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Theorem subrfn 27766
Description: Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
Assertion
Ref Expression
subrfn  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A - r B )  Fn  RR )

Proof of Theorem subrfn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ovex 6142 . . 3  |-  ( ( A `  x )  -  ( B `  x ) )  e. 
_V
2 eqid 2443 . . 3  |-  ( x  e.  RR  |->  ( ( A `  x )  -  ( B `  x ) ) )  =  ( x  e.  RR  |->  ( ( A `
 x )  -  ( B `  x ) ) )
31, 2fnmpti 5608 . 2  |-  ( x  e.  RR  |->  ( ( A `  x )  -  ( B `  x ) ) )  Fn  RR
4 subrval 27760 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A - r B )  =  ( x  e.  RR  |->  ( ( A `  x
)  -  ( B `
 x ) ) ) )
54fneq1d 5571 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( A - r B )  Fn  RR  <->  ( x  e.  RR  |->  ( ( A `  x
)  -  ( B `
 x ) ) )  Fn  RR ) )
63, 5mpbiri 226 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A - r B )  Fn  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1728    e. cmpt 4297    Fn wfn 5484   ` cfv 5489  (class class class)co 6117   RRcr 9027    - cmin 9329   - rcminusr 27751
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 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pr 4438  ax-cnex 9084  ax-resscn 9085
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 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-subr 27757
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