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Theorem dvhvscaval 31289
Description: The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)
Hypothesis
Ref Expression
dvhvscaval.s  |-  .x.  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )
Assertion
Ref Expression
dvhvscaval  |-  ( ( U  e.  E  /\  F  e.  ( T  X.  E ) )  -> 
( U  .x.  F
)  =  <. ( U `  ( 1st `  F ) ) ,  ( U  o.  ( 2nd `  F ) )
>. )
Distinct variable groups:    f, s, E    T, s, f
Allowed substitution hints:    .x. ( f, s)    U( f, s)    F( f, s)

Proof of Theorem dvhvscaval
Dummy variables  t 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq1 5524 . . 3  |-  ( t  =  U  ->  (
t `  ( 1st `  g ) )  =  ( U `  ( 1st `  g ) ) )
2 coeq1 4841 . . 3  |-  ( t  =  U  ->  (
t  o.  ( 2nd `  g ) )  =  ( U  o.  ( 2nd `  g ) ) )
31, 2opeq12d 3804 . 2  |-  ( t  =  U  ->  <. (
t `  ( 1st `  g ) ) ,  ( t  o.  ( 2nd `  g ) )
>.  =  <. ( U `
 ( 1st `  g
) ) ,  ( U  o.  ( 2nd `  g ) ) >.
)
4 fveq2 5525 . . . 4  |-  ( g  =  F  ->  ( 1st `  g )  =  ( 1st `  F
) )
54fveq2d 5529 . . 3  |-  ( g  =  F  ->  ( U `  ( 1st `  g ) )  =  ( U `  ( 1st `  F ) ) )
6 fveq2 5525 . . . 4  |-  ( g  =  F  ->  ( 2nd `  g )  =  ( 2nd `  F
) )
76coeq2d 4846 . . 3  |-  ( g  =  F  ->  ( U  o.  ( 2nd `  g ) )  =  ( U  o.  ( 2nd `  F ) ) )
85, 7opeq12d 3804 . 2  |-  ( g  =  F  ->  <. ( U `  ( 1st `  g ) ) ,  ( U  o.  ( 2nd `  g ) )
>.  =  <. ( U `
 ( 1st `  F
) ) ,  ( U  o.  ( 2nd `  F ) ) >.
)
9 dvhvscaval.s . . 3  |-  .x.  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )
109dvhvscacbv 31288 . 2  |-  .x.  =  ( t  e.  E ,  g  e.  ( T  X.  E )  |->  <.
( t `  ( 1st `  g ) ) ,  ( t  o.  ( 2nd `  g
) ) >. )
11 opex 4237 . 2  |-  <. ( U `  ( 1st `  F ) ) ,  ( U  o.  ( 2nd `  F ) )
>.  e.  _V
123, 8, 10, 11ovmpt2 5983 1  |-  ( ( U  e.  E  /\  F  e.  ( T  X.  E ) )  -> 
( U  .x.  F
)  =  <. ( U `  ( 1st `  F ) ) ,  ( U  o.  ( 2nd `  F ) )
>. )
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:  dvhvsca  31291  dvhopspN  31305
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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