Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  subrfv Structured version   Unicode version

Theorem subrfv 29651
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 29648 . . . 4  |-  ( ( A  e.  E  /\  B  e.  D )  ->  ( A -r
B )  =  ( x  e.  RR  |->  ( ( A `  x
)  -  ( B `
 x ) ) ) )
21fveq1d 5690 . . 3  |-  ( ( A  e.  E  /\  B  e.  D )  ->  ( ( A -r B ) `  C )  =  ( ( x  e.  RR  |->  ( ( A `  x )  -  ( B `  x )
) ) `  C
) )
3 fveq2 5688 . . . . 5  |-  ( x  =  C  ->  ( A `  x )  =  ( A `  C ) )
4 fveq2 5688 . . . . 5  |-  ( x  =  C  ->  ( B `  x )  =  ( B `  C ) )
53, 4oveq12d 6108 . . . 4  |-  ( x  =  C  ->  (
( A `  x
)  -  ( B `
 x ) )  =  ( ( A `
 C )  -  ( B `  C ) ) )
6 eqid 2441 . . . 4  |-  ( x  e.  RR  |->  ( ( A `  x )  -  ( B `  x ) ) )  =  ( x  e.  RR  |->  ( ( A `
 x )  -  ( B `  x ) ) )
7 ovex 6115 . . . 4  |-  ( ( A `  C )  -  ( B `  C ) )  e. 
_V
85, 6, 7fvmpt 5771 . . 3  |-  ( C  e.  RR  ->  (
( x  e.  RR  |->  ( ( A `  x )  -  ( B `  x )
) ) `  C
)  =  ( ( A `  C )  -  ( B `  C ) ) )
92, 8sylan9eq 2493 . 2  |-  ( ( ( A  e.  E  /\  B  e.  D
)  /\  C  e.  RR )  ->  ( ( A -r B ) `  C )  =  ( ( A `
 C )  -  ( B `  C ) ) )
1093impa 1177 1  |-  ( ( A  e.  E  /\  B  e.  D  /\  C  e.  RR )  ->  ( ( A -r B ) `  C )  =  ( ( A `  C
)  -  ( B `
 C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    e. cmpt 4347   ` cfv 5415  (class class class)co 6090   RRcr 9277    - cmin 9591   -rcminusr 29639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-cnex 9334  ax-resscn 9335
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-subr 29645
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator