Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvhvaddval Unicode version

Theorem dvhvaddval 31205
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 5668 . . . 4  |-  ( h  =  F  ->  ( 1st `  h )  =  ( 1st `  F
) )
21coeq1d 4974 . . 3  |-  ( h  =  F  ->  (
( 1st `  h
)  o.  ( 1st `  i ) )  =  ( ( 1st `  F
)  o.  ( 1st `  i ) ) )
3 fveq2 5668 . . . 4  |-  ( h  =  F  ->  ( 2nd `  h )  =  ( 2nd `  F
) )
43oveq1d 6035 . . 3  |-  ( h  =  F  ->  (
( 2nd `  h
)  .+^  ( 2nd `  i
) )  =  ( ( 2nd `  F
)  .+^  ( 2nd `  i
) ) )
52, 4opeq12d 3934 . 2  |-  ( h  =  F  ->  <. (
( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >.  =  <. ( ( 1st `  F
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  i
) ) >. )
6 fveq2 5668 . . . 4  |-  ( i  =  G  ->  ( 1st `  i )  =  ( 1st `  G
) )
76coeq2d 4975 . . 3  |-  ( i  =  G  ->  (
( 1st `  F
)  o.  ( 1st `  i ) )  =  ( ( 1st `  F
)  o.  ( 1st `  G ) ) )
8 fveq2 5668 . . . 4  |-  ( i  =  G  ->  ( 2nd `  i )  =  ( 2nd `  G
) )
98oveq2d 6036 . . 3  |-  ( i  =  G  ->  (
( 2nd `  F
)  .+^  ( 2nd `  i
) )  =  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) )
107, 9opeq12d 3934 . 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 31204 . 2  |-  .+  =  ( h  e.  ( T  X.  E ) ,  i  e.  ( T  X.  E )  |->  <.
( ( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >. )
13 opex 4368 . 2  |-  <. (
( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >.  e.  _V
145, 10, 12, 13ovmpt2 6148 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 1649    e. wcel 1717   <.cop 3760    X. cxp 4816    o. ccom 4822   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   1stc1st 6286   2ndc2nd 6287
This theorem is referenced by:  dvhvadd  31207  dvhopaddN  31229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025
  Copyright terms: Public domain W3C validator