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Theorem addrval 27647
Description: Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)
Assertion
Ref Expression
addrval  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A + r B )  =  ( v  e.  RR  |->  ( ( A `  v
)  +  ( B `
 v ) ) ) )
Distinct variable groups:    v, A    v, B
Allowed substitution hints:    C( v)    D( v)

Proof of Theorem addrval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2964 . 2  |-  ( A  e.  C  ->  A  e.  _V )
2 elex 2964 . 2  |-  ( B  e.  D  ->  B  e.  _V )
3 fveq1 5727 . . . . 5  |-  ( x  =  A  ->  (
x `  v )  =  ( A `  v ) )
4 fveq1 5727 . . . . 5  |-  ( y  =  B  ->  (
y `  v )  =  ( B `  v ) )
53, 4oveqan12d 6100 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x `  v )  +  ( y `  v ) )  =  ( ( A `  v )  +  ( B `  v ) ) )
65mpteq2dv 4296 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( v  e.  RR  |->  ( ( x `  v )  +  ( y `  v ) ) )  =  ( v  e.  RR  |->  ( ( A `  v
)  +  ( B `
 v ) ) ) )
7 df-addr 27644 . . 3  |-  + r  =  ( x  e. 
_V ,  y  e. 
_V  |->  ( v  e.  RR  |->  ( ( x `
 v )  +  ( y `  v
) ) ) )
8 reex 9081 . . . 4  |-  RR  e.  _V
98mptex 5966 . . 3  |-  ( v  e.  RR  |->  ( ( A `  v )  +  ( B `  v ) ) )  e.  _V
106, 7, 9ovmpt2a 6204 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A + r B )  =  ( v  e.  RR  |->  ( ( A `  v
)  +  ( B `
 v ) ) ) )
111, 2, 10syl2an 464 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A + r B )  =  ( v  e.  RR  |->  ( ( A `  v
)  +  ( B `
 v ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   RRcr 8989    + caddc 8993   + rcplusr 27638
This theorem is referenced by:  addrfv  27650  addrfn  27653
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-cnex 9046  ax-resscn 9047
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-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  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-iun 4095  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-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-addr 27644
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