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Theorem dvhopaddN 31926
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 4737 . . 3  |-  ( ( F  e.  T  /\  U  e.  E )  -> 
<. F ,  U >.  e.  ( T  X.  E
) )
2 opelxpi 4737 . . 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 31902 . . 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 6148 . . . . 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 6148 . . . . 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 4864 . . 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 6149 . . . 4  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 2nd `  <. F ,  U >. )  =  U )
12 op2ndg 6149 . . . 4  |-  ( ( G  e.  T  /\  V  e.  E )  ->  ( 2nd `  <. G ,  V >. )  =  V )
1311, 12oveqan12d 5893 . . 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 3820 . 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 2328 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 1632    e. wcel 1696   <.cop 3656    X. cxp 4703    o. ccom 4709   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  dvhopN  31928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139
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