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Theorem dvhopspN 34754
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 4871 . . 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 34738 . . 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 482 . 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 6824 . . . . 5  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 1st `  <. F ,  U >. )  =  F )
65fveq2d 5883 . . . 4  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( R `  ( 1st `  <. F ,  U >. ) )  =  ( R `  F ) )
7 op2ndg 6825 . . . . 5  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 2nd `  <. F ,  U >. )  =  U )
87coeq2d 5002 . . . 4  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( R  o.  ( 2nd `  <. F ,  U >. ) )  =  ( R  o.  U ) )
96, 8opeq12d 4166 . . 3  |-  ( ( F  e.  T  /\  U  e.  E )  -> 
<. ( R `  ( 1st `  <. F ,  U >. ) ) ,  ( R  o.  ( 2nd `  <. F ,  U >. ) ) >.  =  <. ( R `  F ) ,  ( R  o.  U ) >. )
109adantl 473 . 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 2505 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 376    = wceq 1452    e. wcel 1904   <.cop 3965    X. cxp 4837    o. ccom 4843   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813
This theorem is referenced by:  dvhopN  34755
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