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

Theorem dvhvscaval 31834
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 5719 . . 3  |-  ( t  =  U  ->  (
t `  ( 1st `  g ) )  =  ( U `  ( 1st `  g ) ) )
2 coeq1 5022 . . 3  |-  ( t  =  U  ->  (
t  o.  ( 2nd `  g ) )  =  ( U  o.  ( 2nd `  g ) ) )
31, 2opeq12d 3984 . 2  |-  ( t  =  U  ->  <. (
t `  ( 1st `  g ) ) ,  ( t  o.  ( 2nd `  g ) )
>.  =  <. ( U `
 ( 1st `  g
) ) ,  ( U  o.  ( 2nd `  g ) ) >.
)
4 fveq2 5720 . . . 4  |-  ( g  =  F  ->  ( 1st `  g )  =  ( 1st `  F
) )
54fveq2d 5724 . . 3  |-  ( g  =  F  ->  ( U `  ( 1st `  g ) )  =  ( U `  ( 1st `  F ) ) )
6 fveq2 5720 . . . 4  |-  ( g  =  F  ->  ( 2nd `  g )  =  ( 2nd `  F
) )
76coeq2d 5027 . . 3  |-  ( g  =  F  ->  ( U  o.  ( 2nd `  g ) )  =  ( U  o.  ( 2nd `  F ) ) )
85, 7opeq12d 3984 . 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 31833 . 2  |-  .x.  =  ( t  e.  E ,  g  e.  ( T  X.  E )  |->  <.
( t `  ( 1st `  g ) ) ,  ( t  o.  ( 2nd `  g
) ) >. )
11 opex 4419 . 2  |-  <. ( U `  ( 1st `  F ) ) ,  ( U  o.  ( 2nd `  F ) )
>.  e.  _V
123, 8, 10, 11ovmpt2 6201 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 359    = wceq 1652    e. wcel 1725   <.cop 3809    X. cxp 4868    o. ccom 4874   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340
This theorem is referenced by:  dvhvsca  31836  dvhopspN  31850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078
  Copyright terms: Public domain W3C validator