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Theorem dvhvadd 31100
Description: The vector sum operation for the constructed full vector space H. (Contributed by NM, 11-Feb-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Hypotheses
Ref Expression
dvhvadd.h  |-  H  =  ( LHyp `  K
)
dvhvadd.t  |-  T  =  ( ( LTrn `  K
) `  W )
dvhvadd.e  |-  E  =  ( ( TEndo `  K
) `  W )
dvhvadd.u  |-  U  =  ( ( DVecH `  K
) `  W )
dvhvadd.f  |-  D  =  (Scalar `  U )
dvhvadd.s  |-  .+  =  ( +g  `  U )
dvhvadd.p  |-  .+^  =  ( +g  `  D )
Assertion
Ref Expression
dvhvadd  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( F  .+  G
)  =  <. (
( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >. )

Proof of Theorem dvhvadd
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvhvadd.h . . . 4  |-  H  =  ( LHyp `  K
)
2 dvhvadd.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
3 dvhvadd.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
4 dvhvadd.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
5 dvhvadd.f . . . 4  |-  D  =  (Scalar `  U )
6 dvhvadd.p . . . 4  |-  .+^  =  ( +g  `  D )
7 eqid 2316 . . . 4  |-  ( 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 ) ) ,  ( ( 2nd `  f )  .+^  ( 2nd `  g ) ) >.
)
8 dvhvadd.s . . . 4  |-  .+  =  ( +g  `  U )
91, 2, 3, 4, 5, 6, 7, 8dvhfvadd 31099 . . 3  |-  ( ( 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
) ) >. )
)
109oveqd 5917 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( F  .+  G
)  =  ( F ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E ) 
|->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f )  .+^  ( 2nd `  g ) ) >.
) G ) )
117dvhvaddval 31098 . 2  |-  ( ( F  e.  ( T  X.  E )  /\  G  e.  ( T  X.  E ) )  -> 
( F ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E )  |->  <. (
( 1st `  f
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f
)  .+^  ( 2nd `  g
) ) >. ) G )  =  <. ( ( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >. )
1210, 11sylan9eq 2368 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( F  .+  G
)  =  <. (
( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   <.cop 3677    X. cxp 4724    o. ccom 4730   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   1stc1st 6162   2ndc2nd 6163   +g cplusg 13255  Scalarcsca 13258   HLchlt 29358   LHypclh 29991   LTrncltrn 30108   TEndoctendo 30759   DVecHcdvh 31086
This theorem is referenced by:  dvhopvadd  31101  dvhvaddcl  31103  dvhvaddcomN  31104  dvhvaddass  31105  dvhlveclem  31116  diblss  31178  dicvaddcl  31198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-plusg 13268  df-mulr 13269  df-sca 13271  df-vsca 13272  df-edring 30764  df-dvech 31087
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