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Theorem dvhvscaval 36113
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 5865 . . 3  |-  ( t  =  U  ->  (
t `  ( 1st `  g ) )  =  ( U `  ( 1st `  g ) ) )
2 coeq1 5160 . . 3  |-  ( t  =  U  ->  (
t  o.  ( 2nd `  g ) )  =  ( U  o.  ( 2nd `  g ) ) )
31, 2opeq12d 4221 . 2  |-  ( t  =  U  ->  <. (
t `  ( 1st `  g ) ) ,  ( t  o.  ( 2nd `  g ) )
>.  =  <. ( U `
 ( 1st `  g
) ) ,  ( U  o.  ( 2nd `  g ) ) >.
)
4 fveq2 5866 . . . 4  |-  ( g  =  F  ->  ( 1st `  g )  =  ( 1st `  F
) )
54fveq2d 5870 . . 3  |-  ( g  =  F  ->  ( U `  ( 1st `  g ) )  =  ( U `  ( 1st `  F ) ) )
6 fveq2 5866 . . . 4  |-  ( g  =  F  ->  ( 2nd `  g )  =  ( 2nd `  F
) )
76coeq2d 5165 . . 3  |-  ( g  =  F  ->  ( U  o.  ( 2nd `  g ) )  =  ( U  o.  ( 2nd `  F ) ) )
85, 7opeq12d 4221 . 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 36112 . 2  |-  .x.  =  ( t  e.  E ,  g  e.  ( T  X.  E )  |->  <.
( t `  ( 1st `  g ) ) ,  ( t  o.  ( 2nd `  g
) ) >. )
11 opex 4711 . 2  |-  <. ( U `  ( 1st `  F ) ) ,  ( U  o.  ( 2nd `  F ) )
>.  e.  _V
123, 8, 10, 11ovmpt2 6423 1  |-  ( ( U  e.  E  /\  F  e.  ( T  X.  E ) )  -> 
( U  .x.  F
)  =  <. ( U `  ( 1st `  F ) ) ,  ( U  o.  ( 2nd `  F ) )
>. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4033    X. cxp 4997    o. ccom 5003   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   1stc1st 6783   2ndc2nd 6784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290
This theorem is referenced by:  dvhvsca  36115  dvhopspN  36129
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