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Theorem subrfv 27778
Description: Vector subtraction at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)
Assertion
Ref Expression
subrfv  |-  ( ( A  e.  E  /\  B  e.  D  /\  C  e.  RR )  ->  ( ( A - r B ) `  C
)  =  ( ( A `  C )  -  ( B `  C ) ) )

Proof of Theorem subrfv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 subrval 27775 . . . 4  |-  ( ( A  e.  E  /\  B  e.  D )  ->  ( A - r B )  =  ( x  e.  RR  |->  ( ( A `  x
)  -  ( B `
 x ) ) ) )
21fveq1d 5543 . . 3  |-  ( ( A  e.  E  /\  B  e.  D )  ->  ( ( A - r B ) `  C
)  =  ( ( x  e.  RR  |->  ( ( A `  x
)  -  ( B `
 x ) ) ) `  C ) )
3 fveq2 5541 . . . . 5  |-  ( x  =  C  ->  ( A `  x )  =  ( A `  C ) )
4 fveq2 5541 . . . . 5  |-  ( x  =  C  ->  ( B `  x )  =  ( B `  C ) )
53, 4oveq12d 5892 . . . 4  |-  ( x  =  C  ->  (
( A `  x
)  -  ( B `
 x ) )  =  ( ( A `
 C )  -  ( B `  C ) ) )
6 eqid 2296 . . . 4  |-  ( x  e.  RR  |->  ( ( A `  x )  -  ( B `  x ) ) )  =  ( x  e.  RR  |->  ( ( A `
 x )  -  ( B `  x ) ) )
7 ovex 5899 . . . 4  |-  ( ( A `  C )  -  ( B `  C ) )  e. 
_V
85, 6, 7fvmpt 5618 . . 3  |-  ( C  e.  RR  ->  (
( x  e.  RR  |->  ( ( A `  x )  -  ( B `  x )
) ) `  C
)  =  ( ( A `  C )  -  ( B `  C ) ) )
92, 8sylan9eq 2348 . 2  |-  ( ( ( A  e.  E  /\  B  e.  D
)  /\  C  e.  RR )  ->  ( ( A - r B ) `  C )  =  ( ( A `
 C )  -  ( B `  C ) ) )
1093impa 1146 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   RRcr 8752    - cmin 9053   - rcminusr 27766
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-subr 27772
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