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Theorem addrfv 27588
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 27585 . . . 4  |-  ( ( A  e.  E  /\  B  e.  D )  ->  ( A + r B )  =  ( x  e.  RR  |->  ( ( A `  x
)  +  ( B `
 x ) ) ) )
21fveq1d 5721 . . 3  |-  ( ( A  e.  E  /\  B  e.  D )  ->  ( ( A + r B ) `  C
)  =  ( ( x  e.  RR  |->  ( ( A `  x
)  +  ( B `
 x ) ) ) `  C ) )
3 fveq2 5719 . . . . 5  |-  ( x  =  C  ->  ( A `  x )  =  ( A `  C ) )
4 fveq2 5719 . . . . 5  |-  ( x  =  C  ->  ( B `  x )  =  ( B `  C ) )
53, 4oveq12d 6090 . . . 4  |-  ( x  =  C  ->  (
( A `  x
)  +  ( B `
 x ) )  =  ( ( A `
 C )  +  ( B `  C
) ) )
6 eqid 2435 . . . 4  |-  ( x  e.  RR  |->  ( ( A `  x )  +  ( B `  x ) ) )  =  ( x  e.  RR  |->  ( ( A `
 x )  +  ( B `  x
) ) )
7 ovex 6097 . . . 4  |-  ( ( A `  C )  +  ( B `  C ) )  e. 
_V
85, 6, 7fvmpt 5797 . . 3  |-  ( C  e.  RR  ->  (
( x  e.  RR  |->  ( ( A `  x )  +  ( B `  x ) ) ) `  C
)  =  ( ( A `  C )  +  ( B `  C ) ) )
92, 8sylan9eq 2487 . 2  |-  ( ( ( A  e.  E  /\  B  e.  D
)  /\  C  e.  RR )  ->  ( ( A + r B ) `  C )  =  ( ( A `
 C )  +  ( B `  C
) ) )
1093impa 1148 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    e. cmpt 4258   ` cfv 5445  (class class class)co 6072   RRcr 8978    + caddc 8982   + rcplusr 27576
This theorem is referenced by:  addrcom  27594
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-cnex 9035  ax-resscn 9036
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-addr 27582
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