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Theorem dvhopaddN 35122
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 4982 . . 3  |-  ( ( F  e.  T  /\  U  e.  E )  -> 
<. F ,  U >.  e.  ( T  X.  E
) )
2 opelxpi 4982 . . 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 35098 . . 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 477 . 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 6702 . . . . 5  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 1st `  <. F ,  U >. )  =  F )
76adantr 465 . . . 4  |-  ( ( ( F  e.  T  /\  U  e.  E
)  /\  ( G  e.  T  /\  V  e.  E ) )  -> 
( 1st `  <. F ,  U >. )  =  F )
8 op1stg 6702 . . . . 5  |-  ( ( G  e.  T  /\  V  e.  E )  ->  ( 1st `  <. G ,  V >. )  =  G )
98adantl 466 . . . 4  |-  ( ( ( F  e.  T  /\  U  e.  E
)  /\  ( G  e.  T  /\  V  e.  E ) )  -> 
( 1st `  <. G ,  V >. )  =  G )
107, 9coeq12d 5115 . . 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 6703 . . . 4  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 2nd `  <. F ,  U >. )  =  U )
12 op2ndg 6703 . . . 4  |-  ( ( G  e.  T  /\  V  e.  E )  ->  ( 2nd `  <. G ,  V >. )  =  V )
1311, 12oveqan12d 6222 . . 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 4178 . 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 2495 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 369    = wceq 1370    e. wcel 1758   <.cop 3994    X. cxp 4949    o. ccom 4955   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691
This theorem is referenced by:  dvhopN  35124
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