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Theorem dvhvaddval 31727
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 5719 . . . 4  |-  ( h  =  F  ->  ( 1st `  h )  =  ( 1st `  F
) )
21coeq1d 5025 . . 3  |-  ( h  =  F  ->  (
( 1st `  h
)  o.  ( 1st `  i ) )  =  ( ( 1st `  F
)  o.  ( 1st `  i ) ) )
3 fveq2 5719 . . . 4  |-  ( h  =  F  ->  ( 2nd `  h )  =  ( 2nd `  F
) )
43oveq1d 6087 . . 3  |-  ( h  =  F  ->  (
( 2nd `  h
)  .+^  ( 2nd `  i
) )  =  ( ( 2nd `  F
)  .+^  ( 2nd `  i
) ) )
52, 4opeq12d 3984 . 2  |-  ( h  =  F  ->  <. (
( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >.  =  <. ( ( 1st `  F
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  i
) ) >. )
6 fveq2 5719 . . . 4  |-  ( i  =  G  ->  ( 1st `  i )  =  ( 1st `  G
) )
76coeq2d 5026 . . 3  |-  ( i  =  G  ->  (
( 1st `  F
)  o.  ( 1st `  i ) )  =  ( ( 1st `  F
)  o.  ( 1st `  G ) ) )
8 fveq2 5719 . . . 4  |-  ( i  =  G  ->  ( 2nd `  i )  =  ( 2nd `  G
) )
98oveq2d 6088 . . 3  |-  ( i  =  G  ->  (
( 2nd `  F
)  .+^  ( 2nd `  i
) )  =  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) )
107, 9opeq12d 3984 . 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 31726 . 2  |-  .+  =  ( h  e.  ( T  X.  E ) ,  i  e.  ( T  X.  E )  |->  <.
( ( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >. )
13 opex 4419 . 2  |-  <. (
( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >.  e.  _V
145, 10, 12, 13ovmpt2 6200 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 359    = wceq 1652    e. wcel 1725   <.cop 3809    X. cxp 4867    o. ccom 4873   ` cfv 5445  (class class class)co 6072    e. cmpt2 6074   1stc1st 6338   2ndc2nd 6339
This theorem is referenced by:  dvhvadd  31729  dvhopaddN  31751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-iota 5409  df-fun 5447  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077
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