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Theorem dvhopaddN 31304
Description: Sum of  DVecH vectors expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
dvhopadd.a  |-  A  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E ) 
|->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f ) P ( 2nd `  g ) ) >. )
Assertion
Ref Expression
dvhopaddN  |-  ( ( ( F  e.  T  /\  U  e.  E
)  /\  ( G  e.  T  /\  V  e.  E ) )  -> 
( <. F ,  U >. A <. G ,  V >. )  =  <. ( F  o.  G ) ,  ( U P V ) >. )
Distinct variable groups:    f, g, E    P, f, g    T, f, g
Allowed substitution hints:    A( f, g)    U( f, g)    F( f, g)    G( f, g)    V( f, g)

Proof of Theorem dvhopaddN
StepHypRef Expression
1 opelxpi 4721 . . 3  |-  ( ( F  e.  T  /\  U  e.  E )  -> 
<. F ,  U >.  e.  ( T  X.  E
) )
2 opelxpi 4721 . . 3  |-  ( ( G  e.  T  /\  V  e.  E )  -> 
<. G ,  V >.  e.  ( T  X.  E
) )
3 dvhopadd.a . . . 4  |-  A  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E ) 
|->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f ) P ( 2nd `  g ) ) >. )
43dvhvaddval 31280 . . 3  |-  ( (
<. F ,  U >.  e.  ( T  X.  E
)  /\  <. G ,  V >.  e.  ( T  X.  E ) )  ->  ( <. F ,  U >. A <. G ,  V >. )  =  <. ( ( 1st `  <. F ,  U >. )  o.  ( 1st `  <. G ,  V >. )
) ,  ( ( 2nd `  <. F ,  U >. ) P ( 2nd `  <. G ,  V >. ) ) >.
)
51, 2, 4syl2an 463 . 2  |-  ( ( ( F  e.  T  /\  U  e.  E
)  /\  ( G  e.  T  /\  V  e.  E ) )  -> 
( <. F ,  U >. A <. G ,  V >. )  =  <. (
( 1st `  <. F ,  U >. )  o.  ( 1st `  <. G ,  V >. )
) ,  ( ( 2nd `  <. F ,  U >. ) P ( 2nd `  <. G ,  V >. ) ) >.
)
6 op1stg 6132 . . . . 5  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 1st `  <. F ,  U >. )  =  F )
76adantr 451 . . . 4  |-  ( ( ( F  e.  T  /\  U  e.  E
)  /\  ( G  e.  T  /\  V  e.  E ) )  -> 
( 1st `  <. F ,  U >. )  =  F )
8 op1stg 6132 . . . . 5  |-  ( ( G  e.  T  /\  V  e.  E )  ->  ( 1st `  <. G ,  V >. )  =  G )
98adantl 452 . . . 4  |-  ( ( ( F  e.  T  /\  U  e.  E
)  /\  ( G  e.  T  /\  V  e.  E ) )  -> 
( 1st `  <. G ,  V >. )  =  G )
107, 9coeq12d 4848 . . 3  |-  ( ( ( F  e.  T  /\  U  e.  E
)  /\  ( G  e.  T  /\  V  e.  E ) )  -> 
( ( 1st `  <. F ,  U >. )  o.  ( 1st `  <. G ,  V >. )
)  =  ( F  o.  G ) )
11 op2ndg 6133 . . . 4  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 2nd `  <. F ,  U >. )  =  U )
12 op2ndg 6133 . . . 4  |-  ( ( G  e.  T  /\  V  e.  E )  ->  ( 2nd `  <. G ,  V >. )  =  V )
1311, 12oveqan12d 5877 . . 3  |-  ( ( ( F  e.  T  /\  U  e.  E
)  /\  ( G  e.  T  /\  V  e.  E ) )  -> 
( ( 2nd `  <. F ,  U >. ) P ( 2nd `  <. G ,  V >. )
)  =  ( U P V ) )
1410, 13opeq12d 3804 . 2  |-  ( ( ( F  e.  T  /\  U  e.  E
)  /\  ( G  e.  T  /\  V  e.  E ) )  ->  <. ( ( 1st `  <. F ,  U >. )  o.  ( 1st `  <. G ,  V >. )
) ,  ( ( 2nd `  <. F ,  U >. ) P ( 2nd `  <. G ,  V >. ) ) >.  =  <. ( F  o.  G ) ,  ( U P V )
>. )
155, 14eqtrd 2315 1  |-  ( ( ( F  e.  T  /\  U  e.  E
)  /\  ( G  e.  T  /\  V  e.  E ) )  -> 
( <. F ,  U >. A <. G ,  V >. )  =  <. ( F  o.  G ) ,  ( U P V ) >. )
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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