Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvhopaddN Structured version   Unicode version

Theorem dvhopaddN 31912
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 4910 . . 3  |-  ( ( F  e.  T  /\  U  e.  E )  -> 
<. F ,  U >.  e.  ( T  X.  E
) )
2 opelxpi 4910 . . 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 31888 . . 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 464 . 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 6359 . . . . 5  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 1st `  <. F ,  U >. )  =  F )
76adantr 452 . . . 4  |-  ( ( ( F  e.  T  /\  U  e.  E
)  /\  ( G  e.  T  /\  V  e.  E ) )  -> 
( 1st `  <. F ,  U >. )  =  F )
8 op1stg 6359 . . . . 5  |-  ( ( G  e.  T  /\  V  e.  E )  ->  ( 1st `  <. G ,  V >. )  =  G )
98adantl 453 . . . 4  |-  ( ( ( F  e.  T  /\  U  e.  E
)  /\  ( G  e.  T  /\  V  e.  E ) )  -> 
( 1st `  <. G ,  V >. )  =  G )
107, 9coeq12d 5037 . . 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 6360 . . . 4  |-  ( ( F  e.  T  /\  U  e.  E )  ->  ( 2nd `  <. F ,  U >. )  =  U )
12 op2ndg 6360 . . . 4  |-  ( ( G  e.  T  /\  V  e.  E )  ->  ( 2nd `  <. G ,  V >. )  =  V )
1311, 12oveqan12d 6100 . . 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 3992 . 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 2468 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 359    = wceq 1652    e. wcel 1725   <.cop 3817    X. cxp 4876    o. ccom 4882   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348
This theorem is referenced by:  dvhopN  31914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350
  Copyright terms: Public domain W3C validator