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Theorem dvhvaddval 30653
Description: The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.)
Hypothesis
Ref Expression
dvhvaddval.a  |-  .+  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E ) 
|->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f )  .+^  ( 2nd `  g ) ) >.
)
Assertion
Ref Expression
dvhvaddval  |-  ( ( F  e.  ( T  X.  E )  /\  G  e.  ( T  X.  E ) )  -> 
( F  .+  G
)  =  <. (
( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >. )
Distinct variable groups:    f, g, E   
.+^ , f, g    T, f, g
Allowed substitution hints:    .+ ( f, g)    F( f, g)    G( f, g)

Proof of Theorem dvhvaddval
Dummy variables  h  i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . 4  |-  ( h  =  F  ->  ( 1st `  h )  =  ( 1st `  F
) )
21coeq1d 4845 . . 3  |-  ( h  =  F  ->  (
( 1st `  h
)  o.  ( 1st `  i ) )  =  ( ( 1st `  F
)  o.  ( 1st `  i ) ) )
3 fveq2 5525 . . . 4  |-  ( h  =  F  ->  ( 2nd `  h )  =  ( 2nd `  F
) )
43oveq1d 5873 . . 3  |-  ( h  =  F  ->  (
( 2nd `  h
)  .+^  ( 2nd `  i
) )  =  ( ( 2nd `  F
)  .+^  ( 2nd `  i
) ) )
52, 4opeq12d 3804 . 2  |-  ( h  =  F  ->  <. (
( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >.  =  <. ( ( 1st `  F
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  i
) ) >. )
6 fveq2 5525 . . . 4  |-  ( i  =  G  ->  ( 1st `  i )  =  ( 1st `  G
) )
76coeq2d 4846 . . 3  |-  ( i  =  G  ->  (
( 1st `  F
)  o.  ( 1st `  i ) )  =  ( ( 1st `  F
)  o.  ( 1st `  G ) ) )
8 fveq2 5525 . . . 4  |-  ( i  =  G  ->  ( 2nd `  i )  =  ( 2nd `  G
) )
98oveq2d 5874 . . 3  |-  ( i  =  G  ->  (
( 2nd `  F
)  .+^  ( 2nd `  i
) )  =  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) )
107, 9opeq12d 3804 . 2  |-  ( i  =  G  ->  <. (
( 1st `  F
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  i
) ) >.  =  <. ( ( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >. )
11 dvhvaddval.a . . 3  |-  .+  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E ) 
|->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f )  .+^  ( 2nd `  g ) ) >.
)
1211dvhvaddcbv 30652 . 2  |-  .+  =  ( h  e.  ( T  X.  E ) ,  i  e.  ( T  X.  E )  |->  <.
( ( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >. )
13 opex 4237 . 2  |-  <. (
( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >.  e.  _V
145, 10, 12, 13ovmpt2 5983 1  |-  ( ( 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 1623    e. wcel 1684   <.cop 3643    X. cxp 4687    o. ccom 4693   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  dvhvadd  30655  dvhopaddN  30677
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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