Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvhfvadd Unicode version

Theorem dvhfvadd 31281
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 2283 . . . . 5  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
5 dvhfvadd.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
61, 2, 3, 4, 5dvhset 31271 . . . 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 5529 . . 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 31272 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
1110fveq2d 5529 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( +g  `  D
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
128, 11syl5eq 2327 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
.+^  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
1312oveqd 5875 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( 2nd `  f
)  .+^  ( 2nd `  g
) )  =  ( ( 2nd `  f
) ( +g  `  (
( EDRing `  K ) `  W ) ) ( 2nd `  g ) ) )
14133ad2ant1 976 . . . . . . 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 6150 . . . . . . . . . 10  |-  ( f  e.  ( T  X.  E )  ->  ( 2nd `  f )  e.  E )
16 xp2nd 6150 . . . . . . . . . 10  |-  ( g  e.  ( T  X.  E )  ->  ( 2nd `  g )  e.  E )
1715, 16anim12i 549 . . . . . . . . 9  |-  ( ( f  e.  ( T  X.  E )  /\  g  e.  ( T  X.  E ) )  -> 
( ( 2nd `  f
)  e.  E  /\  ( 2nd `  g )  e.  E ) )
18 eqid 2283 . . . . . . . . . 10  |-  ( +g  `  ( ( EDRing `  K
) `  W )
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
)
191, 2, 3, 4, 18erngplus 30992 . . . . . . . . 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 460 . . . . . . . 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 1147 . . . . . . 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 2315 . . . . . 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 3803 . . . . 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 5912 . . . 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 5539 . . . . . . . 8  |-  ( (
LTrn `  K ) `  W )  e.  _V
262, 25eqeltri 2353 . . . . . . 7  |-  T  e. 
_V
27 fvex 5539 . . . . . . . 8  |-  ( (
TEndo `  K ) `  W )  e.  _V
283, 27eqeltri 2353 . . . . . . 7  |-  E  e. 
_V
2926, 28xpex 4801 . . . . . 6  |-  ( T  X.  E )  e. 
_V
3029, 29mpt2ex 6198 . . . . 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 2283 . . . . . 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 13274 . . . . 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 2332 . . 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 2315 . 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 2340 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .+  =  .+b  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150   {csn 3640   {ctp 3642   <.cop 3643    e. cmpt 4077    X. cxp 4687    o. ccom 4693   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   ndxcnx 13145   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941   EDRingcedring 30942   DVecHcdvh 31268
This theorem is referenced by:  dvhvadd  31282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-edring 30946  df-dvech 31269
  Copyright terms: Public domain W3C validator