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Theorem dvhopaddN 34115
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 4974 . . 3  |-  ( ( F  e.  T  /\  U  e.  E )  -> 
<. F ,  U >.  e.  ( T  X.  E
) )
2 opelxpi 4974 . . 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 34091 . . 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 475 . 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 6750 . . . . 5  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 1st `  <. F ,  U >. )  =  F )
76adantr 463 . . . 4  |-  ( ( ( F  e.  T  /\  U  e.  E
)  /\  ( G  e.  T  /\  V  e.  E ) )  -> 
( 1st `  <. F ,  U >. )  =  F )
8 op1stg 6750 . . . . 5  |-  ( ( G  e.  T  /\  V  e.  E )  ->  ( 1st `  <. G ,  V >. )  =  G )
98adantl 464 . . . 4  |-  ( ( ( F  e.  T  /\  U  e.  E
)  /\  ( G  e.  T  /\  V  e.  E ) )  -> 
( 1st `  <. G ,  V >. )  =  G )
107, 9coeq12d 5109 . . 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 6751 . . . 4  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 2nd `  <. F ,  U >. )  =  U )
12 op2ndg 6751 . . . 4  |-  ( ( G  e.  T  /\  V  e.  E )  ->  ( 2nd `  <. G ,  V >. )  =  V )
1311, 12oveqan12d 6253 . . 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 4166 . 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 2443 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   <.cop 3977    X. cxp 4940    o. ccom 4946   ` cfv 5525  (class class class)co 6234    |-> cmpt2 6236   1stc1st 6736   2ndc2nd 6737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-iota 5489  df-fun 5527  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739
This theorem is referenced by:  dvhopN  34117
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