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Theorem dvhfvadd 35075
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 2454 . . . . 5  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
5 dvhfvadd.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
61, 2, 3, 4, 5dvhset 35065 . . . 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 5804 . . 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 35066 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
1110fveq2d 5804 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( +g  `  D
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
128, 11syl5eq 2507 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
.+^  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
1312oveqd 6218 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( 2nd `  f
)  .+^  ( 2nd `  g
) )  =  ( ( 2nd `  f
) ( +g  `  (
( EDRing `  K ) `  W ) ) ( 2nd `  g ) ) )
14133ad2ant1 1009 . . . . . . 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 6718 . . . . . . . . . 10  |-  ( f  e.  ( T  X.  E )  ->  ( 2nd `  f )  e.  E )
16 xp2nd 6718 . . . . . . . . . 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 2454 . . . . . . . . . 10  |-  ( +g  `  ( ( EDRing `  K
) `  W )
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
)
191, 2, 3, 4, 18erngplus 34786 . . . . . . . . 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 1184 . . . . . . 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 2495 . . . . . 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 4175 . . . . 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 6260 . . . 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 5810 . . . . . . . 8  |-  ( (
LTrn `  K ) `  W )  e.  _V
262, 25eqeltri 2538 . . . . . . 7  |-  T  e. 
_V
27 fvex 5810 . . . . . . . 8  |-  ( (
TEndo `  K ) `  W )  e.  _V
283, 27eqeltri 2538 . . . . . . 7  |-  E  e. 
_V
2926, 28xpex 6619 . . . . . 6  |-  ( T  X.  E )  e. 
_V
3029, 29mpt2ex 6761 . . . . 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 2454 . . . . . 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 14424 . . . . 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 2512 . . 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 2495 . 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 2520 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .+  =  .+b  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3078    u. cun 3435   {csn 3986   {ctp 3990   <.cop 3992    |-> cmpt 4459    X. cxp 4947    o. ccom 4953   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203   1stc1st 6686   2ndc2nd 6687   ndxcnx 14290   Basecbs 14293   +g cplusg 14358  Scalarcsca 14361   .scvsca 14362   HLchlt 33334   LHypclh 33967   LTrncltrn 34084   TEndoctendo 34735   EDRingcedring 34736   DVecHcdvh 35062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-plusg 14371  df-mulr 14372  df-sca 14374  df-vsca 14375  df-edring 34740  df-dvech 35063
This theorem is referenced by:  dvhvadd  35076
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