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 2468 . . . . . 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 3991 . . . . 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 6138 . . . 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 5742 . . . . . . . 8  |-  ( (
LTrn `  K ) `  W )  e.  _V
262, 25eqeltri 2506 . . . . . . 7  |-  T  e. 
_V
27 fvex 5742 . . . . . . . 8  |-  ( (
TEndo `  K ) `  W )  e.  _V
283, 27eqeltri 2506 . . . . . . 7  |-  E  e. 
_V
2926, 28xpex 4990 . . . . . 6  |-  ( T  X.  E )  e. 
_V
3029, 29mpt2ex 6425 . . . . 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 2436 . . . . . 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 13595 . . . . 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 8 . . . 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 2485 . . 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 2468 . 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 2493 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .+  =  .+b  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956    u. cun 3318   {csn 3814   {ctp 3816   <.cop 3817    e. cmpt 4266    X. cxp 4876    o. ccom 4882   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348   ndxcnx 13466   Basecbs 13469   +g cplusg 13529  Scalarcsca 13532   .scvsca 13533   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   TEndoctendo 31549   EDRingcedring 31550   DVecHcdvh 31876
This theorem is referenced by:  dvhvadd  31890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-edring 31554  df-dvech 31877
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