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Theorem dvhvscaval 31911
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 5540 . . 3  |-  ( t  =  U  ->  (
t `  ( 1st `  g ) )  =  ( U `  ( 1st `  g ) ) )
2 coeq1 4857 . . 3  |-  ( t  =  U  ->  (
t  o.  ( 2nd `  g ) )  =  ( U  o.  ( 2nd `  g ) ) )
31, 2opeq12d 3820 . 2  |-  ( t  =  U  ->  <. (
t `  ( 1st `  g ) ) ,  ( t  o.  ( 2nd `  g ) )
>.  =  <. ( U `
 ( 1st `  g
) ) ,  ( U  o.  ( 2nd `  g ) ) >.
)
4 fveq2 5541 . . . 4  |-  ( g  =  F  ->  ( 1st `  g )  =  ( 1st `  F
) )
54fveq2d 5545 . . 3  |-  ( g  =  F  ->  ( U `  ( 1st `  g ) )  =  ( U `  ( 1st `  F ) ) )
6 fveq2 5541 . . . 4  |-  ( g  =  F  ->  ( 2nd `  g )  =  ( 2nd `  F
) )
76coeq2d 4862 . . 3  |-  ( g  =  F  ->  ( U  o.  ( 2nd `  g ) )  =  ( U  o.  ( 2nd `  F ) ) )
85, 7opeq12d 3820 . 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 31910 . 2  |-  .x.  =  ( t  e.  E ,  g  e.  ( T  X.  E )  |->  <.
( t `  ( 1st `  g ) ) ,  ( t  o.  ( 2nd `  g
) ) >. )
11 opex 4253 . 2  |-  <. ( U `  ( 1st `  F ) ) ,  ( U  o.  ( 2nd `  F ) )
>.  e.  _V
123, 8, 10, 11ovmpt2 5999 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 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:  dvhvsca  31913  dvhopspN  31927
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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