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Theorem addrfv 27664
Description: Vector addition at a value. The operation takes each vector  A and  B and forms a new vector whose values are the sum of each of the values of  A and  B. (Contributed by Andrew Salmon, 27-Jan-2012.)
Assertion
Ref Expression
addrfv  |-  ( ( A  e.  E  /\  B  e.  D  /\  C  e.  RR )  ->  ( ( A + r B ) `  C
)  =  ( ( A `  C )  +  ( B `  C ) ) )

Proof of Theorem addrfv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 addrval 27661 . . . 4  |-  ( ( A  e.  E  /\  B  e.  D )  ->  ( A + r B )  =  ( x  e.  RR  |->  ( ( A `  x
)  +  ( B `
 x ) ) ) )
21fveq1d 5733 . . 3  |-  ( ( A  e.  E  /\  B  e.  D )  ->  ( ( A + r B ) `  C
)  =  ( ( x  e.  RR  |->  ( ( A `  x
)  +  ( B `
 x ) ) ) `  C ) )
3 fveq2 5731 . . . . 5  |-  ( x  =  C  ->  ( A `  x )  =  ( A `  C ) )
4 fveq2 5731 . . . . 5  |-  ( x  =  C  ->  ( B `  x )  =  ( B `  C ) )
53, 4oveq12d 6102 . . . 4  |-  ( x  =  C  ->  (
( A `  x
)  +  ( B `
 x ) )  =  ( ( A `
 C )  +  ( B `  C
) ) )
6 eqid 2438 . . . 4  |-  ( x  e.  RR  |->  ( ( A `  x )  +  ( B `  x ) ) )  =  ( x  e.  RR  |->  ( ( A `
 x )  +  ( B `  x
) ) )
7 ovex 6109 . . . 4  |-  ( ( A `  C )  +  ( B `  C ) )  e. 
_V
85, 6, 7fvmpt 5809 . . 3  |-  ( C  e.  RR  ->  (
( x  e.  RR  |->  ( ( A `  x )  +  ( B `  x ) ) ) `  C
)  =  ( ( A `  C )  +  ( B `  C ) ) )
92, 8sylan9eq 2490 . 2  |-  ( ( ( A  e.  E  /\  B  e.  D
)  /\  C  e.  RR )  ->  ( ( A + r B ) `  C )  =  ( ( A `
 C )  +  ( B `  C
) ) )
1093impa 1149 1  |-  ( ( A  e.  E  /\  B  e.  D  /\  C  e.  RR )  ->  ( ( A + r B ) `  C
)  =  ( ( A `  C )  +  ( B `  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    e. cmpt 4269   ` cfv 5457  (class class class)co 6084   RRcr 8994    + caddc 8998   + rcplusr 27652
This theorem is referenced by:  addrcom  27670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-cnex 9051  ax-resscn 9052
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-addr 27658
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