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Theorem dvhopspN 37239
Description: Scalar product of  DVecH vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
dvhopsp.s  |-  S  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )
Assertion
Ref Expression
dvhopspN  |-  ( ( R  e.  E  /\  ( F  e.  T  /\  U  e.  E
) )  ->  ( R S <. F ,  U >. )  =  <. ( R `  F ) ,  ( R  o.  U ) >. )
Distinct variable groups:    f, s, E    T, f, s
Allowed substitution hints:    R( f, s)    S( f, s)    U( f, s)    F( f, s)

Proof of Theorem dvhopspN
StepHypRef Expression
1 opelxpi 5020 . . 3  |-  ( ( F  e.  T  /\  U  e.  E )  -> 
<. F ,  U >.  e.  ( T  X.  E
) )
2 dvhopsp.s . . . 4  |-  S  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )
32dvhvscaval 37223 . . 3  |-  ( ( R  e.  E  /\  <. F ,  U >.  e.  ( T  X.  E
) )  ->  ( R S <. F ,  U >. )  =  <. ( R `  ( 1st ` 
<. F ,  U >. ) ) ,  ( R  o.  ( 2nd `  <. F ,  U >. )
) >. )
41, 3sylan2 472 . 2  |-  ( ( R  e.  E  /\  ( F  e.  T  /\  U  e.  E
) )  ->  ( R S <. F ,  U >. )  =  <. ( R `  ( 1st ` 
<. F ,  U >. ) ) ,  ( R  o.  ( 2nd `  <. F ,  U >. )
) >. )
5 op1stg 6785 . . . . 5  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 1st `  <. F ,  U >. )  =  F )
65fveq2d 5852 . . . 4  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( R `  ( 1st `  <. F ,  U >. ) )  =  ( R `  F ) )
7 op2ndg 6786 . . . . 5  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 2nd `  <. F ,  U >. )  =  U )
87coeq2d 5154 . . . 4  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( R  o.  ( 2nd `  <. F ,  U >. ) )  =  ( R  o.  U ) )
96, 8opeq12d 4211 . . 3  |-  ( ( F  e.  T  /\  U  e.  E )  -> 
<. ( R `  ( 1st `  <. F ,  U >. ) ) ,  ( R  o.  ( 2nd `  <. F ,  U >. ) ) >.  =  <. ( R `  F ) ,  ( R  o.  U ) >. )
109adantl 464 . 2  |-  ( ( R  e.  E  /\  ( F  e.  T  /\  U  e.  E
) )  ->  <. ( R `  ( 1st ` 
<. F ,  U >. ) ) ,  ( R  o.  ( 2nd `  <. F ,  U >. )
) >.  =  <. ( R `  F ) ,  ( R  o.  U ) >. )
114, 10eqtrd 2495 1  |-  ( ( R  e.  E  /\  ( F  e.  T  /\  U  e.  E
) )  ->  ( R S <. F ,  U >. )  =  <. ( R `  F ) ,  ( R  o.  U ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   <.cop 4022    X. cxp 4986    o. ccom 4992   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   1stc1st 6771   2ndc2nd 6772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774
This theorem is referenced by:  dvhopN  37240
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