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Theorem addrval 27774
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 2809 . 2  |-  ( A  e.  C  ->  A  e.  _V )
2 elex 2809 . 2  |-  ( B  e.  D  ->  B  e.  _V )
3 fveq1 5540 . . . . 5  |-  ( x  =  A  ->  (
x `  v )  =  ( A `  v ) )
4 fveq1 5540 . . . . 5  |-  ( y  =  B  ->  (
y `  v )  =  ( B `  v ) )
53, 4oveqan12d 5893 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x `  v )  +  ( y `  v ) )  =  ( ( A `  v )  +  ( B `  v ) ) )
65mpteq2dv 4123 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( v  e.  RR  |->  ( ( x `  v )  +  ( y `  v ) ) )  =  ( v  e.  RR  |->  ( ( A `  v
)  +  ( B `
 v ) ) ) )
7 df-addr 27771 . . 3  |-  + r  =  ( x  e. 
_V ,  y  e. 
_V  |->  ( v  e.  RR  |->  ( ( x `
 v )  +  ( y `  v
) ) ) )
8 reex 8844 . . . 4  |-  RR  e.  _V
98mptex 5762 . . 3  |-  ( v  e.  RR  |->  ( ( A `  v )  +  ( B `  v ) ) )  e.  _V
106, 7, 9ovmpt2a 5994 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A + r B )  =  ( v  e.  RR  |->  ( ( A `  v
)  +  ( B `
 v ) ) ) )
111, 2, 10syl2an 463 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   RRcr 8752    + caddc 8756   + rcplusr 27765
This theorem is referenced by:  addrfv  27777  addrfn  27780
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-cnex 8809  ax-resscn 8810
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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  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-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-addr 27771
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