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Theorem dvhopspN 35069
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 4972 . . 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 35053 . . 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 474 . 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 6692 . . . . 5  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 1st `  <. F ,  U >. )  =  F )
65fveq2d 5796 . . . 4  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( R `  ( 1st `  <. F ,  U >. ) )  =  ( R `  F ) )
7 op2ndg 6693 . . . . 5  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 2nd `  <. F ,  U >. )  =  U )
87coeq2d 5103 . . . 4  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( R  o.  ( 2nd `  <. F ,  U >. ) )  =  ( R  o.  U ) )
96, 8opeq12d 4168 . . 3  |-  ( ( F  e.  T  /\  U  e.  E )  -> 
<. ( R `  ( 1st `  <. F ,  U >. ) ) ,  ( R  o.  ( 2nd `  <. F ,  U >. ) ) >.  =  <. ( R `  F ) ,  ( R  o.  U ) >. )
109adantl 466 . 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 2492 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 369    = wceq 1370    e. wcel 1758   <.cop 3984    X. cxp 4939    o. ccom 4945   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195   1stc1st 6678   2ndc2nd 6679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681
This theorem is referenced by:  dvhopN  35070
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