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Theorem dvhvaddval 31962
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 5731 . . . 4  |-  ( h  =  F  ->  ( 1st `  h )  =  ( 1st `  F
) )
21coeq1d 5037 . . 3  |-  ( h  =  F  ->  (
( 1st `  h
)  o.  ( 1st `  i ) )  =  ( ( 1st `  F
)  o.  ( 1st `  i ) ) )
3 fveq2 5731 . . . 4  |-  ( h  =  F  ->  ( 2nd `  h )  =  ( 2nd `  F
) )
43oveq1d 6099 . . 3  |-  ( h  =  F  ->  (
( 2nd `  h
)  .+^  ( 2nd `  i
) )  =  ( ( 2nd `  F
)  .+^  ( 2nd `  i
) ) )
52, 4opeq12d 3994 . 2  |-  ( h  =  F  ->  <. (
( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >.  =  <. ( ( 1st `  F
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  i
) ) >. )
6 fveq2 5731 . . . 4  |-  ( i  =  G  ->  ( 1st `  i )  =  ( 1st `  G
) )
76coeq2d 5038 . . 3  |-  ( i  =  G  ->  (
( 1st `  F
)  o.  ( 1st `  i ) )  =  ( ( 1st `  F
)  o.  ( 1st `  G ) ) )
8 fveq2 5731 . . . 4  |-  ( i  =  G  ->  ( 2nd `  i )  =  ( 2nd `  G
) )
98oveq2d 6100 . . 3  |-  ( i  =  G  ->  (
( 2nd `  F
)  .+^  ( 2nd `  i
) )  =  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) )
107, 9opeq12d 3994 . 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 31961 . 2  |-  .+  =  ( h  e.  ( T  X.  E ) ,  i  e.  ( T  X.  E )  |->  <.
( ( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >. )
13 opex 4430 . 2  |-  <. (
( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >.  e.  _V
145, 10, 12, 13ovmpt2 6212 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 360    = wceq 1653    e. wcel 1726   <.cop 3819    X. cxp 4879    o. ccom 4885   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351
This theorem is referenced by:  dvhvadd  31964  dvhopaddN  31986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089
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