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Theorem dvhopspN 31232
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 4852 . . 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 31216 . . 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 461 . 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 6300 . . . . 5  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 1st `  <. F ,  U >. )  =  F )
65fveq2d 5674 . . . 4  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( R `  ( 1st `  <. F ,  U >. ) )  =  ( R `  F ) )
7 op2ndg 6301 . . . . 5  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 2nd `  <. F ,  U >. )  =  U )
87coeq2d 4977 . . . 4  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( R  o.  ( 2nd `  <. F ,  U >. ) )  =  ( R  o.  U ) )
96, 8opeq12d 3936 . . 3  |-  ( ( F  e.  T  /\  U  e.  E )  -> 
<. ( R `  ( 1st `  <. F ,  U >. ) ) ,  ( R  o.  ( 2nd `  <. F ,  U >. ) ) >.  =  <. ( R `  F ) ,  ( R  o.  U ) >. )
109adantl 453 . 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 2421 1  |-  ( ( R  e.  E  /\  ( F  e.  T  /\  U  e.  E
) )  ->  ( R S <. F ,  U >. )  =  <. ( R `  F ) ,  ( R  o.  U ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   <.cop 3762    X. cxp 4818    o. ccom 4824   ` cfv 5396  (class class class)co 6022    e. cmpt2 6024   1stc1st 6288   2ndc2nd 6289
This theorem is referenced by:  dvhopN  31233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291
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