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

Theorem dvhvscaval 34420
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 5871 . . 3  |-  ( t  =  U  ->  (
t `  ( 1st `  g ) )  =  ( U `  ( 1st `  g ) ) )
2 coeq1 5003 . . 3  |-  ( t  =  U  ->  (
t  o.  ( 2nd `  g ) )  =  ( U  o.  ( 2nd `  g ) ) )
31, 2opeq12d 4189 . 2  |-  ( t  =  U  ->  <. (
t `  ( 1st `  g ) ) ,  ( t  o.  ( 2nd `  g ) )
>.  =  <. ( U `
 ( 1st `  g
) ) ,  ( U  o.  ( 2nd `  g ) ) >.
)
4 fveq2 5872 . . . 4  |-  ( g  =  F  ->  ( 1st `  g )  =  ( 1st `  F
) )
54fveq2d 5876 . . 3  |-  ( g  =  F  ->  ( U `  ( 1st `  g ) )  =  ( U `  ( 1st `  F ) ) )
6 fveq2 5872 . . . 4  |-  ( g  =  F  ->  ( 2nd `  g )  =  ( 2nd `  F
) )
76coeq2d 5008 . . 3  |-  ( g  =  F  ->  ( U  o.  ( 2nd `  g ) )  =  ( U  o.  ( 2nd `  F ) ) )
85, 7opeq12d 4189 . 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 34419 . 2  |-  .x.  =  ( t  e.  E ,  g  e.  ( T  X.  E )  |->  <.
( t `  ( 1st `  g ) ) ,  ( t  o.  ( 2nd `  g
) ) >. )
11 opex 4677 . 2  |-  <. ( U `  ( 1st `  F ) ) ,  ( U  o.  ( 2nd `  F ) )
>.  e.  _V
123, 8, 10, 11ovmpt2 6437 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 370    = wceq 1437    e. wcel 1867   <.cop 3999    X. cxp 4843    o. ccom 4849   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6796   2ndc2nd 6797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5556  df-fun 5594  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301
This theorem is referenced by:  dvhvsca  34422  dvhopspN  34436
  Copyright terms: Public domain W3C validator