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Theorem dvhopspN 31305
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 4721 . . 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 31289 . . 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 460 . 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 6132 . . . . 5  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 1st `  <. F ,  U >. )  =  F )
65fveq2d 5529 . . . 4  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( R `  ( 1st `  <. F ,  U >. ) )  =  ( R `  F ) )
7 op2ndg 6133 . . . . 5  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 2nd `  <. F ,  U >. )  =  U )
87coeq2d 4846 . . . 4  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( R  o.  ( 2nd `  <. F ,  U >. ) )  =  ( R  o.  U ) )
96, 8opeq12d 3804 . . 3  |-  ( ( F  e.  T  /\  U  e.  E )  -> 
<. ( R `  ( 1st `  <. F ,  U >. ) ) ,  ( R  o.  ( 2nd `  <. F ,  U >. ) ) >.  =  <. ( R `  F ) ,  ( R  o.  U ) >. )
109adantl 452 . 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 2315 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 358    = wceq 1623    e. wcel 1684   <.cop 3643    X. cxp 4687    o. ccom 4693   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  dvhopN  31306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123
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