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Theorem dvhvaddval 31902
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 5541 . . . 4  |-  ( h  =  F  ->  ( 1st `  h )  =  ( 1st `  F
) )
21coeq1d 4861 . . 3  |-  ( h  =  F  ->  (
( 1st `  h
)  o.  ( 1st `  i ) )  =  ( ( 1st `  F
)  o.  ( 1st `  i ) ) )
3 fveq2 5541 . . . 4  |-  ( h  =  F  ->  ( 2nd `  h )  =  ( 2nd `  F
) )
43oveq1d 5889 . . 3  |-  ( h  =  F  ->  (
( 2nd `  h
)  .+^  ( 2nd `  i
) )  =  ( ( 2nd `  F
)  .+^  ( 2nd `  i
) ) )
52, 4opeq12d 3820 . 2  |-  ( h  =  F  ->  <. (
( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >.  =  <. ( ( 1st `  F
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  i
) ) >. )
6 fveq2 5541 . . . 4  |-  ( i  =  G  ->  ( 1st `  i )  =  ( 1st `  G
) )
76coeq2d 4862 . . 3  |-  ( i  =  G  ->  (
( 1st `  F
)  o.  ( 1st `  i ) )  =  ( ( 1st `  F
)  o.  ( 1st `  G ) ) )
8 fveq2 5541 . . . 4  |-  ( i  =  G  ->  ( 2nd `  i )  =  ( 2nd `  G
) )
98oveq2d 5890 . . 3  |-  ( i  =  G  ->  (
( 2nd `  F
)  .+^  ( 2nd `  i
) )  =  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) )
107, 9opeq12d 3820 . 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 31901 . 2  |-  .+  =  ( h  e.  ( T  X.  E ) ,  i  e.  ( T  X.  E )  |->  <.
( ( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >. )
13 opex 4253 . 2  |-  <. (
( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >.  e.  _V
145, 10, 12, 13ovmpt2 5999 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 1632    e. wcel 1696   <.cop 3656    X. cxp 4703    o. ccom 4709   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  dvhvadd  31904  dvhopaddN  31926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879
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