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Theorem subrfn 27678
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 5883 . . 3  |-  ( ( A `  x )  -  ( B `  x ) )  e. 
_V
2 eqid 2283 . . 3  |-  ( x  e.  RR  |->  ( ( A `  x )  -  ( B `  x ) ) )  =  ( x  e.  RR  |->  ( ( A `
 x )  -  ( B `  x ) ) )
31, 2fnmpti 5372 . 2  |-  ( x  e.  RR  |->  ( ( A `  x )  -  ( B `  x ) ) )  Fn  RR
4 subrval 27672 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A - r B )  =  ( x  e.  RR  |->  ( ( A `  x
)  -  ( B `
 x ) ) ) )
54fneq1d 5335 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( A - r B )  Fn  RR  <->  ( x  e.  RR  |->  ( ( A `  x
)  -  ( B `
 x ) ) )  Fn  RR ) )
63, 5mpbiri 224 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A - r B )  Fn  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684    e. cmpt 4077    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   RRcr 8736    - cmin 9037   - rcminusr 27663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-subr 27669
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